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MM IN* •»•» # ff.j'.sr & Contract No.: 5331984031 MPR Reference No.: 8207050 STATE ESTIMATES OF IN*ANTS AND CHILDREN INCOME ELIGIBLE FOR THE WIC PROGRAM IN 1993 May 15,1995 Author AJIenLSchinn Submitted to: U.S. Department of Agriculture Food and Consumer Service 3101 Park Center Drive Room 214 Alexandria, VA 22302 Attention: Cindy Long Submitted by: Matbematica Policy Research, Inc. 600 Maryland Avenue, S.W. Suite 550 Washington, D.C 20024 \qs0wf67 <*/ Nttd. 1 SUMMARY: STATE ESTIMATES OF INFANTS AND CHILDREN INCOME ELIGD3LE FOR THE WIC PROGRAM IN 1992 This report presents the methodology and data used for developing statelevel estimates of infants and children income eligible for WIC in 1992. These estimates were released in September 1994 and were used in calculating FY1995 WIC food grants. These estimates were developed using a statistical technique known as "shrinkage". For the eligibles estimates, shrinkage was used to develop estimates of the proportion of children age 04 in each state in households with incomes below 185% of poverty. These proportions were then applied to 1992 state population estimates of infants (01) and children (14) provided by the Bureau of the Census. The shrinkage approach combines estimates obtained directly through surveys with estimates produced by an econometric model which uses statelevel economic data. The direct sample estimates used were the March 1993 Current Population Survey (CPS) estimates of the proportion of children 04 below 185% of poverty in each state in 1992. The model estimates were developed using a regression model which estimated the proportion of children 04 below 185% of poverty using the following statelevel economic variables': • Food Stamp participation • Unemployment Insurance (receipt of first benefits for a period of unemployment) • Per capita income The model used was a "change" model, which estimated the change in the proportion of children (04) below 185% of poverty from 1989 (based on 1990 Census data) to 1992 as a function of the change in the variables noted above for the same time period. This "change" model provided better estimates than models which directly estimated the proportion of children below 185% of poverty. The shrinkage technique averages the sample and the model estimates using weights that reflect the relative precision of each estimate Thus, in cases where the CPS estimate has a high standard error, the model estimate would tend to receive more weight than it does in cases where the CPS standard error is lower. The shrinkage estimates are, for many purposes, superior to either the direct sample estimate or the model estimate. In particular, the shrinkage estimates have lower standard errors than the direct sample estimates. £ ACKNOWLEDGMENTS I thank Matthew McKearn and Cindy Long of the Food and Consumer Service, Alan Zaslavsky of Harvard University, and John Czajka of Mathematica Policy Research for helpful comments and guidance. Aleda Freeman provided skillful programming assistance. Deborah Patterson assisted in the preparation of the report, and Daryl Hall edited the report ui / m,MB Mi CONTENTS Chapter Page EXECUTIVE SUMMARY xi I INTRODUCTION 1 II A STEPBYSTEP GUIDE TO DERIVING STATE ESTIMATES OF ELIGIBLE INFANTS AND CHILDREN 5 HI STATE ESTIMATES OF WIC ELIGIBLES FOR 1992 21 REFERENCES 27 APPENDIX: THE ESTIMATION PROCEDURE: ADDITIONAL TECHNICAL DETAILS 29 Y\ Q ^ ®m MMK TABLES Tabk II.1 D.2 n.3 II.4 m.i ra.2 Al A2 A3 A4 A5 A6 A7 A8 A.9 Pac PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND CPS SAMPLE ESTIMATES 8 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE 13 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND SHRINKAGE ESTIMATES 17 PRELIMINARY SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 19 FINAL SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 22 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR SHRINKAGE ESTIMATES 23 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE 32 WIC POVERTY GUIDELINES FOR 1992 33 DEFINITIONS AND DATA SOURCES FOR PREDICTOR VARIABLES 38 1989 DATA FOR CALCULATING PREDICTOR VARIABLES 39 1992 DATA FOR CALCULATING PREDICTOR VARIABLES 40 VALUES FOR PREDICTOR VARIABLES IN REGRESSION MODEL 41 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: REGRESSION ESTIMATES 43 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: SHRINKAGE ESTIMATES 47 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 49 vii TABLES (continued) Table Page A10 SAMPLE AND SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 52 A11 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE INFANTS 53 A12 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE CHILDREN 54 Vlll FIGURES Figure P«*e D.1 THE ESTIMATION PROCEDURE 6 IL2 AN ILLUSTRATIVE REGRESSION ESTIMATOR 11 D.3 SHRINKAGE ESTIMATION 15 IX X [EMS Mi EXECUTIVE SUMMARY This report presents state estimates of the numbers of infants and children who were income eligible for the Special Supplemental Nutrition Program for Women, Infants, and Children (WIC) in 1992. These estimates, used to calculate state WIC grants for fiscal year 1995, were derived using "shrinkage" estimation. Drawing on both census and Current Population Survey (CPS) data as well as administrative records data from government program case files and vital statistics systems, we developed shrinkage estimates by averaging CPS sample estimates with predictions of WIC eligibles from a statistical regression model. The predictions were based on observed changes in government program participation and other indicators of socioeconomic conditions. The shrinkage estimates are more timely than census estimates, which had been used for fund allocation in prior years, and substantially more precise than CPS direct sample estimates. I. INTRODUCTION To allocate program funds equitably, the U.S. Department of Agriculture requires timely and accurate state estimates of the number of people eligible for the Special Supplemental Nutrition Program for Women, Infants, and Children (WIC). This report presents state estimates of the numbers of infants (age under 1) and children (ages 1 to 4 inclusive) who were income eligible for the WIC Program in 1992, that is, the numbers whose family incomes were at or below 185 percent of the applicable poverty guidelines. These estimates, used to calculate state WIC grants for fiscal year 1995, were derived using "shrinkage'' estimation. This introductory chapter explains the advantages of shrinkage estimation relative to direct estimation from the census or the Current Population Survey (CPS), the leading data sources for developing state estimates. Chapter II describes how we derived shrinkage estimates, and Chapter III presents our state estimates of WICeligible infants and children for 1992. Technical details and additional information about our estimation method are provided in the Appendix. The census is the most commonly used data source for deriving state, county, and other subnational estimates. However, because the census is conducted only once every 10 years, census estimates may not be timely for many purposes. As suggested by the estimates presented in this report, social and economic conditions change, often rapidly, over time. Therefore, more recent data may better reflect current conditions. The CPS provides the most recent data from which we can develop annual state estimates of WIC eligibies. However, despite their timeliness, CPS sample estimates are typically imprecise because state samples of infants and children are small. For example, although our single best direct estimate from the CPS is that Minnesota bad 114 thousand eligible children in 1992, we are able to state with confidenceaccording to widely accepted statistical standardsonly that we believe the true number lies between 74 and 154 thousand. Such a wide range reflects imprecision and suggests that we are very uncertain about the number of eligible children in Minnesota and that our estimate of 114 thousand may be highly inaccurate. We are also unable to determine how Minnesota compares with other states. The estimates of eligible children for about onethird of the states fall within the 74 to 154 thousand range, even though some of those other states have much bigger or smaller populations than Minnesota. Ranked in terms of eligible children, Minnesota could fall below those states, above them, or somewhere in the middle. Restricting ourselves to direct estimation from the census or the CPS forces us to make a tradeoff between timeliness and precision. We have minimized this tradeoff by using an alternative methodshrinkage estimationto develop state estimates of WIC eligibles. Our shrinkage estimator uses both census and CPS data as well as administrative records data from government program case files and vital statistics systems. We obtained shrinkage estimates by averaging CPS sample estimates with predictions of WIC eligibles made using a statistical regression model. Our predictions are based on observed changes in government program participation and other indicators of socioeconomic conditions. The shrinkage estimates presented in this report are as timely as the CPS sample estimates but substantially more precise. Shrinkage estimators have been used for allocating program funds and other purposes. Fay and Herrioti (1979) developed a shrinkage estimator that combined sample and regression estimates of per capita income for small places (population less than 1,000). Their estimates were used to allocate funds under the General Revenue Sharing Program. State shrinkage estimates of median income for fourperson families are used to administer the Low Income Home Energy Assistance Program (LIHEAP) (Fay, Nelson, and Litow 1993). Schirm, Swearingen, and Hendricks (1992) used a shrinkage estimator similar to the one used for this report to develop state estimates of poverty and Food Stamp Program eligibility and participation. Finally, a shrinkage estimator was used to adjust the 1990 decennial census for the undercount (Hogan 1993), although the secretary of commerce ultimately rejected adjusted figures in favor of unadjusted Ggures as the official 1990 census population estimates. A recent review of shrinkage methods and other techniques for "small area" estimation can be found in Ghosh and Rao (1994). In his evaluation of small area estimators, Schirm (1994) compared the relative accuracy of alternative state poverty estimates and found that shrinkage estimates are substantially more accurate than the estimates obtained from other methods that have been widely used. Those findings give us further confidence in the estimates presented in this report. 1 mm. ME II. A STEPBYSTEP GUIDE TO DERIVING STATE ESTIMATES OF ELIGIBLE INFANTS AND CHILDREN This chapter describes our procedure for estimating the numbers of infants and children who were income eligible for WIC in each state. This procedure, summarized by the flow chart in Figure II. 1, has the following eight steps: 1. From the most recent census (1990), derive state estimates of the percentage of infants and children who were income eligible. 2. From the most recent CPS (March 1993), derive state sample estimates of the percentage of infants and children \vho were income eligible. 3. Construct sample estimates of the change in the percentage eligible between 1989 and 1992. 4. Using a regression model, predict the change in the percentage eligible for each state based on observed changes in (i) Food Sump Program (FSP) participation, (ii) Unemployment Insurance (UI) Program participation, and (iii) per capita income. 5. Using "shrinkage" methods, average the sample estimates of change and the predictions of change. 6. Add the shrinkage estimate of the change between 1989 and 1992 to the census estimate of the percentage eligible in 1989 to get a shrinkage estimate of the percentage eligible in 1992. 7. Multiply the shrinkage estimate of the percentage eligible by the state population of infants and the state population of children to get preliminary shrinkage estimates of the numbers of eligible infants and children. 8. Control the preliminary state shrinkage estimates of the numbers of eligible infants and children to sum to the national totals for eligible infants and children obtained from the CPS. Each step is described below, and additional technical details are provided in the Appendix. FIGURE n.l THE ESTIMATION PROCEDURE ^ 1990 Census ^ 1. State estimates of percentage eligible in 1989 5. Shrinkage estimates of change in percentage eligible (obtained by averaging) 6. Shrinkage estimates of percentage eligible in 1992 3. Estimates of change in percentage eligible 4. Regression predictions of change in percentage eligible QMarch 1993 CPS^ i 2. State estimates of percentage eligible in 1992 Administrative data on FSP and UI Frogram participation and per capita income Independent state population estimates for infants and children 7. Preliminary shrinkage estimates of numbers eligible 8. Final shrinkage estimates of numbers eligible (controlled to national sample estimates) 1. From the most recent census (1990), derive state estimates of the percentage of infants and children who were income eligible. Table II. 1 presents 1990 decennial census estimates of the percentage of infants and children who were income eligible in each state. Because the family income data collected in the census pertain to the preceding calendar year, the eligibility estimates in Table II. 1 are for 1989. According to the table, 28.543 percent of all infants and children in Delaware, for example, were income eligible for WIC in 1989. We estimated the percentages, rather than the numbers, of infants and children who were income eligible for a simple technical reason. Percentages standardize for state size, in contrast to counts where one state may have more eligible infants and children than another state simply because the first state has a larger population. Such standardization is required for the regression and shrinkage estimation performed in subsequent steps. We derived the estimated percentages in Table II. 1 from estimates developed by Sigma One Corporation (1993). Because census samples for states are very large, the estimates are precise. However, they may quickly become "old" if economic conditions have changed substantially in the years since the census. 2. From the most recent CPS (March 1993), derive state sample estimates of the percentage of infants and children who were income eligible. The most recent CPS that has income data for families provides more timely information than the census. That CPS was the March 1993 CPS when we were developing eligibles estimates to be used in allocating funds for fiscal year 1995. Table II. 1 displays sample estimates from the March 1993 CPS. Like the census, the CPS collects family income data for the prior year. Thus, the sample estimates pertain to 1992. According to the table, 32.065 percent of all infants and children in Delaware, for example, were income eligible for WIC in 1992, compared with 28.543 percent in 1989. TABLE D.l PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND CPS SAMPLE ESTIMATES State 1989 (Census) 1992 Change Between (CPS) 1989 and 1992 44.476 1.826 41.355 O012 42802 1672 53.929 1.723 46.573 MIS 36.754 0.697 34.766 13366 32065 3.522 67.187 20.946 49.457 9.436 40.193 0422 46.620 9.799 46.822 0.014 40.696 7.513 47.088 11.618 41.081 4235 34.028 •1732 53.568 5.445 55.899 3148 47.455 11927 37.928 13.682 27.620 1533 39.026 1.854 41.236 11.874 59.956 1412 59.946 21.017 53.704 7.065 32.261 5.839 38.135 3.782 30.094 9.563 31.036 1590 54.170 0.175 44.364 9228 45.196 5285 41977 0423 39.171 1123 '^280 4.642 44.718 4.839 33.162 0266 38.819 8.996 51130 8283 40561 6653 60352 16.348 51.141 5306 31884 7.115 29.180 1.984 33.635 2266 34242 0522 60589 8.986 29.870 4.224 36373 4.338 Alabama 46302 Alaska 41367 Arizona 45.474 Arkansas 52206 California 37.760 Colorado 36.057 Connecticut 21200 Delaware 28.543 District of Columbia 46241 Florida 40021 Georgia 40.615 Hawaii 36321 Idaho 46308 Dlinois 33.183 Indiana 35.470 Iowa 36346 Kansas 36.760 Kentucky 48.123 Louisiana 51651 Maine 34328 Maryland 24246 Massachusetts 25.087 Michigan 37.172 Minnesota 29362 Mississippi 57344 Missouri 38.929 Montana 46339 Nebraska 38.100 Nevada 34353 New Hampshire 20331 New Jersey 21446 New Mexico 53.995 New York 35.136 North Carolina 39.911 North Dakota 42354 Ohio 37.048 Oklahoma 47338 Oregon 39379 Pennsylvania 33.428 Rhode Island 29323 South Carolina 43347 South Dakota 47214 Tennessee 44.004 Texas 45335 Utah 39.999 Vermont 31.164 Virginia 31369 Washington 34.764 West Virginia 51303 Wisconsin 34.094 Wyoming 41211 United States 37.789 43380 5.791 8 Although timely compared with the census estimates, the CPS sample estimates are relatively imprecise. The standard errors for the CPS estimates, reported in the Appendix, tend to be large, so our uncertainty is great. For example, according to widely used statistical standards, we can be confident only that the percentage of incomeeligible infants and children in Delaware was between 22.501 percent and 41.629 percent. This range is so wide and our uncertainty so great because the CPS samples of infants and children in each state are small. Indeed, that is why we derived an eligibility estimate for infants and children combined, rather than separate estimates, one for infants and one for children. In the March 1993 CPS, there are data for fewer than 30 infants for most states. 3. Construct sample estimates of the change in the percentage eligible between 1989 and 1992. A sample estimate of the change in the percentage eligible between 1989 and 1992 was calculated by subtracting the census estimate for 1989 from the CPS estimate for 1991 According to Table II. 1, the percentage eligible in Delaware rose by 32.065  28.543 = 3.522 percentage points over the three years. We calculated sample estimates of change for use in the regression and shrinkage estimation described in the next few steps. Focusing on the change in the percentage eligible between 1989 and 1992, rather than just the percentage eligible in 1992, is a simple way to reflect a strong systematic relationship: states with a high percentage eligible in 1989 tend to have a high percentage eligible in 1992, and states with a low percentage eligible in 1989 tend to have a low percentage eligible in 1992. In principle, our shrinkage method obtains better estimates by using information on not only where a state "is," but also where it "began." 4. Using a regression model, predict the change in the percentage eligible for each state based on observed changes in (i) Food Stamp Program (FSP) participation, (ii) Unemployment Insurance (UI) Program participation, and (Hi) per capita income. The main limitation of the sample estimates derived in the previous step is imprecision. Regression can reduce that imprecision. Regression estimates are predictions based on nonsample or highly precise sample data, such as census and administrative records data. The latter include government program case files and vital statistics records. Figure D.2 illustrates how the regression estimator works. The simple example in the Ggure has just nine states and one predictor variablethe change in FSP participationthat will be used to predict each state's change in the percentage of infants and children who were income eligible for WIG The triangles in the figure correspond to sample estimates; a triangle shows the change in FSP participation in a state (on the horizontal axis) and the sample estimate of change in WIC eligibles in that state (on the vertical axis). Not surprisingly, the graph suggests that the change in FSP participation is systematically associated with the change in WIC eligibles. States with larger increases in FSP participation tend to have larger estimated increases in WIC eligibles, although the relationship is far from perfect. To depict this relationship between changes in FSP participation and WIC eligibles, we can use a technique called "least squares regression" to draw a line through the triangles (that is, we "regress" the sample estimates on the predictor variable). Regression estimates of WIC eligibles are points on that line, the circles in Figure II.2. The predicted change in WIC eligibles for a particular state is obtained by moving vertically from the state's sample estimate (the triangle) to the regression line (where there is a circle) and reading the value off the vertical axis. For example, the regression estimator predicts about a 6 percentage point change in WIC eligibles for both of the states with increases in FSP participation just under 3 percentage points. In contrast, for the state with a 1 percentage point increase in FSP participation, the predicted increase in WIC eligibles is under 2 percentage points. 10 g5 LU 0) O) FIGURE 11.2 AN ILLUSTRATIVE REGRESSION ESTIMATOR 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Change in Percentage Receiving Food Stamps ii Table II.2 displays the sample estimates calculated in Step 3 and the regression estimates calculated in this step. To derive the regression estimates in Table II.2, we included all of the states, not just nine as in our illustrative example, and we used three predictor variables, not just one. Adding two predictor variables improves our predictions. The three predictor variables used measure the changes between 1989 and 1992 in (1) FSP participation, (2) UI Program participation, and (3) per capita income. These three were selected as the best predictors from a longer list presented in the Appendix, which also provides complete definitions and data for calculating values for the three best predictors. As expected, the estimated regression displayed in the Appendix shows that states with relatively large increases in FSP and UI Program participation and large decreases in per capita income tend to have relatively large increases in the percentage of infants and children eligible for WIG The Appendix also presents standard errors for the regression estimates. Because they are much smaller than the standard errors for the sample estimates, the regression estimates are more precise than the sample estimates. Comparing how the sample and regression estimators use data reveals how the regression estimator "borrows strength" to improve precision. When we derived sample estimates in Step 3, we used only data from Delaware to estimate the change in the percentage of infants and children eligible for WIC in Delaware, even though Delaware, like nearly all states, has a small CPS sample. Deriving regression estimates in this step, we estimated a regression line from sample and administrative records data for all the states and used the estimated line (with administrative records data for Delaware) to predict the change in WIC eligibles for Delaware. In other words, the regression estimator not only uses the sample estimates from every state to develop a regression estimate for a single state but also incorporates data from outside the sample, namely, data in administrative records systems. The regression estimator improves precision by using more data to identify states with sample estimates that seem too high or too low because of sampling error, that is, error from drawing a sample that has a higher or lower percentage of eligible infants and children 12 TABLEIU CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE Sample Regression Shrinkage SUte Estimate Estimate Estimate Alabama •1.826 1760 2306 Alaska 0.012 9343 8.238 Arizona 1672 7.439 6336 Arkansas 1.723 4.142 3.977 California 8.813 8.377 8419 Colorado 0.697 0.068 0.133 Connecticut 13366 8.805 9.044 Delaware 3.522 7.174 6.788 District of Columbia 20.946 1L814 11032 Florida 9.436 1L670 10.907 Georgia •0.422 5.290 4.785 Hawaii 9.799 7.058 7325 Idaho 0.014 1256 1020 Dlinois 7.513 3486 4.638 Indiana 11.618 5411 6.221 Iowa 4.235 1070 1634 Kansas 1732 1.051 0.620 Kentucky 5.445 1226 1409 Louisiana 3.248 0395 0.977 Maine 11927 10.428 10310 Maryland 13.682 6.785 8J68 Massachusetts 1533 1.295 1396 Michigan 1.854 4.737 4331 Minnesota 1L874 0.905 1.454 Mississippi 1412 1432 1436 Missouri 21.017 4.710 5.911 Montana 7.065 1260 2366 Nebraska 5.839 1.151 0.191 Nevada 3.782 8.055 7.788 New Hampshire 9.563 6.979 7.436 New Jersey 8.590 5.436 6309 New Mexico ans 6.441 6.123 New York 9.228 5.750 6.917 North Carolina 5.285 4.403 4406 North Dakota 0.423 0.215 •0.184 Ohio 1123 1614 2322 Oklahoma 4.642 5.182 5.145 Oregon 4.839 4.817 4430 Pennsylvania 0.266 5.439 3.904 Rhode Island 8.996 10305 10390 South Carolina 8.283 6347 6.717 South Dakota 6.653 •OtlS 1.148 TCODCMCC 16348 6.438 7473 Texas 5306 6.472 6306 Utah •7.115 0.721 0227 Vermont 1.984 9.782 9.442 Virginia 1266 3.137 3433 Washington 0522 3.625 3441 West Virginia 8.986 4.681 4468 Wkconsin •4234 3313 1413 Wyoming •4338 1397 1.212 United States 5.791 5303 5390 13 than the entire state population has. For example, suppose a state bad experienced stable FSP and UI Program participation and rising per capita income. Our regression estimator would predict a stable or declining percentage of eligible infants and children, implying that a sample estimate showing a large increase in WIC eligibles is too high. The regression estimate will be lower than the sample estimate for such a state. On the other hand, if the sample data for a state show a much smaller increase in eligible infants and children than expected in light of the observed changes in FSP and UI Program participation and per capita income, the regression estimate for that state will be higher than the sample estimate. 5. Using "shrinkage" methods, average the sample estimates of change and the predictions of change. As noted, the limitation of the sample estimator is imprecision. The limitation of the regression estimator is called "bias." Some states really have larger or smaller increases in WIC eligibles than we expect (and predict with the regression estimator) based on changes in FSP and UI Program participation and per capita income. Such errors in regression estimates reflect bias. These limitations arise for the following reasons. The sample estimator uses only sample data for one state to obtain an estimate for that state. It does not use sample data for other states or administrative records data. Although the regression estimator borrows strength, using data from all the states and administrative records data, it makes no further use of the sample data after estimating the regression line. It assumes that the entire difference between the sample and regression estimates is sampling error, that is, error in the sample estimate. No allowance is made for prediction error, that is, error in the regression estimate. Although not all, if any, true state values lie on the regression line, the regression estimator assumes they do. Using all of the information at hand, a shrinkage estimator addresses the limitations of the sample and regression estimators by combining the sample and regression estimates, striking a compromise. As illustrated in Figure II3, a shrinkage estimator takes a weighted average of the 14 FIGURE 11.3 SHRINKAGE ESTIMATION More Precise Sample Estimate, Worse Fitting Regression Line =* More Weight on Sample Estimate Sample Shrinkage Regression Estimate Estimate Estimate 4% 5% 8% Less Precise Sample Estimate, Better Fitting Regression Line ■> Less Weight on Sample Estimate Sample Shrinkage Regression Estimate Estimate Estimate 4% 7% 8% 15 sample and regression estimates. Generally, the more precise the sample estimate for a state, the closer the shrinkage estimate will be to it. Hie larger samples drawn in large states support more precise sample estimates, so shrinkage estimates tend to be closer to the sample estimates for large states. Given the precision of the sample estimate for a state, the weight given to the regression estimate depends on how well the regression line "fits." If the regression estimator cannot find good predictors reflecting why some states have larger increases in WIC eligibles than other states, we say that the regression line "fits poorly." The shrinkage estimate will be farther from the regression estimate and closer to the sample estimate when the regression line fits poorly. In contrast, the shrinkage estimate will be closer to the regression estimate and farther from the sample estimate when the regression line fits well. Striking a compromise between the sample and regression estimators, the shrinkage estimator strikes a compromise between imprecision and bias. The sample and regression estimates are optimally weighted to improve accuracy by minimizing a measure of error that reflects both imprecision and bias. By accepting a little bias, the shrinkage estimator may be substantially more precise than the sample estimator. By sacrificing a little precision, the shrinkage estimator may be substantially less biased than the regression estimator. Table II.2 presents state shrinkage estimates of the change between 1989 and 1992 in the percentage of infants and children who were income eligible for WIC. Table II.2 also displays the sample and regression estimates from Steps 3 and 4. 6. Add the shrinkage estimate of the change between 1989 and 1992 to the census estimate of the percentage eligible in 1989 to get a shrinkage estimate of the percentage eligible in 1992. Table II.3 presents census estimates of the percentage eligible in 1989 from Step 1, shrinkage estimates of the change in the percentage eligible between 1989 and 1992 from Step 5, and shrinkage estimates of the percentage eligible in 1992 from this step. The shrinkage estimate of change added to the census estimate for 1989 gives the shrinkage estimate for 1992. In other words, where a state starts plus how much it changes tells us where the state ends up. For example, 28.543 percent of 16 TABLE U3 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND SHRINKAGE ESTIMATES Stale 1989 (Census) Shrinkage Estimate of Change Between 1989 and 1992 Shrinkage Estimate for 1992 Alabama 46302 Alaska 41.367 Arizona 45.474 Arkansas 51206 California 37.760 Colorado 36.057 Connecticut 21.200 Delaware 28343 District of Columbia 46141 Florida 40.021 Georgia 40.615 Hawaii 36421 Idaho 46.808 Elinois 33.183 Indiana 35.470 Iowa 36446 Kansas 36.760 Kentucky 48.123 Louisiana 51651 Maine 34328 Maryland 24146 Massachusetts 25.087 Michigan 37.172 Minnesota 29362 Mississippi 57344 Missouri 38.929 Montana 46.639 Nebraska 38.100 Nevada 34353 New Hampshire 20331 New Jersey 21446 New Mexico 53.995 New York 35.136 North Carolina 39.911 North Dakota 42354 Ohio 37.048 Oklahoma 47438 Oregon 39479 Pennsylvania 33.428 Rhode Island 29423 South Carolina 43447 South Dakota 47114 Tennessee 44.004 Texas 45435 Utah 39.999 Vermont 3L164 Virginia 31369 Washington 34.764 West Virginia 51403 Wisconsin 34.094 Wyoming 41111 United States 37.789 2306 8138 6336 3.977 8419 0.133 9.044 6.788 11032 10.907 4.785 7325 1020 4.638 6121 1634 0420 1409 0.977 10310 8168 1396 4331 1.454 1436 5.911 2366 0.191 7.788 7.436 6309 6.123 6.917 4408 a 184 2322 5.145 4430 3.904 10390 6.717 •1.148 7473 6306 0127 9.442 3433 3441 4468 1413 1112 5390 48408 49405 51410 56.183 46379 36.190 30144 35331 58173 50.928 45.400 44.146 48428 37421 41491 39.480 37380 50332 53428 45.038 32314 26483 41303 30416 59.980 44440 49.005 38191 41141 27.967 28.955 60.118 41053 44319 42370 39370 51783 44.709 37332 40113 50364 46466 51477 51141 40126 40406 34.402 37405 56.471 36307 41423 43379 17 infants and children were income eligible in Delaware in 1989, and that Ggure rose by 6.788 percentage points between 1989 and 1992 according to our shrinkage estimator. Therefore, we estimate that 28.543 + 6.788 = 35.331 percent of infants and children were income eligible in Delaware in 1992. 7. Multiply the shrinkage estimate of the percentage eligible by the state population or infants and the state population of children to get preliminary shrinkage estimates of the numbers of eligible infants awl children. To obtain separate estimates for infants and children, we have assumed that the percentage of infants who were income eligible in a state is the same as the percentage of children who were income eligible. Our estimate of that percentage was obtained in Step 6. To obtain estimated numbers from estimated percentages, we require state population estimates for both infants and children. The population estimates we used pertain to the resident population on July 1,1992 and were developed by the U.S. Bureau of the Census from census and administrative records (mainly vital statistics) data. These estimates are often called "independent" estimates because they are not based on CPS or other sample data. In broad terms, they were derived by subtracting from census counts persons "exiting" the population between April 1, 1990 and July 1, 1992 (due to death or net outmigration) and adding persons "entering" the population (due to birth or net inmigration). Because infants in the Jury 1, 1992 population had not yet been born on April 1, 1990, census data have no bearing on the population estimates for infants. Those estimates are based entirely on vital statistics data and other administrative records data needed to account for migration. Likewise, census data are irrelevant to the population estimates for children age 1 and some children age 2. (The population estimates for children ages 1 through 4 were obtained by summing estimates for each year in that range.) Table II.4 displays preliminary shrinkage estimates of the number of infants and the number of children who were income eligible for WIC in 1992. It also shows shrinkage estimates of the percentages eligible from Step 6 and state population estimates for infants and children developed 18 TABLE D.4 PRELIMINARY SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 State Shrinkage Estimate of Percentage Eligible Population Preliminary Shrinkage Estimate of Number Eligible Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode bland South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming United States 48.808 49.605 51.810 56.183 46379 36.190 30244 35331 58.273 50.928 45.400 44.146 48.828 37.821 41.691 39.480 37380 50332 53.628 45X88 32314 26.683 41303 30.816 59.980 44.840 49.005 38.291 42.141 27.967 28.955 60.118 42.053 44319 42370 39370 52.783 44.709 37332 40.213 50364 46.066 5L877 52.141 40226 40.606 34.402 37.805 56.471 36307 42423 43379 Infants Children 61,680 11313 65,911 34^57 592344 53389 46380 10,769 8321 190419 109,227 19.608 17.069 168.287 82321 37,743 36,797 52301 70356 15395 75332 86,239 138.700 64,757 42,485 74,623 11367 22333 21,921 15,487 117.976 27316 281322 101,190 8,722 164,409 47301 41,279 162326 14379 55,711 11,029 72329 317,748 36313 7332 95368 78349 21356 69318 6,718 4300022 236.274 45,998 254,228 135,906 2,169,211 209,645 189,045 41,456 31388 751382 422,780 71,900 67348 713,032 323.759 155,417 150458 206.213 270028 66,707 302339 340993 575390 269,688 160775 301,457 46,728 95334 84381 66056 455,968 105,955 1379.972 393,791 36,102 634.255 185,746 169,723 649372 56348 217352 43357 281,056 1,183390 141.484 32366 371346 312385 86392 289,261 27326 Infants Children 30,105 5,711 34.148 19359 274,677 19394 14388 3305 4349 96377 49389 8356 8334 71,212 34320 14.901 13,755 26.732 37,731 7324 24356 233H 57365 19,956 25.483 33,461 5370 8,743 9^38 4331 34,160 16.722 118388 45349 3396 65357 24367 18355 60600 5363 28,170 5381 37333 165377 14388 3358 32377 29320 12342 25306 2350 15312,163 1,737337 115321 22317 13L716 76356 1306358 75371 57,175 14347 18.466 382317 191.942 31,741 32,738 269,676 134,978 61359 56241 104J04 144311 30343 98.465 90,987 238387 83,107 96.433 135,173 22399 36396 35343 18330 132,026 63398 454,161 175312 15396 250975 98342 75381 242,498 22359 109,902 20203 145303 617392 56313 13,143 127354 118.173 48,786 105301 11,720 6.72L734 19 by the Census Bureau. According to Table II.4, there were 10,769 infants and 41,456 children living in Delaware in 1992. Our shrinkage estimate is that 35.331 percent of those infants and children were income eligible. Therefore, our preliminary shrinkage estimates of the numbers eligible are (35.331 + 100) x 10,769 = 3,805 infants and (35.331 + 100) x 41,456 = 14,647 children. 8. Control the preliminary state shrinkage estimates of the numbers of eligible infants and children to sum to the national totals for eligible infants and children obtained from the CPS. The preliminary state shrinkage estimates derived in Step 7 sum to 1,737,837 eligible infants and 6,721,734 eligible children nationwide. According to the March 1993 CPS, there were 1,717,743 eligible infants and 6,925,815 eligible children in the entire U.S. The most recent national sample estimates are typically used to develop the budget for the WIC Program. To obtain final shrinkage estimates for states that sum (aside from rounding error) to the national totals from the most recent CPS (March 1993), we multiply each of the preliminary state shrinkage estimates for infants by 1,717,743 + 1,737,837 ( m 0.9884) and each of the preliminary state shrinkage estimates for children by 6,925,815 + 6,721,734 (  1.0304). This ensures that the estimates used to allocate funds are consistent with the estimates generally used to determine total program funding. The final shrinkage estimates are presented in the next chapter. 20 m. STATE ESTIMATES OF WIC EUGIBLES FOR 1992 Table m.l presents our final state shrinkage estimates of the number of infants and the number of children who were income eligible for WIC in 1992. The strength of these estimates is that they are timely relative to census estimates and precise relative to CPS estimates. As documented in the appendix, the shrinkage estimates have much smaller standard errors and narrower confidence intervals than the CPS sample estimates. Table m.2 displays approximate 90percent confidence intervals showing the uncertainty remaining after using shrinkage estimation. One interpretation of a 90percent confidence interval is that there is a 90 percent chance that the true valuethat is, the true number of eligibleslies in the estimated interval. A wide interval means that we are very uncertain about the true value. According to our calculations, a shrinkage confidence interval is, on average, only about 39 percent as wide as the corresponding sample confidence interval. Thus, shrinkage substantially reduces our uncertainty. The Food and Consumer Service (FCS) of the U.S. Department of Agriculture used the final shrinkage estimates of infants and children incomeeligible for WIC in 1992 to determine state WIC food grants for fiscal year 1995. From the final shrinkage estimates in Table III.l, FCS calculated each state's "fair share" of total fiscal year 1995 WIC food funds. A state's fair share is its percentage share of the national number of eligible infants and children. Thus, for example, Delawarewhich has about 0.2 percent ((3,761 + 15,092) + (1,717,746 + 6,925,819)) of all eligible infants and childrenhas a fair share of about 0.2 percent of total WIC food funds. According to the WIC food funding formula (7 C.F.R. §246.16), a state's WIC food grant is determined by comparing the fair share amount to the prior year food grant. If the prior year grant equals or exceeds the fair share amount, the state is entitled to receive only the prior year amount, adjusted for inflation (if total food funds are adequate to provide inflation increases to all states). If the prior year grant is below the fair share amount, the state is entitled to received an inflation 21 TABLE III1 FINAL SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 State Infants Children Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida 29.757 5,645 33.754 19.135 271.501 19.170 13.925 3,761 4,793 95355 118,822 23,510 135,715 78374 1.036304 78,174 58.911 15392 19.026 394.439 Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine 49.016 8356 8038 70389 33.924 14.729 13396 26.423 37^94 6.942 197,770 32.705 33.732 277,864 139,077 63022 57.949 107367 149.207 30.956 Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire 24371 22,745 56399 19,725 25.188 33.074 5306 8342 9.131 4381 101.455 93.750 246,140 85330 99361 139,277 23394 37310 36,725 19,092 New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming United States 33,765 16329 117.020 44328 3353 64304 24378 18342 59399 5.795 27344 5.022 37396 163,761 14318 3323 32,497 29377 12300 23313 2317 1.717,746 136,034 65332 467,950 180,635 15.761 258395 101.019 78.185 249361 23347 113339 20317 150330 636,034 58341 13342 131,735 12L761 50368 108307 12376 6325319 22 TABLE HU APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR SHRINKAGE ESTIMATES Number of Eligible Infants Number of Eligible Children State Lower Bound Upper Bound Lower Bound Upper Bound Alabama 27,266 32348 108374 128.770 Alaska 5,135 6,155 21386 25334 Arizona 30335 36373 123,977 147,453 Arkansas 17345 20,425 73371 83377 California 257,990 285,012 985,020 1388.188 Colorado 17,073 21367 69321 86,727 Connecticut 12373 15,777 51377 66.745 Delaware 3357 4,165 13,470 16,714 District of Columbia 4337 5349 17315 20337 Florida 89,771 101,939 369,404 419.474 Georgia 44322 53,410 180,042 215.498 Hawaii 7398 9314 29342 36368 Idaho 7314 8362 31,176 36388 Dlinois 64,462 76316 254,466 301362 Indiana 31337 36311 127340 150.914 Iowa 13377 15381 58378 68.166 Kansas 12314 14,978 52358 63340 Kentucky 24375 28,471 99345 115389 Louisiana 34,423 40.165 137,719 160.695 Maine 6305 7379 28,116 33,796 Maryland 21,706 27336 90363 112347 Massachusetts 18391 26,799 77341 110.459 Michigan 52316 61,782 225,018 267362 Minnesota 17,135 22315 74385 96375 Mississippi 23310 26.766 93,137 105385 Missouri 30,141 36307 126,927 151327 Montana 5365 5347 2L703 25.485 Nebraska 7,765 9319 33375 41345 Nevada 8381 9381 33307 40,143 New Hampshire 3.788 4,774 16392 21392 New Jersey 30361 36369 123.128 148,940 New Mexico 15314 17344 60,411 70353 New York 109312 124,728 437,125 498,775 North Carolina 41,469 47387 168327 193343 North Dakota 3363 4343 14378 17344 Ohio 59338 69370 239,427 277,763 Oklahoma 23316 26340 94316 107322 Oregon 16,713 19,771 71332 84,738 Pennsylvania 54319 65,479 226386 273.136 Rhode Island 5310 6380 20389 25,705 South Carolina 25348 29340 105,123 121355 South Dakota 4348 5396 18352 22,782 Tennessee 34,165 40327 137349 163311 Texas 151,491 176331 588377 683391 Utah 13363 15373 52,763 64319 Vermont 2,711 3335 12,146 14338 Vagina 28350 36444 116351 146319 Washington 26314 32340 110369 133353 West Virginia 11383 13,117 46392 34344 Wisconsin 22368 27358 97300 120314 Wyoming 2,497 3,137 10,708 13346 23 increase plus additional funds for program growth (if program growth funds are available after providing all states with inflation increases). In the initial fiscal year 1995 fund allocation, 19 states were below fair share and received program growth funds. Eight Indian Tribal Organizations (ITOs), which are authorized to participate in the WIC Program as state agencies, were also identified as below fair share. The eligibles estimates used to determine WIC food grants for ITOs were derived from 1990 decennial census data and March 1993 CPS data, but were not developed using the shrinkage estimation procedure described in Chapter II. Using the shrinkage estimator described in Chapter II, we are able to substantially reduce our uncertainty about the numbers of infants and children who were eligible for WIC. In the future, there may be an opportunity to reduce uncertainty even further by enhancing our shrinkage estimator to use still more data. The estimator now uses census estimates for the "base" year (1989) and CPS estimates for the "current" year (1992 in this reportthe year for which we are developing shrinkage estimates). Estimates for intervening years are not used, although CPS data for obtaining such estimates are available. With each intervening year, we are ignoring more information that could be relevant. An unusually large increase in WIC eligibles over three years, for example, would be more plausible if it appeared to consist of a series of modest increases rather than two small decreases followed by one enormous jump. An advantage of shrinkage methods is that they are powerful enough to allow such information to be taken into account in a systematic, rather than an ad hoc, way. Although the estimation procedure would be more complicated, an enhanced shrinkage estimator would be conceptually the same as the current estimator and might yield even better state estimates of WIC eligibles. Accuracy might also be improved by using data that incorporate an adjustment for the census undercount. Before CPS data are released, they are made consistent with Census Bureau population estimates. When 1992 eligibles estimates were needed for calculating fiscal year 1995 WIC grants, the available CPS data were consistent with population estimates based on unadjusted decennial 24 census data (as well as vital statistics and other administrative records data). CPS data released subsequently are consistent with adjusted population estimates. Therefore, it is expected that future estimates of WIC eligibles will reflect an adjustment for the census undercount. 25 # \m mm REFERENCES DuMouchel, William H., and Jeffrey E Harris. "Bayes Methods for Combining the Results of Cancer Studies in Humans and Other Species." Journal of the American Statistical Association, vol. 78, no. 382, June 1983, pp. 293315. Ericksen, Eugene P., and Joseph B. Kadane. "Estimating the Population in a Census Year: 1980 and Beyond." Journal of the American Statistical Association, vol. 80, no. 389, March 1985, pp. 98 131. Fay, Robert E, and Roger Herriott. "Estimates of Incomes for SmallPlaces: An Application of JamesStein Procedures to Census Data." Journal ofthe American StatisticalAssociation, vol. 74, no. 366, June 1979, pp. 269277. Fay, Robert E, Charles T. Nelson, and Leon Litow. "Estimation of Median Income for 4Person Families by State." In Indirect Estimators in Federal Programs. Statistical Policy Working Paper no. 21. Washington, DC: Office of Management and Budget, July 1993. Fisher, Gordon M. "Poverty Guidelines for 1992." Social Security Bulletin, vol. 55, no. 1, Spring 1992, pp. 4346. Ghosh, M. and J.N.K. Rao. "Small Area Estimation: An Appraisal" (with comments). Statistical Science, vol. 9, no. 1, February 1994, pp. 5593. Hogan, Howard. The 1990 PostEnumeration Survey: Operations and Results." Journal of the American Statistical Association, vol. 88, no. 423, September 1993, pp. 10471060. Rao, J.N.K., C.FJ. Wu, and K. Yue. "Some Recent Work on Resampling Methods for Complex Surveys." Survey Methodology, vol. 18, no. 2, December 1992, pp. 209217. Schirm, Allen L. "The Relative Accuracy of Direct and Indirect Estimators of State Poverty Rates." 1994 Proceedings of the Section on Survey Research Methods. Alexandria, VA American Statistical Association, 1994. Schirm, Allen L., Gary D. Swearingen, and Cara S. Hendricks. "Development and Evaluation of Alternative State Estimates of Poverty, Food Stamp Program Eligibility, and Food Stamp Program Participation." Washington, DC: Mathematica Policy Research, December 1992. Sigma One Corporation. Estimates of Persons Income Eligible for the Special Supplemental Food Program for Women, Infants, and Children (WIC) in 1989: National and State Tables. Alexandria, VA: U.S. Department of Agriculture, Food and Nutrition Service, Office of Analysis and Evaluation, August 1993. U.S. Department of Commerce, Bureau of the Census. Statistical Abstract of the United States. Washington, DC: U.S. Government Printing Office, 1993a. U.S. Department of Commerce, Bureau of Economic Analysis. Survey of Current Business, vol. 73, no. 9. Washington, DC: U.S. Government Printing Office, September 1993b. 27 U.S. Department of Commerce, Bureau of the Census. Statistical Abstract of the United States. Washington, DC: U.S. Government Printing Office, 1991a. U.S. Department of Commerce, Bureau of Economic Analysis. Survey of Current Business, vol. 71, no. 8. Washington, DC: U.S. Government Printing Office, August 1991b. APPENDIX THE ESTIMATION PROCEDURE: ADDITIONAL TECHNICAL DETAILS P 50 A This appendix provides additional information and technical details for several of the steps in our estimation procedure. For Step 2, we discuss how we calculated sample estimates and their standard errors. For Step 4, we provide complete definitions and data for calculating values for the three predictor variables in our regression model. We also list the other variables that we considered as potential predictors. For Step 5, we present the equations used to calculate shrinkage estimates and their standard errors. We also discuss at the end of this Appendix how we derived confidence intervals. For some steps, we provide, as needed, few or no additional details. 1. From the most recent census (1990), derive state estimates of the percentage of infants and children who were income eligible. 2. From the most recent CPS (March 1993), derive state sample estimates of the percentage of infants and children who were income eligible. Table Al displays sample estimates and estimated standard errors. We obtained CPS sample eligibility estimates with the same methodology used by the Census Bureau to calculate poverty estimates for individuals except (1) we compared a family's income to 185 percent, rather than 100 percent, of the applicable poverty guideline; (2) we used the poverty guidelines shown in Table A.2. rather than the poverty thresholds developed by the Census Bureau for official government statistical (as opposed to administrative) purposes; and (3) we counted secondary individuals under age IS (if they fell in the age ranges for infants and children) as poor/eligible, rather than excluding them.1 An infant or child is income eligible for WIC if his or her family's income is less than or equal to 185 percent of the poverty guideline for that family. The WIC poverty guidelines for 1992 in Table A2 were obtained by averaging "HHS" poverty guidelines for 1991 and 1992. We averaged poverty guidelines for consecutive calendar years because the WIC program year runs from Jury 1 of one calendar year to June 30 of the following calendar 'Previous research suggests that most of these young secondary individuals are foster children. F or determining WIC eligibility, a foster child who is the legal responsibility of a court or state welfare agency is a family of one individual. Although the CPS does not collect income data for a secondary individual under age 15, it is likely that such a person has little, if any, income. 31 TABLE A.1 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE 1992 Change Between 1909 (CPS) 1989 and 1992 Estimate Standard Standard State (Census) Estimate Error Estimate Error AUbama 46302 44.476 8.422 •1426 8.422 Alaska 4L367 41455 5430 •0.012 5430 Arizona 45.474 42402 5473 1672 5473 Arkansas 51206 53.929 7.405 1.723 7.405 California 37.760 46473 1487 8413 1487 Colorado 36.057 36.754 6431 0497 6431 Connecticut 21.200 34.766 9440 13466 9440 Delaware 28.543 31065 5414 3422 5414 District of Columbia 46.241 67.187 13.154 20446 13.154 Florida 40.021 49.457 1446 9.436 1446 Georgia 40615 40.193 6.443 •0422 6.443 Hawaii 36.821 46420 6.187 9.799 6.187 Idaho 46.808 46422 5448 0414 5448 Dlinoii 33.183 40496 3.736 7413 3.736 Indiana 35.470 47.068 5.924 11418 5.924 Iowa 36446 41.081 3407 4435 3407 Kansas 36.760 34428 5489 •1732 5489. Kentucky 48.123 53468 8.401 5.445 8.401 Louisiana 51651 55499 11.137 3448 11.137 Maine 34.528 47.455 11415 11927 11415 Maryland 24.246 37.928 3410 13482 3410 Massachusetts 25.087 27420 3.700 1533 3.700 Michigan 37.172 39426 4413 1454 4413 Minnesota 29.362 41.236 9409 11474 9409 Mississippi 57.544 59.956 8470 1412 8470 Missouri 38.929 59.946 7.180 21417 7.180 Montana 46439 53.704 14.153 7465 14.153 Nebraska 38.100 31261 4424 •5439 4424 Nevada 34.353 38.135 7.787 3.782 7.787 New Hampshire 20.531 30.094 4.184 9463 4.184 New Jersey 21446 31436 1461 8490 1461 New Mexico 53.995 54.170 8.942 0175 8.942 New York 35.136 44464 1461 9428 1461 Norm Carolina 39.911 45.196 3412 5485 3.412 Worth Dakota 41554 41977 9417 0423 9417 Ohio 37.048 39.171 3.114 1123 3.114 Oklahoma 47438 51280 7444 4442 7444 Oregon 39479 44.718 7.123 4439 7.123 33.428 33.162 1952 0.266 1952 Rhode Island 29423 38419 6.903 8496 6403 South Carolina 43447 51130 5499 8483 5499 South Dakota 47.214 40461 8.148 6453 8.148 Tennessee 44404 60452 4492 16448 4492 Taxat 4S435 51.141 4481 5406 4481 Utah 39.999 31884 7.756 7.115 7.756 Vermont 31.164 29.180 11009 1.984 11009 Virginia 31469 33435 4442 1266 4442 Washington 34.764 34.242 4.757 •0522 4.757 Watt Virginia 51403 60489 9473 8486 9473 Wnconsin 34494 29470 3430 4434 5430 Wyoming 41.211 36473 11.290 •4438 11490 United States 37.789 43480 0.795 5.791 0795 32 TABLEAU WIC POVERTY GUIDELINES FOR 1992 (Dollars) HHS Poverty Guidelines WIC Poverty Guidelines State and Family Size 1991 1992 1992 Alaska Oneperson family 8^90 8,500 8395 Each extra person 2,820 2,980 2,900 Hawaii Oneperson family 7,610 7,830 7,720 Each extra person 2,600 2,740 2,670 Other States and DC Oneperson family 6,620 6,810 6,715 Each extra person 2,260 2380 2,320 NOTE: The WIC Dovertv guidelines are simnle arithm ctic averasei of the*IHS novertv guidelines 33 year. Therefore, eligibility workers determined a family's eligibility for WIC using the 1991 HHS poverty guidelines during the first six months of 1992 and the 1992 HHS poverty guidelines during the last six months of 1992. The Office of the Secretary, Department of Health and Human Services, is responsible for developing the HHS poverty guidelines. The HHS poverty guidelines are derived from the Census Bureau poverty thresholds (Fisher 1992). We estimated standard errors for our sample estimates using the jackknife estimator proposed by Rao, Wu, and Yue (1992), treating CPS rotation groups as clusters. A rotation group, about oneeighth of a monthly CPS sample, consists of a group of households that begin the CPS at the same time. They are in the CPS for four months, rotate out for eight months, and rotate back in for four months, after which they are dropped from the CPS. To obtain jackknife standard errors, we let Z, equal the CPS sample estimate of the number of eligible infants and children in state i (i = 1,2, .... 51) and Z^ equal the contribution of rotation group r (r = 1,2,..., 8) to that estimate. In other words: 0) Zi  E zi, ■ r  1 If we were to exclude the observations in rotation group r, we could estimate the number of poor persons in state / by: (2) Zl(f)  • (Z,  Zf>) . The "(/■)" subscript indicates that rotation group r has been excluded. The factor 8/7 enters the expression because when (approximately) 1/8 of the sample is removed, an estimate from the remaining 7/8 of the sample needs to be inflated to get an estimate for the whole. By excluding each of the eight rotation groups in turn, we can get eight alternative estimates for the number of poor 34 persons in state i. Then, we can assess the degree of sampling variability (estimate the variance of Z,) by measuring the variability among the eight estimates according to: (3) var(Z,)  1 £ (Zi(r)  Zf . 8,^ I The factor 7/8 enters this expression because the Zi(r) are obtained from samples that are only 7/8 the size of the full CPS sample for state i and, hence, are expected to be more variable than Z, (by a factor of 8/7). If Y, equals the CPS sample estimate of the percentage of infants and children eligible in state /: (4) y.  100 % , where N, is the CPS sample estimate of the population of infants and children in state i. We estimate the variance of Yt by. , var(Z) (5) var(y,)  lOO2 _1£ where var(Z,) is calculated according to Equation (3). Our jackknife estimate of the standard error of yj is obtained by taking the square root of var(yi). Estimated jackknife standard errors for the CPS sample estimates for 1992 are presented in Table A.1. 3. CoBStroct sample estimates of the esuutge ia the pcrceatafe eligible betwcea 1989 aad 1992. A state's sample estimate of the change between 1989 and 1992 in the percentage of infants and children who were income eligible was obtained by subtracting the census estimate for 1989 from the CPS estimate for 1992. Sample estimates of change and their standard errors are presented in Table A. 1. We assumed that the sampling error associated with a census estimate is negligible. Therefore, the standard errors for the estimates of change in the percentage eligible equal the standard errors for the 1992 estimates of the percentage eligible. 4. Using a regression model, predict the change in the percentage eligible for each state based on observed changes in (i) Food Stamp Program (FSP) participation, (ii) Unemployment Insurance (UI) Program participation, and (ill) per capita income. Our "best'' regression model has three predictors that measure the changes between 1989 and 1992 in: • FSP participation • UI Program participation • Per capita income These three predictors were selected from a list that included variables measuring the changes in: • National School Lunch Program (NSLP) participation (number of students approved for free or reducedprice meals relative to the size of the schoolage populationages 5 through 17) • Supplemental Security Income (SSI) Program participation (number of recipients relative to the size of the population) • Aid to Families with Dependent Children (AFDC) Program participation (number of recipients relative to the size of the population) • Head Start Program participation (enrollment relative to the size of the preschoolage populationages 0 through 4) • Chapter 1 (Compensatory Education) Program funding (basic grant, in dollars, relative to the size of the schoolage population) • Per capita residential construction (in dollars) • Per capita nonresidential construction (in dollars) • Crime rate • Population density 36 We considered these variables because (1) we believed that they might indicate differences among states in the incidence of poverty (especially child poverty), socioeconomic conditions related to poverty, or the health of the state economy and (2) they could be measured uniformly across states for 1992 from nonsample or highly precise sample data. Variables measuring vital events (e.g., infant deaths), WIC participation, and Medicaid participation were rejected as potential predictors because they are often used as outcome measures in analyses of the effectiveness of the WIC Program.2 We selected our best regression model on the basis of its consistently strong relative performance in predicting changes in WIC eligibles for three time periods: 1989 to 1990, 1989 to 1991, and 1989 to 1992. We judged performance by examining numerous functions of the regression residuals, including R2 as well as measures that adjust for the loss in degrees of freedom from adding predictor variables.3 Definitions and data sources for the three predictor variables in our best regression model are given in Table A.3. Tables A.4 and A.5 provide the raw data for 1989 and 1992, respectively, used to calculate the predictor variables, and Table A.6 displays the calculated predictor variables for each state. Following the estimation procedure described in Step 5, we obtained the estimated regression equation shown below: Change in percentage eligible =  1.899 + 1.617 x Change in FSP participation + 4.644 x Change in UI Program participation  6.498 x Change in per capita income ^Estimating the numbers of WIC eligibles and, implicitly, WIC participation rates using the infant mortality rate (IMR), for example, as a predictor would have "built in" a relationship between WIC participation and the IMR, therefore biasing analyses of the effectiveness of WIC in reducing infant deaths. 3The residual for a state is the difference between the sample estimate and the regression prediction. Our best model tended to produce smaller residuals than did alternative models. 37 BLANK PAGE TABLE A.3 DEFINITIONS AND DATA SOURCES FOR PREDICTOR VARIABLES 00 Predictor Variable Definition: Change between 1989 and 1992 in Principal Data Sources' FSP participation* trw v Number of participants during August Resident population FSP participation data are population counts of participants from state program operations data and were obtained electronically from the Food and Consumer Service, U.S. Department of Agriculture. UI Program participation0 im v Number of first payment beneficiaries during year Resident population UI data for 1992 were obtained electronically from the Unemployment Insurance Service, U.S. Department of Labor. Data for 1989 are from Table 603, "State Unemployment Insurance, by State and Other Areas: 1989,' in U.S. Department of Commerce (1991a, p. 367). Per capita income4 (Total personal income + Resident population) WIC poverty guideline for oneperson family Total personal income data are from Table 1, Total and Per Capita Personal Income by State and Region, 198590," in U.S. Department of Commerce (1991b, p. 30) and Table 1, "Total and Per Capita Personal Income by State and Region, 198792," in US Department of Commerce (1993b, p. 74). ■Data on the resident population as of Jury 1 are from Table 26, "Resident PopulationStates and Puerto Rico: 1960 to 1990," in U.S. Department of Commerce (1991a, pp. 2021) and Table 31, "Resident PopulationStates: 1970 to 1992," in U.S. Department of Commerce (1993a, pp. 2829). *Data for August are often used to measure FSP participation. See, for example, Schirm, Swearingen, and Hendricks (1992). *A first payment beneficiary is a person receiving a UI payment for the first time in more than a year. *We measure per capita income relative to the WIC poverty guideline for a oneperson family to account for inflation. Poverty guidelines are adjusted annually based on the Consumer Price Index (CPI). The 1992 WIC poverty guidelines are displayed in Table A.2. The 1989 guidelines for a oneperson family are $7345, $6760, and $5875 for Alaska, Hawaii, and the rest of the U.S., respectively. >/ TABLE A.4 1989 DATA FOR CALCULATING PREDICTOR VARIABLES UI First Total Personal Resident Population FSP Recipients Payment Income on Jury 1 State in August Beneficiaries ($1*00.000) (1.000) Alabama 428*80 152,000 56*98 4.118 Alaska 23.766 33*00 11*76 527 Arizona 276*62 74*00 55352 3356 Arkansas 226,262 83*00 31*90 2,406 California 1327,414 1*24,000 576,489 29*63 Colorado 206384 74*00 58*15 3317 Connecticut 117396 119,000 80*09 3*39 Delaware 30286 22*00 12*93 673 District of Columbia 58.903 19*00 13*00 604 Florida 691.285 187,000 225*61 12371 Georgia 486.762 210*00 104,107 6*36 Hawaii 79.135 19*00 20,417 1,112 Idaho 57378 37,000 14.153 1*14 Dlinois 973*76 303,000 220*89 11358 Indiana 282.643 116.000 88308 5393 Iowa 163*10 73*00 44356 2*40 Kansas 132.794 69*00 41,916 2313 Kentucky 446.171 112,000 51396 3,727 Louisiana 725332 99*00 56320 4*82 Maine 84.185 44*00 20*81 1*22 Maryland 248.688 89*00 98*31 4*94 Massachusetts 319341 261,000 131.403 5*13 Michigan 875,425 393,000 163*69 9*73 Minnesota 248354 123,000 77334 4*53 Mississippi 483.489 72*00 31*89 2*21 Missouri 402*92 161.000 85,163 5,159 Montana 52313 22*00 11*48 806 Nebraska 91*87 27*00 25.772 1*11 Nevada 44*04 36*00 20.919 1.111 New Hampshire 23*22 32,000 22*46 1.107 New Jersey 357,935 268,000 182*82 7,736 New Mexico 150*28 28.000 20*40 1328 New York 1,409,738 544,000 374*92 17*50 North Carolina 381*99 211.000 101.440 6371 North Dakota 37*36 15*00 9*47 660 Ohio 1,072380 305.000 180,197 10*07 Oklahoma 253,921 50*00 43*91 3*24 Oregon 208*95 106,000 45.409 2320 Pennsylvania 901.156 406,000 209*00 12*40 Rhode Island 57*80 46*00 18*92 998 South Carolina 249*51 97*00 48*44 3*12 South Dakota 48*00 8*00 10*22 715 Tennessee 500,159 164,000 72*12 4*40 Tax* 1*81*21 340,000 263*58 16*91 Utah 93.793 31*00 22*87 L707 Vermont 34*92 19*00 9*34 367 Virginia 325,167 131*00 113*46 6*98 Washington 319*47 169*00 84*08 4,761 West Virginia 257,470 33*00 23*41 1*37 WsKonsin 280*11 172*00 80*79 4*67 Wyoming 26*19 10*00 6*44 475 TABLE AJ 1992 DATA FOR CALCULATING PREDICTOR VARIABLES UI First Total Personal Resident Population FSP Recipients Payment Income on July 1 State in August Beneficiaries ($1,000,000) (1.000) Alabama 555.232 157,084 68321 4.136 Alaska 40,477 44394 13.157 587 Arizona 475,882 90.486 66386 3332 Aifcaiuat 278,876 99322 37317 2399 California 2358340 1,443,782 662,786 30367 Colorado 264,118 79360 71354 3.470 Connecticut 207380 157319 89336 3381 Delaware 54360 28,787 15301 689 District of Columbia 86,135 26331 15390 589 Florida 1.398.057 339388 262,929 13,488 Georgia 777,194 231.957 124303 6.751 Hawaii 95.484 39381 25355 1.160 Idaho 71321 46.156 17334 1367 Dlinois 1.158311 390,904 255351 11331 Indiana 464394 149343 104304 5362 Iowa 191,727 88304 52,103 2312 Kansas 179,183 70323 48307 2323 Kentucky 527,608 127,034 63361 3,755 Louisiana 773335 109,968 68355 4387 Maine 133330 58340 22360 1335 Maryland 355,947 144326 114,115 4,908 Massachusetts 430,034 249341 142328 5.998 Michigan 1,002,451 487346 185,713 9,437 Minnesota 317332 133306 91312 4,480 Mississippi 540,061 79,145 36336 2314 Missouri 558.861 184.467 98363 5.193 Montana 66.965 25,147 13397 824 Nebraska 109353 33,436 30,438 1306 Nevada 83.417 60368 28354 1327 New Hampshire 57302 39315 25,100 1.111 New Jersey 510,070 339337 210,059 7,789 New Mexico 233334 31,702 24309 1381 New York 1,921386 673398 432,001 18,119 North Carolina 608,734 243,700 123.074 6343 North Dakota 47324 14336 10334 636 Ohio 1347.751 357397 207,769 11316 Oklahoma 352,129 65369 52347 3312 Oregon 258,457 141,756 54340 2377 Pennsylvania 1,157341 517,8i0 244314 12309 Rhode Island 88.795 60,746 19396 1305 South Carolina 380309 125.030 58362 3303 South Dakota 55300 8368 12,147 711 Tennessee 725.074 189367 88384 5324 Taut 2305.165 429,726 323387 17356 Utah 122358 37385 28328 1313 Vermont 53326 26377 10,732 570 Virginia 518397 137398 135.003 6377 Washington 439,451 219317 108301 5,136 West Virginia 310,970 60322 27.784 1312 Wisconsin 339.986 215369 95336 5307 Wyoming 33317 12322 8345 466 40 TABLEAU VALUES FOR PREDICTOR VARIABLES IN REGRESSION MODEL Change Between 1989 and 1992 in FSP UI Per Capita State Participation Participation Income Alabama 3.024 0.107 0.112 Alack* 2386 1350 •0.243 Arizona 4.633 0380 0.084 Arkansas 2.221 0.715 0.134 California 2325 1.154 •0.178 Colorado 1389 0.056 0.083 Connecticut 2.681 1.121 ■0.179 Delaware 3390 0.909 0.097 District of Columbia 4.872 1.409 0.109 Florida 4.909 1.039 •a 124 Georgia 3.949 0.173 0.000 Hawaii 1.115 1.686 0.104 Idaho 0.967 0.677 0.085 Dliuois 1.610 0.762 0.055 Indiana 3.151 0373 0.038 Iowa 1.071 0381 0.071 Kansas L818 0.061 0.042 Kentucky 2.080 0378 0.153 Louisiana 1.480 0306 0.157 Maine 3.923 L147 a 101 Maryland 1.954 1.051 •0.099 Massachusetts 1.761 0.257 0.237 Michigan 1.182 0.925 0.066 Minnesota 1364 0.154 0.018 Mississippi 1213 0381 0.085 Missouri 2.962 0.431 0.028 Montana 1375 0322 0.018 Nebraska 1.118 0.406 0.099 Nevada 1280 1302 •0.034 New Hampshire 3.060 0.702 0.103 New Jersey 1.922 0399 •0.008 New Mexico 4.914 0.173 0.063 New York 2.752 0.686 0.002 North Carolina 3.093 0350 0.050 North Dakota 1.738 0.075 0327 Ohio 1.492 0.448 0.003 Oklahoma 3.087 0303 0.038 Oregon 1181 1.003 0X02 Pennsylvania 1152 0.940 0.078 Rhode Island 3.055 1.435 ■a123 South Carolina 3.467 0.708 O069 South Dakota 0.939 a128 0.158 Tennessee 4307 0.455 0.114 Texas 4295 0.433 0.090 Utah. 1387 0363 0.105 Vermont 3395 1394 0.028 Virginia 1796 0.010 0.072 Washington 1344 0.718 0.122 West Virginia 3397 0308 0.171 Wisconsin 1.026 0.773 0021 Wyoming 1.672 0318 0379 41 As expected, the signs of the regression coefficients imply that, all else equal, states with (1) larger increases in FSP participation, (2) larger increases in UI Program participation, or (3) larger decreases in per capita income tend to have larger increases in the percentage of infants and children eligible for WIG4 Table A. 7 presents regression estimates and their standard errors for each state.' 5. Using "shrinkage" methods, average the sample estimates of change and the predictions of change. We have used a shrinkage estimator based on the Empirical Bayes estimator proposed by DuMouchel and Harris (1983). Their estimator was used by Ericksen and Kadane (1985) to estimate population undercounts in the 1980 census for 66 areas covering the entire U.S. and by Schirm, Swearingen, and Hendricks (1992) to estimate state poverty rates and FSP participation rates. The Empirical Bayes shrinkage estimator proposed by DuMouchel and Harris (1983) is: (6) Y,c.EB 1 D ♦ 1M u2 DYS, where K,ff is a (51 x 1) vector of Empirical Bayes shrinkage estimates, and 7, is a (51 x 1) vector of direct sample estimates. D is a (51 x 51) diagonal matrix with diagonal element (i,i) equal to one divided by the variance (standard error squared) of the direct sample estimate for state i.* M = I  X(X'X)~XX', where / is a (51 x 51) identity matrix and A' is a (51 x K) matrix containing data for 4This equsjon does not express a causal relationship. It does not imply that more FSP participants cause more WIC eligibles. Rather the equation implies only a statistical association: states with more FSP participants typically have more WIC eligibles than states with fewer FSP participants. For this reason, predictors are often called "symptomatic indicators." They are symptomatic of differences among states in conditions associated with having more or fewer WIC eligibles. 5As shown in the next step, we do not have to calculate regression estimates as a separate step, although we do have to select a best regression model before we can calculate shrinkage estimates. 6The fourth column of numbers in Table A1 is Yr while D can be obtained from the last column in that table. 42 TABLE A.7 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE REGRESSION ESTIMATES Stale Estimate Standard Error Alabama Alaaka Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida 2.760 9.343 7.439 4.142 8377 0.068 8.805 7.174 1L814 11470 2427 1868 1888 2.478 1618 2449 2389 1573 3.450 1898 Georgia Hawaii Idaho Elinois Indiana Iowa Kansas Kentucky Louisiana Maine 5290 7.058 1256 3.886 5.611 1070 L051 1226 (X895 10.428 1685 3.350 1525 1362 1337 1476 1575 1531 1607 1607 Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island 6.785 1295 4.737 0.905 1432 4.710 1260 1.151 8.055 6.979 5.436 6.441 5.750 4.403 0.215 1614 5.182 4417 5.439 10.505 1437 3.693 1505 1577 1421 1336 1450 1527 1540 1362 1314 3.072 1254 1406 1891 1390 1339 1477 1422 1702 South Carolina South Dakota Tennessee Utah Vermont Virginia Washington Wast Virginia Wisconsin Wyoming United States 6347 0L813 6.438 6.472 0721 9.782 3.137 3425 4481 3213 L397 5303 1454 1785 1822 1765 2354 1607 1728 1459 1699 1468 3.028 0499 43 each state on a set of k = K  1 symptomatic indicators. (The other column of A" consists of all ones and allows for an intercept in the regression model.)7 u2, a scalar reflecting the lack of fit of the regression model, is estimated by maximizing the likelihood function: (7) L  \W\W \X>WX\xri exp V\fr*\ , where W = (D~x + u2/)1 and S = W  WX(X'WX)XX'W. The variancecovariance matrix of the Empirical Bayes shrinkage estimator is: l (8) K. c.EB D * ±M u2 This estimator treats the maximum likelihood estimate of u2. once it is calculated, as known. We have taken a more fully Bayesian approach, treating u2 as estimated. Ifwe specify flat prior distributions for both B~the (K x 1) vector of regression coefficientsand U, that is, distributions proportional to one, the posterior density of u, evaluated at Uj, is proportional to: (9) P;  n)iw i*'»;.*r,/2exp( J(y,  tyywp,  &,)), where Wj^ (Z)'1 ♦ ufl)'1 and 6..  (X'WjX)1XtW.Yt. Under this formulation treating u as unknown but following a particular distribution, there is no closedform expression for our shrinkage estimator. Instead, we must numerically integrate over u. 7Except for a column of ones to allow for an intercept in the regression model, Table A.6 is the X matrix To perform the numerical integration, we selected a grid of 701 equally spaced values of u, starting with 0.00 and incrementing by 0.01. For each value u; * 0.00, 0.01 7.00 of u, we calculated a vector of shrinkage estimates: (10) n, D ♦ 1M i DYt, and a variancecovariancc matrix: (") Vcj D + 1M i These expressions for the shrinkage estimates and the variancecovariance matrix are the same as when u is treated as known.8 For each u;, we also calculated p* according to Equation (9). After calculating Yc , Vc , and p* 701 times (once for each value of u;), we calculated the probability of Uf. <>2) n ■ T^ yi which is also an estimate of the probability that the shrinkage estimates V are the true values. As Equation (12) suggests, the/>; are obtained by normalizing the p* to sum to one.9 •For a,  0, we set K,  XQ(fDK)'xX!B¥t and VeJ  XtfDX)^, the limiting values derived by DuMouchel and" Harris (1983). *The pi should approach 0 as u approaches the upper limit of the grid over which we integrate. If that does not occur, the grid should be extended, and the calculations repeated. 45 To complete the numerical integration over u and obtain a single set of shrinkage estimates, we calculated a weighted sum of the 701 sets of shrinkage estimates, weighting each set YCJ by its associated probability/^. Thus, our shrinkage estimates are: 701 j • i The variancecovariance matrix is: 701 701 (14) Ve  V PjVc. ♦ £ Pj(Ycj  Yc)(YcJ  Yc)' > 1 / • l The first term on the right side of this expression reflects the error from sampling variability and the lack of fit of the regression model. The second term captures how the shrinkage estimates vary as our estimate of u varies. Thus, the second term accounts for the variability from not being able to estimate u very well. Our shrinkage estimates and their standard errors are displayed in Table A.8.10 Our regression estimates, which were presented in the previous step, were similarly obtained. They are: 701 (i5) r,  EPjYrJ, where Y ■ X& is the vector of regression estimates obtained when u * u, The variancecovariance matrix is: 10The standard errors were calculated by taking the square roots of the diagonal elements of Vc. 46 TABLE A3 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE. SHRINKAGE ESTIMATES Sute Ffimur Standard Error Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida 2306 8.238 6336 3.977 8.619 0.133 9.044 6.7S8 11032 1O907 2.484 2.724 2.724 2302 1.403 1407 2.445 2308 3.372 1365 Georgia Hawaii Idaho Dlinoa Indiana Iowa Kansas Kentucky Louisiana Maine 4.785 7325 2.020 4.638 6.221 2.634 0.620 2.409 a977 10310 2.474 3.006 1249 1.936 1157 1.877 1310 1381 1510 1512 Maryland Massachusetts Michigan Minnesota Mississippi Missoun Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island 8.268 1396 4331 1.454 1436 5.911 2366 0.191 7.788 7.436 6309 6.123 6317 4.608 0.184 2322 5.145 4.830 3.904 10390 1161 1891 1165 1460 1284 1417 2388 2361 2384 1.959 1.670 1907 1.684 1J59 1750 1.783 1161 1278 1114 1469 South Carolina South Dakota Te Ta Utah Ve Virginia Washington West Virginia Wisconsin Wyoming United States 6.717 •L148 7373 6306 0227 9.442 3.033 3.041 4368 1413 1.212 5390 1203 1643 1725 2375 1451 2345 2347 1169 2379 2347 1925 0383 47 701 701 (16) vr  £ PjvrJ ♦ £ j^c*^i  w„  n>'. i 1 > 1 wbcrc Kf « XiX'WXj'W * up. We can estimate the regression coefficient vector by: 701 (17) 6  £ />,*, . i 1 Estimated values for the regression coefficients were displayed in the previous step. 6. Add the shrinkage estimate of the change between 1989 and 1992 to the census estimate of the percentage eligible in 1989 to get a shrinkage estimate of the percentage eligible in 1992. To facilitate a comparison of the alternative estimates, we have displayed in Table A. 9 not only shrinkage estimates but also sample and regression estimates of the percentage eligible in 1992. These estimates were obtained by adding the census estimates for 1989 from Step 1 to the estimates of change derived in Steps 3, 4, and 5. The estimates of change were displayed together in Chapter II, Table II.2 The sample estimates in Table A.9 are, of course, equal to the sample estimates obtained in Step 2 because we have just added and then subtracted the census estimates. 7. Multiply the shrinkage estimate of the perceatage eligible by the state population of infants and the state population of children to get preliminary shrinkage estimates of the numbers of eligible infants and children. As we stated in Chapter II, we assumed in this step that the percentage of infants who were income eligible equals the percentage of children who were income eligible. Census estimates show that this assumption is reasonable. Nationwide, 37.982 percent of infants and 37.739 percent of children were income eligible in 1989. TABLE A.9 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 Sample Regression Shrinkage Slate Estimate Estimate Estimate Alabama 44.476 49.062 48308 Alaska 41.355 50.710 49305 Arizona 42J0S 51913 51310 Arkansas 53.929 56348 56.183 California 46.573 46.137 46379 Colorado 36.754 36.125 36.190 Connecticut 34.766 30,005 30344 Delaware 32.065 35.717 35331 District of Columbia 67.187 58355 58373 Florida 49.457 51391 50.928 Georgia 40.193 45.905 45.400 Hawaii 46.620 43379 44.146 Idaho 46.822 49.064 48328 Dlkiois 40.696 37369 37321 Indiana 47.088 41381 41391 Iowa 41.081 38.916 39.480 Kansas 34.028 37311 37380 Kentucky 53.568 50349 50332 Louisiana 55.899 53346 53328 Maine 47.455 44.956 45.038 Maryland 37.928 31331 31514 Massachusetts 27.620 26382 26383 Michigan 39.026 41.909 41303 Minnesota 41236 30367 30316 Mississippi 59.956 59.976 59.980 Missouri 59.946 43339 44340 Montana 53.704 48399 49.005 Nebraska 31261 39351 38391 Nevada 38.135 41408 41141 New Hampshire 30.094 27310 27.967 New Jersey 31.036 27382 28.955 New Mexico 54.170 60.436 60.118 New York 44364 40386 41053 North Carolina 45.196 44314 44319 North Dakota 42.977 42339 42370 Ohio 39.171 39362 39370 Oklahoma 52380 51820 51783 Oregon 44.718 44396 44.709 Pennsylvania 33.162 38367 37332 Rhode bland 38319 40328 40313 South Carolina 51130 50394 50364 South Dakota 40361 46.401 46366 Tennessee 60352 50342 51377 Texas 51.141 52307 51141 Utah 32384 40.720 40326 Vermont 29.180 40.946 40306 Virginia 33335 34306 34.402 Washington 34342 38389 37305 Watt Virginia 60389 56384 56.471 Wisconsin 29370 37307 36307 Wyoming 36373 42308 41423 United States 43380 43392 43379 The independent population estimates used in this step and displayed in Chapter II, Table 11.4 were obtained electronically from the U.S. Bureau of the Census. The release date for the estimtics was March IS, 1994. 8. Control the preliminary state shrinkage estimates of the numbers of eligible infants and children to sum to the national totals for eligible infants and children obtained from the CPS. In Chapter III, we presented approximate 90percent confidence intervals for our final shrinkage estimates. The upper and lower bounds of the confidence intervals were calculated according to: (18) Upper Bound. = Eci * 1.645 eci and (19) Lower Bound.  Eci  1.645 eci , where Eci is the final shrinkage estimate (for infants or children) for state i and eci is the standard error of that estimate. That standard error is: (20) ^.r.H.ljJpL. where rc equals 1,717,743 + 1,737,837 (for infants) or 6,925,815 + 6,721,734 (for children), N„ is the independent population estimate of either infants or children in state i, and Vc(ii) is the (L,i) diagonal element of Ve, which was calculated according to Equation (14). In other words, the square root of Vc(ii) is the standard error of the shrinkage estimate of the percentage of infants and children eligible 50 in state i in 1992.11 We can find values for Eci, Nti, and the square root of Vc(ii) in Tables III. 1, 114, and A.8, respectively. In addition to presenting the confidence intervals for our shrinkage estimates in Chapter III. we discussed the relative precision of sample and shrinkage estimates. To inform that discussion, we derived "final" sample estimates in the same way as we derived our final shrinkage estimates.12 Both sets of final estimates appear in Table A. 10. In Tables A. 11 and A. 12, we present confidence intervals for sample and shrinkage estimates of eligible infants and children, respectively. We calculated bounds for confidence intervals of sample estimates according to Equations (18) and (19), replacing shrinkage estimates by sample estimates. The standard error for a sample estimate is given by: (2.) «,,»,«. where r, equals 1,717,743 + 1,746,319 (for infants) or 6,925,815 + 6,754,737 (for children), and the square root of MD(ii) is in the third column of numbers in Table Al. 11As in Step 3, we assumed that the sampling error associated with a census estimate is negligible. Therefore, the standard error for the shrinkage estimate of the proportion eligible is the same as the standard error for the shrinkage estimate of the change in the proportion eligible. Our estimate of ecl does not take account of the correlation between rc and our estimate of the proportion eligible. Instead, rc is treated as a constant "Beginning with the sample estimates of the percentage eligible in 1992, we used the independent population estimates of infants and children to obtain preliminary sample estimates of the numbers eligible and, then, controlled those preliminary estimates to the national totals. The preliminary estimates summed to 1,746319 infants and 6,754,737 children. 51 TABLE A.10 SAMPLE AND SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 •Final" Sample Estimates Final.Shrinkage Estimates Sutc Infants Children Infants Children Alabama 26,984 107,747 29,757 118322 Alaska 4,683 19304 5345 23310 Arizona 27,750 111371 33.754 135.715 Arkansas 18,278 75.149 19,135 78374 California 271412 1335354 271301 1336304 Colorado 19374 79304 19,170 78,174 Connecticut 15.929 67388 13325 58311 Delaware 3397 13330 3,761 15392 District of Columbia 5,499 21329 4,793 19326 Florida 92,634 381.175 95355 394,439 Georgia 43.183 174332 49316 197,770 Hawaii 8.992 34369 8356 32,705 Idaho 7361 32.188 8338 33,732 Illinois 75371 297325 70389 277364 Indiana 38,129 156313 33324 139,077 Iowa 15351 65.464 14,729 63322 Kansas 12316 52.495 13396 57349 Kentucky 27374 113362 26,423 107367 Louisiana 38385 154.766 37394 149307 Maine 7380 32.458 6342 30356 Maryland 28391 117,770 24371 101,455 Massachusetts 23.429 96368 22,745 93.750 Michigan 53343 230319 56399 246.140 Minnesota 26366 114.025 19,725 85330 Mississippi 25355 98336 25,188 99361 Missouri 44302 185388 33374 139377 Montana 6305 25.730 5306 23394 Nebraska 7346 31.700 8342 37310 Nevada 8323 33072 9,131 36.725 New Hampshire 4384 20.444 4381 19392 New Jersey 36316 145,098 33,765 136.034 New Mexico 14321 58349 16329 65332 New York 122351 491353 117,020 467,950 North Carolina 44,985 182.485 44328 180335 North Dakota 3387 15309 3353 15.761 Ohio 63347 254,736 64304 258395 Oklahoma 24324 99367 24378 101.019 Oregon 18,157 77319 18342 78,185 Pennsylvania 52350 220367 59399 249361 Rhode Island 5367 22328 5.795 23347 South Carolina 28367 116.175 27344 113339 South Dakota 4.400 18339 5322 20317 Tennessee 43394 173.919 37396 150230 Taut 159340 620,788 163.761 636334 Utah 11310 47.704 14318 58341 Vermont 2.162 9384 3323 13342 Virginia 31318 128,169 32,497 131.735 Washington 26389 109.746 29377 121,761 West Virginia 13326 53370 12300 50368 WaMoaan 20366 88391 25313 108307 Wyoming 2337 10445 2317 12376 United States 1.717,740 6325316 1,717,746 6325319 52 TABLE A.11 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE INFANTS Sample Estimates Shrinkage Estimates State Lower Bound Upper Bound Lower Bound Upper Bound Alabami 18,579 35389 27366 32348 AlMfc. 3.690 5376 5,135 6.155 Arizona 21,700 33300 30335 36373 Arkansas 14,149 22,407 17345 20325 California 255.145 287.479 257,990 285312 Colorado 13.624 25124 17373 21367 Connecticut 9.116 22,742 12373 15,777 Delaware 2384 4310 3357 4.165 District of Columbia 3,728 7370 4337 5349 Florida 85,098 100.170 89,771 101.939 Georgia 31,796 54370 44322 53.410 Hawaii 7,029 10355 7398 9314 Idaho 6329 9393 7314 8362 Illinois 63.989 86.753 64362 76316 Indiana 30.238 46320 31337 36311 Iowa 13,292 17310 13377 15381 Kansas 9,107 15325 12314 14378 Kentucky 20.683 35365 24375 28,471 Louisiana 26.006 51364 34,423 40,165 Maine 4.450 10.110 6305 7379 Maryland 23361 32.721 21.706 27336 Massachusetts 18.266 28392 18391 26,799 Michigan 42390 63396 52316 61,782 Minnesota 16326 35.706 17,135 22315 Mississippi 19,095 31315 23310 26,766 Missouri 35332 52372 30.141 36307 Montana 3.402 8308 5365 5347 Nebraska 5.464 9328 7,765 9319 Nevada 5.461 10385 8381 9381 New Hampshire 3336 5332 3.788 4,774 New Jersey 31318 40,714 30361 36369 New Mexico 10,796 18346 15314 17344 New York 111341 134.061 109312 124,728 North Carolina 39398 50372 41.469 47387 North Dakota 2330 5344 3363 4343 Ohio 55.063 71331 59338 69370 Oklahoma 18,780 29368 23316 26340 Oregon 13399 22315 16,713 19,771 Pennsylvania 45.196 60.704 54319 65379 Rhode Island 3339 7,195 5310 6380 South Carolina 23,159 33375 25348 29340 South Dakota 2346 5354 4348 5396 Tennessee 37,757 48331 34,165 40327 Texas 134331 185349 15L491 176331 Utah 7328 16392 13363 15373 Vermont 698 3326 2.711 3335 Vnginia 23376 39360 28350 36,144 Washington 20358 32320 26314 32340 West Virgkua 9341 1M11 11383 13,117 Wkcoosin 14331 26381 22368 27358 Wyoming L210 3364 2397 3,137 53 TABLE A12 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE CHILDREN Sample Estimates Shrinkage Estimates State Lower Bound Upper Bound Lower Bound Upper Bound Alabama 74,184 141310 108374 128,770 Alaska 15.369 23339 21486 25334 Arizona 87J45 135397 123,977 147.453 Arkansas 58,175 92,123 73471 83477 California 974,131 1397477 985.020 1388.188 Colorado 55,557 102,451 69321 86,727 Connecticut 38463 96413 51377 66,745 Delaware 9^65 17395 13,470 16,714 District of Columbia 14,799 28359 17415 20837 Florida 350,164 412.186 369,404 419,474 Georgia 128,288 220176 180,042 215.498 Hawaii 26366 41372 29342 36468 Idaho 25.914 38,462 31,176 36488 Dlinois 252494 342,456 254.466 301462 Indiana 123,964 188,662 127440 150914 Iowa 57,057 73371 58478 68.166 Kansas 38319 66.171 52358 63340 Kentucky 84,042 142,482 99345 115.689 Louisiana 104,043 205,489 137.719 160695 Maine 19340 45376 28.116 33,796 Maryland 99331 136409 90463 112447 Massachusetts 75,288 117348 77341 110459 Michigan 185335 275.103 225,018 267462 Minnesota 73,046 155.004 74485 96375 Mississippi 75325 122447 93,137 105485 Missouri 148,781 221.795 126,927 151327 Montana 14375 36385 21,703 25.485 Nebraska 23,903 39.497 33475 41345 Nevada 21.963 44,181 33407 40143 New Hampshire 15,768 25.120 16392 21492 New Jersey 126,171 164.025 123.128 148.940 New Mexico 42369 74329 60.411 70353 New York 446.425 536.081 437,125 498.775 North Carolina 159323 205,147 168427 193,043 North Dakota 10053 21.765 14378 17,444 Ohio 221,423 288.049 239,427 277,763 Oklahoma 76372 122462 94416 107322 Oregon 57,428 98410 71332 84,738 Pennsylvania 188325 253409 226486 273,136 Rhode Island 15367 28489 20489 25.705 South Carolina 94.183 138,167 105,123 121455 South Dakota 12412 24466 18352 22,782 Tennessee 151377 196,161 137449 163411 Texas 521326 720450 588477 683391 Utah 29,195 66413 52,763 64419 Vermont 3428 16440 12,146 14438 Virginia 97,190 159.148 116451 146419 Washington 84366 134326 110469 133453 West Virginia 39.721 67319 46,492 54344 Wisconsin 61.123 116,059 974OO 120414 Wyoming 5484 15,706 10706 13446 54
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Title  State estimates of infants and children income eligible for the WIC Program in 1992 / 
Date  1995 
Contributors (individual)  Schirm, Allen L. 
Contributors (group) 
United States Dept. of Agriculture Food and Consumer Service. Mathematica Policy Research, Inc. 
Subject headings 
Food reliefUnited StatesStatesMathematical models Food reliefUnited StatesStatesStatistics 
Type  Text 
Format  Pamphlets 
Physical description  xi, 54 p. :ill. 
Publisher  Washington, D.C. : Mathematica Policy Research 
Language  en 
Contributing institution  Martha Blakeney Hodges Special Collections and University Archives, UNCG University Libraries 
Source collection  Government Documents Collection (UNCG University Libraries) 
Rights statement  http://rightsstatements.org/vocab/NoCUS/1.0/ 
Additional rights information  NO COPYRIGHT  UNITED STATES. This item has been determined to be free of copyright restrictions in the United States. The user is responsible for determining actual copyright status for any reuse of the material. 
SUDOC number  A 98.2:St 2 
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Fulltext  MM IN* •»•» # ff.j'.sr & Contract No.: 5331984031 MPR Reference No.: 8207050 STATE ESTIMATES OF IN*ANTS AND CHILDREN INCOME ELIGIBLE FOR THE WIC PROGRAM IN 1993 May 15,1995 Author AJIenLSchinn Submitted to: U.S. Department of Agriculture Food and Consumer Service 3101 Park Center Drive Room 214 Alexandria, VA 22302 Attention: Cindy Long Submitted by: Matbematica Policy Research, Inc. 600 Maryland Avenue, S.W. Suite 550 Washington, D.C 20024 \qs0wf67 <*/ Nttd. 1 SUMMARY: STATE ESTIMATES OF INFANTS AND CHILDREN INCOME ELIGD3LE FOR THE WIC PROGRAM IN 1992 This report presents the methodology and data used for developing statelevel estimates of infants and children income eligible for WIC in 1992. These estimates were released in September 1994 and were used in calculating FY1995 WIC food grants. These estimates were developed using a statistical technique known as "shrinkage". For the eligibles estimates, shrinkage was used to develop estimates of the proportion of children age 04 in each state in households with incomes below 185% of poverty. These proportions were then applied to 1992 state population estimates of infants (01) and children (14) provided by the Bureau of the Census. The shrinkage approach combines estimates obtained directly through surveys with estimates produced by an econometric model which uses statelevel economic data. The direct sample estimates used were the March 1993 Current Population Survey (CPS) estimates of the proportion of children 04 below 185% of poverty in each state in 1992. The model estimates were developed using a regression model which estimated the proportion of children 04 below 185% of poverty using the following statelevel economic variables': • Food Stamp participation • Unemployment Insurance (receipt of first benefits for a period of unemployment) • Per capita income The model used was a "change" model, which estimated the change in the proportion of children (04) below 185% of poverty from 1989 (based on 1990 Census data) to 1992 as a function of the change in the variables noted above for the same time period. This "change" model provided better estimates than models which directly estimated the proportion of children below 185% of poverty. The shrinkage technique averages the sample and the model estimates using weights that reflect the relative precision of each estimate Thus, in cases where the CPS estimate has a high standard error, the model estimate would tend to receive more weight than it does in cases where the CPS standard error is lower. The shrinkage estimates are, for many purposes, superior to either the direct sample estimate or the model estimate. In particular, the shrinkage estimates have lower standard errors than the direct sample estimates. £ ACKNOWLEDGMENTS I thank Matthew McKearn and Cindy Long of the Food and Consumer Service, Alan Zaslavsky of Harvard University, and John Czajka of Mathematica Policy Research for helpful comments and guidance. Aleda Freeman provided skillful programming assistance. Deborah Patterson assisted in the preparation of the report, and Daryl Hall edited the report ui / m,MB Mi CONTENTS Chapter Page EXECUTIVE SUMMARY xi I INTRODUCTION 1 II A STEPBYSTEP GUIDE TO DERIVING STATE ESTIMATES OF ELIGIBLE INFANTS AND CHILDREN 5 HI STATE ESTIMATES OF WIC ELIGIBLES FOR 1992 21 REFERENCES 27 APPENDIX: THE ESTIMATION PROCEDURE: ADDITIONAL TECHNICAL DETAILS 29 Y\ Q ^ ®m MMK TABLES Tabk II.1 D.2 n.3 II.4 m.i ra.2 Al A2 A3 A4 A5 A6 A7 A8 A.9 Pac PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND CPS SAMPLE ESTIMATES 8 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE 13 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND SHRINKAGE ESTIMATES 17 PRELIMINARY SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 19 FINAL SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 22 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR SHRINKAGE ESTIMATES 23 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE 32 WIC POVERTY GUIDELINES FOR 1992 33 DEFINITIONS AND DATA SOURCES FOR PREDICTOR VARIABLES 38 1989 DATA FOR CALCULATING PREDICTOR VARIABLES 39 1992 DATA FOR CALCULATING PREDICTOR VARIABLES 40 VALUES FOR PREDICTOR VARIABLES IN REGRESSION MODEL 41 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: REGRESSION ESTIMATES 43 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: SHRINKAGE ESTIMATES 47 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 49 vii TABLES (continued) Table Page A10 SAMPLE AND SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 52 A11 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE INFANTS 53 A12 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE CHILDREN 54 Vlll FIGURES Figure P«*e D.1 THE ESTIMATION PROCEDURE 6 IL2 AN ILLUSTRATIVE REGRESSION ESTIMATOR 11 D.3 SHRINKAGE ESTIMATION 15 IX X [EMS Mi EXECUTIVE SUMMARY This report presents state estimates of the numbers of infants and children who were income eligible for the Special Supplemental Nutrition Program for Women, Infants, and Children (WIC) in 1992. These estimates, used to calculate state WIC grants for fiscal year 1995, were derived using "shrinkage" estimation. Drawing on both census and Current Population Survey (CPS) data as well as administrative records data from government program case files and vital statistics systems, we developed shrinkage estimates by averaging CPS sample estimates with predictions of WIC eligibles from a statistical regression model. The predictions were based on observed changes in government program participation and other indicators of socioeconomic conditions. The shrinkage estimates are more timely than census estimates, which had been used for fund allocation in prior years, and substantially more precise than CPS direct sample estimates. I. INTRODUCTION To allocate program funds equitably, the U.S. Department of Agriculture requires timely and accurate state estimates of the number of people eligible for the Special Supplemental Nutrition Program for Women, Infants, and Children (WIC). This report presents state estimates of the numbers of infants (age under 1) and children (ages 1 to 4 inclusive) who were income eligible for the WIC Program in 1992, that is, the numbers whose family incomes were at or below 185 percent of the applicable poverty guidelines. These estimates, used to calculate state WIC grants for fiscal year 1995, were derived using "shrinkage'' estimation. This introductory chapter explains the advantages of shrinkage estimation relative to direct estimation from the census or the Current Population Survey (CPS), the leading data sources for developing state estimates. Chapter II describes how we derived shrinkage estimates, and Chapter III presents our state estimates of WICeligible infants and children for 1992. Technical details and additional information about our estimation method are provided in the Appendix. The census is the most commonly used data source for deriving state, county, and other subnational estimates. However, because the census is conducted only once every 10 years, census estimates may not be timely for many purposes. As suggested by the estimates presented in this report, social and economic conditions change, often rapidly, over time. Therefore, more recent data may better reflect current conditions. The CPS provides the most recent data from which we can develop annual state estimates of WIC eligibies. However, despite their timeliness, CPS sample estimates are typically imprecise because state samples of infants and children are small. For example, although our single best direct estimate from the CPS is that Minnesota bad 114 thousand eligible children in 1992, we are able to state with confidenceaccording to widely accepted statistical standardsonly that we believe the true number lies between 74 and 154 thousand. Such a wide range reflects imprecision and suggests that we are very uncertain about the number of eligible children in Minnesota and that our estimate of 114 thousand may be highly inaccurate. We are also unable to determine how Minnesota compares with other states. The estimates of eligible children for about onethird of the states fall within the 74 to 154 thousand range, even though some of those other states have much bigger or smaller populations than Minnesota. Ranked in terms of eligible children, Minnesota could fall below those states, above them, or somewhere in the middle. Restricting ourselves to direct estimation from the census or the CPS forces us to make a tradeoff between timeliness and precision. We have minimized this tradeoff by using an alternative methodshrinkage estimationto develop state estimates of WIC eligibles. Our shrinkage estimator uses both census and CPS data as well as administrative records data from government program case files and vital statistics systems. We obtained shrinkage estimates by averaging CPS sample estimates with predictions of WIC eligibles made using a statistical regression model. Our predictions are based on observed changes in government program participation and other indicators of socioeconomic conditions. The shrinkage estimates presented in this report are as timely as the CPS sample estimates but substantially more precise. Shrinkage estimators have been used for allocating program funds and other purposes. Fay and Herrioti (1979) developed a shrinkage estimator that combined sample and regression estimates of per capita income for small places (population less than 1,000). Their estimates were used to allocate funds under the General Revenue Sharing Program. State shrinkage estimates of median income for fourperson families are used to administer the Low Income Home Energy Assistance Program (LIHEAP) (Fay, Nelson, and Litow 1993). Schirm, Swearingen, and Hendricks (1992) used a shrinkage estimator similar to the one used for this report to develop state estimates of poverty and Food Stamp Program eligibility and participation. Finally, a shrinkage estimator was used to adjust the 1990 decennial census for the undercount (Hogan 1993), although the secretary of commerce ultimately rejected adjusted figures in favor of unadjusted Ggures as the official 1990 census population estimates. A recent review of shrinkage methods and other techniques for "small area" estimation can be found in Ghosh and Rao (1994). In his evaluation of small area estimators, Schirm (1994) compared the relative accuracy of alternative state poverty estimates and found that shrinkage estimates are substantially more accurate than the estimates obtained from other methods that have been widely used. Those findings give us further confidence in the estimates presented in this report. 1 mm. ME II. A STEPBYSTEP GUIDE TO DERIVING STATE ESTIMATES OF ELIGIBLE INFANTS AND CHILDREN This chapter describes our procedure for estimating the numbers of infants and children who were income eligible for WIC in each state. This procedure, summarized by the flow chart in Figure II. 1, has the following eight steps: 1. From the most recent census (1990), derive state estimates of the percentage of infants and children who were income eligible. 2. From the most recent CPS (March 1993), derive state sample estimates of the percentage of infants and children \vho were income eligible. 3. Construct sample estimates of the change in the percentage eligible between 1989 and 1992. 4. Using a regression model, predict the change in the percentage eligible for each state based on observed changes in (i) Food Sump Program (FSP) participation, (ii) Unemployment Insurance (UI) Program participation, and (iii) per capita income. 5. Using "shrinkage" methods, average the sample estimates of change and the predictions of change. 6. Add the shrinkage estimate of the change between 1989 and 1992 to the census estimate of the percentage eligible in 1989 to get a shrinkage estimate of the percentage eligible in 1992. 7. Multiply the shrinkage estimate of the percentage eligible by the state population of infants and the state population of children to get preliminary shrinkage estimates of the numbers of eligible infants and children. 8. Control the preliminary state shrinkage estimates of the numbers of eligible infants and children to sum to the national totals for eligible infants and children obtained from the CPS. Each step is described below, and additional technical details are provided in the Appendix. FIGURE n.l THE ESTIMATION PROCEDURE ^ 1990 Census ^ 1. State estimates of percentage eligible in 1989 5. Shrinkage estimates of change in percentage eligible (obtained by averaging) 6. Shrinkage estimates of percentage eligible in 1992 3. Estimates of change in percentage eligible 4. Regression predictions of change in percentage eligible QMarch 1993 CPS^ i 2. State estimates of percentage eligible in 1992 Administrative data on FSP and UI Frogram participation and per capita income Independent state population estimates for infants and children 7. Preliminary shrinkage estimates of numbers eligible 8. Final shrinkage estimates of numbers eligible (controlled to national sample estimates) 1. From the most recent census (1990), derive state estimates of the percentage of infants and children who were income eligible. Table II. 1 presents 1990 decennial census estimates of the percentage of infants and children who were income eligible in each state. Because the family income data collected in the census pertain to the preceding calendar year, the eligibility estimates in Table II. 1 are for 1989. According to the table, 28.543 percent of all infants and children in Delaware, for example, were income eligible for WIC in 1989. We estimated the percentages, rather than the numbers, of infants and children who were income eligible for a simple technical reason. Percentages standardize for state size, in contrast to counts where one state may have more eligible infants and children than another state simply because the first state has a larger population. Such standardization is required for the regression and shrinkage estimation performed in subsequent steps. We derived the estimated percentages in Table II. 1 from estimates developed by Sigma One Corporation (1993). Because census samples for states are very large, the estimates are precise. However, they may quickly become "old" if economic conditions have changed substantially in the years since the census. 2. From the most recent CPS (March 1993), derive state sample estimates of the percentage of infants and children who were income eligible. The most recent CPS that has income data for families provides more timely information than the census. That CPS was the March 1993 CPS when we were developing eligibles estimates to be used in allocating funds for fiscal year 1995. Table II. 1 displays sample estimates from the March 1993 CPS. Like the census, the CPS collects family income data for the prior year. Thus, the sample estimates pertain to 1992. According to the table, 32.065 percent of all infants and children in Delaware, for example, were income eligible for WIC in 1992, compared with 28.543 percent in 1989. TABLE D.l PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND CPS SAMPLE ESTIMATES State 1989 (Census) 1992 Change Between (CPS) 1989 and 1992 44.476 1.826 41.355 O012 42802 1672 53.929 1.723 46.573 MIS 36.754 0.697 34.766 13366 32065 3.522 67.187 20.946 49.457 9.436 40.193 0422 46.620 9.799 46.822 0.014 40.696 7.513 47.088 11.618 41.081 4235 34.028 •1732 53.568 5.445 55.899 3148 47.455 11927 37.928 13.682 27.620 1533 39.026 1.854 41.236 11.874 59.956 1412 59.946 21.017 53.704 7.065 32.261 5.839 38.135 3.782 30.094 9.563 31.036 1590 54.170 0.175 44.364 9228 45.196 5285 41977 0423 39.171 1123 '^280 4.642 44.718 4.839 33.162 0266 38.819 8.996 51130 8283 40561 6653 60352 16.348 51.141 5306 31884 7.115 29.180 1.984 33.635 2266 34242 0522 60589 8.986 29.870 4.224 36373 4.338 Alabama 46302 Alaska 41367 Arizona 45.474 Arkansas 52206 California 37.760 Colorado 36.057 Connecticut 21200 Delaware 28.543 District of Columbia 46241 Florida 40021 Georgia 40.615 Hawaii 36321 Idaho 46308 Dlinois 33.183 Indiana 35.470 Iowa 36346 Kansas 36.760 Kentucky 48.123 Louisiana 51651 Maine 34328 Maryland 24246 Massachusetts 25.087 Michigan 37.172 Minnesota 29362 Mississippi 57344 Missouri 38.929 Montana 46339 Nebraska 38.100 Nevada 34353 New Hampshire 20331 New Jersey 21446 New Mexico 53.995 New York 35.136 North Carolina 39.911 North Dakota 42354 Ohio 37.048 Oklahoma 47338 Oregon 39379 Pennsylvania 33.428 Rhode Island 29323 South Carolina 43347 South Dakota 47214 Tennessee 44.004 Texas 45335 Utah 39.999 Vermont 31.164 Virginia 31369 Washington 34.764 West Virginia 51303 Wisconsin 34.094 Wyoming 41211 United States 37.789 43380 5.791 8 Although timely compared with the census estimates, the CPS sample estimates are relatively imprecise. The standard errors for the CPS estimates, reported in the Appendix, tend to be large, so our uncertainty is great. For example, according to widely used statistical standards, we can be confident only that the percentage of incomeeligible infants and children in Delaware was between 22.501 percent and 41.629 percent. This range is so wide and our uncertainty so great because the CPS samples of infants and children in each state are small. Indeed, that is why we derived an eligibility estimate for infants and children combined, rather than separate estimates, one for infants and one for children. In the March 1993 CPS, there are data for fewer than 30 infants for most states. 3. Construct sample estimates of the change in the percentage eligible between 1989 and 1992. A sample estimate of the change in the percentage eligible between 1989 and 1992 was calculated by subtracting the census estimate for 1989 from the CPS estimate for 1991 According to Table II. 1, the percentage eligible in Delaware rose by 32.065  28.543 = 3.522 percentage points over the three years. We calculated sample estimates of change for use in the regression and shrinkage estimation described in the next few steps. Focusing on the change in the percentage eligible between 1989 and 1992, rather than just the percentage eligible in 1992, is a simple way to reflect a strong systematic relationship: states with a high percentage eligible in 1989 tend to have a high percentage eligible in 1992, and states with a low percentage eligible in 1989 tend to have a low percentage eligible in 1992. In principle, our shrinkage method obtains better estimates by using information on not only where a state "is," but also where it "began." 4. Using a regression model, predict the change in the percentage eligible for each state based on observed changes in (i) Food Stamp Program (FSP) participation, (ii) Unemployment Insurance (UI) Program participation, and (Hi) per capita income. The main limitation of the sample estimates derived in the previous step is imprecision. Regression can reduce that imprecision. Regression estimates are predictions based on nonsample or highly precise sample data, such as census and administrative records data. The latter include government program case files and vital statistics records. Figure D.2 illustrates how the regression estimator works. The simple example in the Ggure has just nine states and one predictor variablethe change in FSP participationthat will be used to predict each state's change in the percentage of infants and children who were income eligible for WIG The triangles in the figure correspond to sample estimates; a triangle shows the change in FSP participation in a state (on the horizontal axis) and the sample estimate of change in WIC eligibles in that state (on the vertical axis). Not surprisingly, the graph suggests that the change in FSP participation is systematically associated with the change in WIC eligibles. States with larger increases in FSP participation tend to have larger estimated increases in WIC eligibles, although the relationship is far from perfect. To depict this relationship between changes in FSP participation and WIC eligibles, we can use a technique called "least squares regression" to draw a line through the triangles (that is, we "regress" the sample estimates on the predictor variable). Regression estimates of WIC eligibles are points on that line, the circles in Figure II.2. The predicted change in WIC eligibles for a particular state is obtained by moving vertically from the state's sample estimate (the triangle) to the regression line (where there is a circle) and reading the value off the vertical axis. For example, the regression estimator predicts about a 6 percentage point change in WIC eligibles for both of the states with increases in FSP participation just under 3 percentage points. In contrast, for the state with a 1 percentage point increase in FSP participation, the predicted increase in WIC eligibles is under 2 percentage points. 10 g5 LU 0) O) FIGURE 11.2 AN ILLUSTRATIVE REGRESSION ESTIMATOR 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Change in Percentage Receiving Food Stamps ii Table II.2 displays the sample estimates calculated in Step 3 and the regression estimates calculated in this step. To derive the regression estimates in Table II.2, we included all of the states, not just nine as in our illustrative example, and we used three predictor variables, not just one. Adding two predictor variables improves our predictions. The three predictor variables used measure the changes between 1989 and 1992 in (1) FSP participation, (2) UI Program participation, and (3) per capita income. These three were selected as the best predictors from a longer list presented in the Appendix, which also provides complete definitions and data for calculating values for the three best predictors. As expected, the estimated regression displayed in the Appendix shows that states with relatively large increases in FSP and UI Program participation and large decreases in per capita income tend to have relatively large increases in the percentage of infants and children eligible for WIG The Appendix also presents standard errors for the regression estimates. Because they are much smaller than the standard errors for the sample estimates, the regression estimates are more precise than the sample estimates. Comparing how the sample and regression estimators use data reveals how the regression estimator "borrows strength" to improve precision. When we derived sample estimates in Step 3, we used only data from Delaware to estimate the change in the percentage of infants and children eligible for WIC in Delaware, even though Delaware, like nearly all states, has a small CPS sample. Deriving regression estimates in this step, we estimated a regression line from sample and administrative records data for all the states and used the estimated line (with administrative records data for Delaware) to predict the change in WIC eligibles for Delaware. In other words, the regression estimator not only uses the sample estimates from every state to develop a regression estimate for a single state but also incorporates data from outside the sample, namely, data in administrative records systems. The regression estimator improves precision by using more data to identify states with sample estimates that seem too high or too low because of sampling error, that is, error from drawing a sample that has a higher or lower percentage of eligible infants and children 12 TABLEIU CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE Sample Regression Shrinkage SUte Estimate Estimate Estimate Alabama •1.826 1760 2306 Alaska 0.012 9343 8.238 Arizona 1672 7.439 6336 Arkansas 1.723 4.142 3.977 California 8.813 8.377 8419 Colorado 0.697 0.068 0.133 Connecticut 13366 8.805 9.044 Delaware 3.522 7.174 6.788 District of Columbia 20.946 1L814 11032 Florida 9.436 1L670 10.907 Georgia •0.422 5.290 4.785 Hawaii 9.799 7.058 7325 Idaho 0.014 1256 1020 Dlinois 7.513 3486 4.638 Indiana 11.618 5411 6.221 Iowa 4.235 1070 1634 Kansas 1732 1.051 0.620 Kentucky 5.445 1226 1409 Louisiana 3.248 0395 0.977 Maine 11927 10.428 10310 Maryland 13.682 6.785 8J68 Massachusetts 1533 1.295 1396 Michigan 1.854 4.737 4331 Minnesota 1L874 0.905 1.454 Mississippi 1412 1432 1436 Missouri 21.017 4.710 5.911 Montana 7.065 1260 2366 Nebraska 5.839 1.151 0.191 Nevada 3.782 8.055 7.788 New Hampshire 9.563 6.979 7.436 New Jersey 8.590 5.436 6309 New Mexico ans 6.441 6.123 New York 9.228 5.750 6.917 North Carolina 5.285 4.403 4406 North Dakota 0.423 0.215 •0.184 Ohio 1123 1614 2322 Oklahoma 4.642 5.182 5.145 Oregon 4.839 4.817 4430 Pennsylvania 0.266 5.439 3.904 Rhode Island 8.996 10305 10390 South Carolina 8.283 6347 6.717 South Dakota 6.653 •OtlS 1.148 TCODCMCC 16348 6.438 7473 Texas 5306 6.472 6306 Utah •7.115 0.721 0227 Vermont 1.984 9.782 9.442 Virginia 1266 3.137 3433 Washington 0522 3.625 3441 West Virginia 8.986 4.681 4468 Wkconsin •4234 3313 1413 Wyoming •4338 1397 1.212 United States 5.791 5303 5390 13 than the entire state population has. For example, suppose a state bad experienced stable FSP and UI Program participation and rising per capita income. Our regression estimator would predict a stable or declining percentage of eligible infants and children, implying that a sample estimate showing a large increase in WIC eligibles is too high. The regression estimate will be lower than the sample estimate for such a state. On the other hand, if the sample data for a state show a much smaller increase in eligible infants and children than expected in light of the observed changes in FSP and UI Program participation and per capita income, the regression estimate for that state will be higher than the sample estimate. 5. Using "shrinkage" methods, average the sample estimates of change and the predictions of change. As noted, the limitation of the sample estimator is imprecision. The limitation of the regression estimator is called "bias." Some states really have larger or smaller increases in WIC eligibles than we expect (and predict with the regression estimator) based on changes in FSP and UI Program participation and per capita income. Such errors in regression estimates reflect bias. These limitations arise for the following reasons. The sample estimator uses only sample data for one state to obtain an estimate for that state. It does not use sample data for other states or administrative records data. Although the regression estimator borrows strength, using data from all the states and administrative records data, it makes no further use of the sample data after estimating the regression line. It assumes that the entire difference between the sample and regression estimates is sampling error, that is, error in the sample estimate. No allowance is made for prediction error, that is, error in the regression estimate. Although not all, if any, true state values lie on the regression line, the regression estimator assumes they do. Using all of the information at hand, a shrinkage estimator addresses the limitations of the sample and regression estimators by combining the sample and regression estimates, striking a compromise. As illustrated in Figure II3, a shrinkage estimator takes a weighted average of the 14 FIGURE 11.3 SHRINKAGE ESTIMATION More Precise Sample Estimate, Worse Fitting Regression Line =* More Weight on Sample Estimate Sample Shrinkage Regression Estimate Estimate Estimate 4% 5% 8% Less Precise Sample Estimate, Better Fitting Regression Line ■> Less Weight on Sample Estimate Sample Shrinkage Regression Estimate Estimate Estimate 4% 7% 8% 15 sample and regression estimates. Generally, the more precise the sample estimate for a state, the closer the shrinkage estimate will be to it. Hie larger samples drawn in large states support more precise sample estimates, so shrinkage estimates tend to be closer to the sample estimates for large states. Given the precision of the sample estimate for a state, the weight given to the regression estimate depends on how well the regression line "fits." If the regression estimator cannot find good predictors reflecting why some states have larger increases in WIC eligibles than other states, we say that the regression line "fits poorly." The shrinkage estimate will be farther from the regression estimate and closer to the sample estimate when the regression line fits poorly. In contrast, the shrinkage estimate will be closer to the regression estimate and farther from the sample estimate when the regression line fits well. Striking a compromise between the sample and regression estimators, the shrinkage estimator strikes a compromise between imprecision and bias. The sample and regression estimates are optimally weighted to improve accuracy by minimizing a measure of error that reflects both imprecision and bias. By accepting a little bias, the shrinkage estimator may be substantially more precise than the sample estimator. By sacrificing a little precision, the shrinkage estimator may be substantially less biased than the regression estimator. Table II.2 presents state shrinkage estimates of the change between 1989 and 1992 in the percentage of infants and children who were income eligible for WIC. Table II.2 also displays the sample and regression estimates from Steps 3 and 4. 6. Add the shrinkage estimate of the change between 1989 and 1992 to the census estimate of the percentage eligible in 1989 to get a shrinkage estimate of the percentage eligible in 1992. Table II.3 presents census estimates of the percentage eligible in 1989 from Step 1, shrinkage estimates of the change in the percentage eligible between 1989 and 1992 from Step 5, and shrinkage estimates of the percentage eligible in 1992 from this step. The shrinkage estimate of change added to the census estimate for 1989 gives the shrinkage estimate for 1992. In other words, where a state starts plus how much it changes tells us where the state ends up. For example, 28.543 percent of 16 TABLE U3 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE: CENSUS AND SHRINKAGE ESTIMATES Stale 1989 (Census) Shrinkage Estimate of Change Between 1989 and 1992 Shrinkage Estimate for 1992 Alabama 46302 Alaska 41.367 Arizona 45.474 Arkansas 51206 California 37.760 Colorado 36.057 Connecticut 21.200 Delaware 28343 District of Columbia 46141 Florida 40.021 Georgia 40.615 Hawaii 36421 Idaho 46.808 Elinois 33.183 Indiana 35.470 Iowa 36446 Kansas 36.760 Kentucky 48.123 Louisiana 51651 Maine 34328 Maryland 24146 Massachusetts 25.087 Michigan 37.172 Minnesota 29362 Mississippi 57344 Missouri 38.929 Montana 46.639 Nebraska 38.100 Nevada 34353 New Hampshire 20331 New Jersey 21446 New Mexico 53.995 New York 35.136 North Carolina 39.911 North Dakota 42354 Ohio 37.048 Oklahoma 47438 Oregon 39479 Pennsylvania 33.428 Rhode Island 29423 South Carolina 43447 South Dakota 47114 Tennessee 44.004 Texas 45435 Utah 39.999 Vermont 3L164 Virginia 31369 Washington 34.764 West Virginia 51403 Wisconsin 34.094 Wyoming 41111 United States 37.789 2306 8138 6336 3.977 8419 0.133 9.044 6.788 11032 10.907 4.785 7325 1020 4.638 6121 1634 0420 1409 0.977 10310 8168 1396 4331 1.454 1436 5.911 2366 0.191 7.788 7.436 6309 6.123 6.917 4408 a 184 2322 5.145 4430 3.904 10390 6.717 •1.148 7473 6306 0127 9.442 3433 3441 4468 1413 1112 5390 48408 49405 51410 56.183 46379 36.190 30144 35331 58173 50.928 45.400 44.146 48428 37421 41491 39.480 37380 50332 53428 45.038 32314 26483 41303 30416 59.980 44440 49.005 38191 41141 27.967 28.955 60.118 41053 44319 42370 39370 51783 44.709 37332 40113 50364 46466 51477 51141 40126 40406 34.402 37405 56.471 36307 41423 43379 17 infants and children were income eligible in Delaware in 1989, and that Ggure rose by 6.788 percentage points between 1989 and 1992 according to our shrinkage estimator. Therefore, we estimate that 28.543 + 6.788 = 35.331 percent of infants and children were income eligible in Delaware in 1992. 7. Multiply the shrinkage estimate of the percentage eligible by the state population or infants and the state population of children to get preliminary shrinkage estimates of the numbers of eligible infants awl children. To obtain separate estimates for infants and children, we have assumed that the percentage of infants who were income eligible in a state is the same as the percentage of children who were income eligible. Our estimate of that percentage was obtained in Step 6. To obtain estimated numbers from estimated percentages, we require state population estimates for both infants and children. The population estimates we used pertain to the resident population on July 1,1992 and were developed by the U.S. Bureau of the Census from census and administrative records (mainly vital statistics) data. These estimates are often called "independent" estimates because they are not based on CPS or other sample data. In broad terms, they were derived by subtracting from census counts persons "exiting" the population between April 1, 1990 and July 1, 1992 (due to death or net outmigration) and adding persons "entering" the population (due to birth or net inmigration). Because infants in the Jury 1, 1992 population had not yet been born on April 1, 1990, census data have no bearing on the population estimates for infants. Those estimates are based entirely on vital statistics data and other administrative records data needed to account for migration. Likewise, census data are irrelevant to the population estimates for children age 1 and some children age 2. (The population estimates for children ages 1 through 4 were obtained by summing estimates for each year in that range.) Table II.4 displays preliminary shrinkage estimates of the number of infants and the number of children who were income eligible for WIC in 1992. It also shows shrinkage estimates of the percentages eligible from Step 6 and state population estimates for infants and children developed 18 TABLE D.4 PRELIMINARY SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 State Shrinkage Estimate of Percentage Eligible Population Preliminary Shrinkage Estimate of Number Eligible Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode bland South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming United States 48.808 49.605 51.810 56.183 46379 36.190 30244 35331 58.273 50.928 45.400 44.146 48.828 37.821 41.691 39.480 37380 50332 53.628 45X88 32314 26.683 41303 30.816 59.980 44.840 49.005 38.291 42.141 27.967 28.955 60.118 42.053 44319 42370 39370 52.783 44.709 37332 40.213 50364 46.066 5L877 52.141 40226 40.606 34.402 37.805 56.471 36307 42423 43379 Infants Children 61,680 11313 65,911 34^57 592344 53389 46380 10,769 8321 190419 109,227 19.608 17.069 168.287 82321 37,743 36,797 52301 70356 15395 75332 86,239 138.700 64,757 42,485 74,623 11367 22333 21,921 15,487 117.976 27316 281322 101,190 8,722 164,409 47301 41,279 162326 14379 55,711 11,029 72329 317,748 36313 7332 95368 78349 21356 69318 6,718 4300022 236.274 45,998 254,228 135,906 2,169,211 209,645 189,045 41,456 31388 751382 422,780 71,900 67348 713,032 323.759 155,417 150458 206.213 270028 66,707 302339 340993 575390 269,688 160775 301,457 46,728 95334 84381 66056 455,968 105,955 1379.972 393,791 36,102 634.255 185,746 169,723 649372 56348 217352 43357 281,056 1,183390 141.484 32366 371346 312385 86392 289,261 27326 Infants Children 30,105 5,711 34.148 19359 274,677 19394 14388 3305 4349 96377 49389 8356 8334 71,212 34320 14.901 13,755 26.732 37,731 7324 24356 233H 57365 19,956 25.483 33,461 5370 8,743 9^38 4331 34,160 16.722 118388 45349 3396 65357 24367 18355 60600 5363 28,170 5381 37333 165377 14388 3358 32377 29320 12342 25306 2350 15312,163 1,737337 115321 22317 13L716 76356 1306358 75371 57,175 14347 18.466 382317 191.942 31,741 32,738 269,676 134,978 61359 56241 104J04 144311 30343 98.465 90,987 238387 83,107 96.433 135,173 22399 36396 35343 18330 132,026 63398 454,161 175312 15396 250975 98342 75381 242,498 22359 109,902 20203 145303 617392 56313 13,143 127354 118.173 48,786 105301 11,720 6.72L734 19 by the Census Bureau. According to Table II.4, there were 10,769 infants and 41,456 children living in Delaware in 1992. Our shrinkage estimate is that 35.331 percent of those infants and children were income eligible. Therefore, our preliminary shrinkage estimates of the numbers eligible are (35.331 + 100) x 10,769 = 3,805 infants and (35.331 + 100) x 41,456 = 14,647 children. 8. Control the preliminary state shrinkage estimates of the numbers of eligible infants and children to sum to the national totals for eligible infants and children obtained from the CPS. The preliminary state shrinkage estimates derived in Step 7 sum to 1,737,837 eligible infants and 6,721,734 eligible children nationwide. According to the March 1993 CPS, there were 1,717,743 eligible infants and 6,925,815 eligible children in the entire U.S. The most recent national sample estimates are typically used to develop the budget for the WIC Program. To obtain final shrinkage estimates for states that sum (aside from rounding error) to the national totals from the most recent CPS (March 1993), we multiply each of the preliminary state shrinkage estimates for infants by 1,717,743 + 1,737,837 ( m 0.9884) and each of the preliminary state shrinkage estimates for children by 6,925,815 + 6,721,734 (  1.0304). This ensures that the estimates used to allocate funds are consistent with the estimates generally used to determine total program funding. The final shrinkage estimates are presented in the next chapter. 20 m. STATE ESTIMATES OF WIC EUGIBLES FOR 1992 Table m.l presents our final state shrinkage estimates of the number of infants and the number of children who were income eligible for WIC in 1992. The strength of these estimates is that they are timely relative to census estimates and precise relative to CPS estimates. As documented in the appendix, the shrinkage estimates have much smaller standard errors and narrower confidence intervals than the CPS sample estimates. Table m.2 displays approximate 90percent confidence intervals showing the uncertainty remaining after using shrinkage estimation. One interpretation of a 90percent confidence interval is that there is a 90 percent chance that the true valuethat is, the true number of eligibleslies in the estimated interval. A wide interval means that we are very uncertain about the true value. According to our calculations, a shrinkage confidence interval is, on average, only about 39 percent as wide as the corresponding sample confidence interval. Thus, shrinkage substantially reduces our uncertainty. The Food and Consumer Service (FCS) of the U.S. Department of Agriculture used the final shrinkage estimates of infants and children incomeeligible for WIC in 1992 to determine state WIC food grants for fiscal year 1995. From the final shrinkage estimates in Table III.l, FCS calculated each state's "fair share" of total fiscal year 1995 WIC food funds. A state's fair share is its percentage share of the national number of eligible infants and children. Thus, for example, Delawarewhich has about 0.2 percent ((3,761 + 15,092) + (1,717,746 + 6,925,819)) of all eligible infants and childrenhas a fair share of about 0.2 percent of total WIC food funds. According to the WIC food funding formula (7 C.F.R. §246.16), a state's WIC food grant is determined by comparing the fair share amount to the prior year food grant. If the prior year grant equals or exceeds the fair share amount, the state is entitled to receive only the prior year amount, adjusted for inflation (if total food funds are adequate to provide inflation increases to all states). If the prior year grant is below the fair share amount, the state is entitled to received an inflation 21 TABLE III1 FINAL SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 State Infants Children Alabama Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida 29.757 5,645 33.754 19.135 271.501 19.170 13.925 3,761 4,793 95355 118,822 23,510 135,715 78374 1.036304 78,174 58.911 15392 19.026 394.439 Georgia Hawaii Idaho Illinois Indiana Iowa Kansas Kentucky Louisiana Maine 49.016 8356 8038 70389 33.924 14.729 13396 26.423 37^94 6.942 197,770 32.705 33.732 277,864 139,077 63022 57.949 107367 149.207 30.956 Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire 24371 22,745 56399 19,725 25.188 33.074 5306 8342 9.131 4381 101.455 93.750 246,140 85330 99361 139,277 23394 37310 36,725 19,092 New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah Vermont Virginia Washington West Virginia Wisconsin Wyoming United States 33,765 16329 117.020 44328 3353 64304 24378 18342 59399 5.795 27344 5.022 37396 163,761 14318 3323 32,497 29377 12300 23313 2317 1.717,746 136,034 65332 467,950 180,635 15.761 258395 101.019 78.185 249361 23347 113339 20317 150330 636,034 58341 13342 131,735 12L761 50368 108307 12376 6325319 22 TABLE HU APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR SHRINKAGE ESTIMATES Number of Eligible Infants Number of Eligible Children State Lower Bound Upper Bound Lower Bound Upper Bound Alabama 27,266 32348 108374 128.770 Alaska 5,135 6,155 21386 25334 Arizona 30335 36373 123,977 147,453 Arkansas 17345 20,425 73371 83377 California 257,990 285,012 985,020 1388.188 Colorado 17,073 21367 69321 86,727 Connecticut 12373 15,777 51377 66.745 Delaware 3357 4,165 13,470 16,714 District of Columbia 4337 5349 17315 20337 Florida 89,771 101,939 369,404 419.474 Georgia 44322 53,410 180,042 215.498 Hawaii 7398 9314 29342 36368 Idaho 7314 8362 31,176 36388 Dlinois 64,462 76316 254,466 301362 Indiana 31337 36311 127340 150.914 Iowa 13377 15381 58378 68.166 Kansas 12314 14,978 52358 63340 Kentucky 24375 28,471 99345 115389 Louisiana 34,423 40.165 137,719 160.695 Maine 6305 7379 28,116 33,796 Maryland 21,706 27336 90363 112347 Massachusetts 18391 26,799 77341 110.459 Michigan 52316 61,782 225,018 267362 Minnesota 17,135 22315 74385 96375 Mississippi 23310 26.766 93,137 105385 Missouri 30,141 36307 126,927 151327 Montana 5365 5347 2L703 25.485 Nebraska 7,765 9319 33375 41345 Nevada 8381 9381 33307 40,143 New Hampshire 3.788 4,774 16392 21392 New Jersey 30361 36369 123.128 148,940 New Mexico 15314 17344 60,411 70353 New York 109312 124,728 437,125 498,775 North Carolina 41,469 47387 168327 193343 North Dakota 3363 4343 14378 17344 Ohio 59338 69370 239,427 277,763 Oklahoma 23316 26340 94316 107322 Oregon 16,713 19,771 71332 84,738 Pennsylvania 54319 65,479 226386 273.136 Rhode Island 5310 6380 20389 25,705 South Carolina 25348 29340 105,123 121355 South Dakota 4348 5396 18352 22,782 Tennessee 34,165 40327 137349 163311 Texas 151,491 176331 588377 683391 Utah 13363 15373 52,763 64319 Vermont 2,711 3335 12,146 14338 Vagina 28350 36444 116351 146319 Washington 26314 32340 110369 133353 West Virginia 11383 13,117 46392 34344 Wisconsin 22368 27358 97300 120314 Wyoming 2,497 3,137 10,708 13346 23 increase plus additional funds for program growth (if program growth funds are available after providing all states with inflation increases). In the initial fiscal year 1995 fund allocation, 19 states were below fair share and received program growth funds. Eight Indian Tribal Organizations (ITOs), which are authorized to participate in the WIC Program as state agencies, were also identified as below fair share. The eligibles estimates used to determine WIC food grants for ITOs were derived from 1990 decennial census data and March 1993 CPS data, but were not developed using the shrinkage estimation procedure described in Chapter II. Using the shrinkage estimator described in Chapter II, we are able to substantially reduce our uncertainty about the numbers of infants and children who were eligible for WIC. In the future, there may be an opportunity to reduce uncertainty even further by enhancing our shrinkage estimator to use still more data. The estimator now uses census estimates for the "base" year (1989) and CPS estimates for the "current" year (1992 in this reportthe year for which we are developing shrinkage estimates). Estimates for intervening years are not used, although CPS data for obtaining such estimates are available. With each intervening year, we are ignoring more information that could be relevant. An unusually large increase in WIC eligibles over three years, for example, would be more plausible if it appeared to consist of a series of modest increases rather than two small decreases followed by one enormous jump. An advantage of shrinkage methods is that they are powerful enough to allow such information to be taken into account in a systematic, rather than an ad hoc, way. Although the estimation procedure would be more complicated, an enhanced shrinkage estimator would be conceptually the same as the current estimator and might yield even better state estimates of WIC eligibles. Accuracy might also be improved by using data that incorporate an adjustment for the census undercount. Before CPS data are released, they are made consistent with Census Bureau population estimates. When 1992 eligibles estimates were needed for calculating fiscal year 1995 WIC grants, the available CPS data were consistent with population estimates based on unadjusted decennial 24 census data (as well as vital statistics and other administrative records data). CPS data released subsequently are consistent with adjusted population estimates. Therefore, it is expected that future estimates of WIC eligibles will reflect an adjustment for the census undercount. 25 # \m mm REFERENCES DuMouchel, William H., and Jeffrey E Harris. "Bayes Methods for Combining the Results of Cancer Studies in Humans and Other Species." Journal of the American Statistical Association, vol. 78, no. 382, June 1983, pp. 293315. Ericksen, Eugene P., and Joseph B. Kadane. "Estimating the Population in a Census Year: 1980 and Beyond." Journal of the American Statistical Association, vol. 80, no. 389, March 1985, pp. 98 131. Fay, Robert E, and Roger Herriott. "Estimates of Incomes for SmallPlaces: An Application of JamesStein Procedures to Census Data." Journal ofthe American StatisticalAssociation, vol. 74, no. 366, June 1979, pp. 269277. Fay, Robert E, Charles T. Nelson, and Leon Litow. "Estimation of Median Income for 4Person Families by State." In Indirect Estimators in Federal Programs. Statistical Policy Working Paper no. 21. Washington, DC: Office of Management and Budget, July 1993. Fisher, Gordon M. "Poverty Guidelines for 1992." Social Security Bulletin, vol. 55, no. 1, Spring 1992, pp. 4346. Ghosh, M. and J.N.K. Rao. "Small Area Estimation: An Appraisal" (with comments). Statistical Science, vol. 9, no. 1, February 1994, pp. 5593. Hogan, Howard. The 1990 PostEnumeration Survey: Operations and Results." Journal of the American Statistical Association, vol. 88, no. 423, September 1993, pp. 10471060. Rao, J.N.K., C.FJ. Wu, and K. Yue. "Some Recent Work on Resampling Methods for Complex Surveys." Survey Methodology, vol. 18, no. 2, December 1992, pp. 209217. Schirm, Allen L. "The Relative Accuracy of Direct and Indirect Estimators of State Poverty Rates." 1994 Proceedings of the Section on Survey Research Methods. Alexandria, VA American Statistical Association, 1994. Schirm, Allen L., Gary D. Swearingen, and Cara S. Hendricks. "Development and Evaluation of Alternative State Estimates of Poverty, Food Stamp Program Eligibility, and Food Stamp Program Participation." Washington, DC: Mathematica Policy Research, December 1992. Sigma One Corporation. Estimates of Persons Income Eligible for the Special Supplemental Food Program for Women, Infants, and Children (WIC) in 1989: National and State Tables. Alexandria, VA: U.S. Department of Agriculture, Food and Nutrition Service, Office of Analysis and Evaluation, August 1993. U.S. Department of Commerce, Bureau of the Census. Statistical Abstract of the United States. Washington, DC: U.S. Government Printing Office, 1993a. U.S. Department of Commerce, Bureau of Economic Analysis. Survey of Current Business, vol. 73, no. 9. Washington, DC: U.S. Government Printing Office, September 1993b. 27 U.S. Department of Commerce, Bureau of the Census. Statistical Abstract of the United States. Washington, DC: U.S. Government Printing Office, 1991a. U.S. Department of Commerce, Bureau of Economic Analysis. Survey of Current Business, vol. 71, no. 8. Washington, DC: U.S. Government Printing Office, August 1991b. APPENDIX THE ESTIMATION PROCEDURE: ADDITIONAL TECHNICAL DETAILS P 50 A This appendix provides additional information and technical details for several of the steps in our estimation procedure. For Step 2, we discuss how we calculated sample estimates and their standard errors. For Step 4, we provide complete definitions and data for calculating values for the three predictor variables in our regression model. We also list the other variables that we considered as potential predictors. For Step 5, we present the equations used to calculate shrinkage estimates and their standard errors. We also discuss at the end of this Appendix how we derived confidence intervals. For some steps, we provide, as needed, few or no additional details. 1. From the most recent census (1990), derive state estimates of the percentage of infants and children who were income eligible. 2. From the most recent CPS (March 1993), derive state sample estimates of the percentage of infants and children who were income eligible. Table Al displays sample estimates and estimated standard errors. We obtained CPS sample eligibility estimates with the same methodology used by the Census Bureau to calculate poverty estimates for individuals except (1) we compared a family's income to 185 percent, rather than 100 percent, of the applicable poverty guideline; (2) we used the poverty guidelines shown in Table A.2. rather than the poverty thresholds developed by the Census Bureau for official government statistical (as opposed to administrative) purposes; and (3) we counted secondary individuals under age IS (if they fell in the age ranges for infants and children) as poor/eligible, rather than excluding them.1 An infant or child is income eligible for WIC if his or her family's income is less than or equal to 185 percent of the poverty guideline for that family. The WIC poverty guidelines for 1992 in Table A2 were obtained by averaging "HHS" poverty guidelines for 1991 and 1992. We averaged poverty guidelines for consecutive calendar years because the WIC program year runs from Jury 1 of one calendar year to June 30 of the following calendar 'Previous research suggests that most of these young secondary individuals are foster children. F or determining WIC eligibility, a foster child who is the legal responsibility of a court or state welfare agency is a family of one individual. Although the CPS does not collect income data for a secondary individual under age 15, it is likely that such a person has little, if any, income. 31 TABLE A.1 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE 1992 Change Between 1909 (CPS) 1989 and 1992 Estimate Standard Standard State (Census) Estimate Error Estimate Error AUbama 46302 44.476 8.422 •1426 8.422 Alaska 4L367 41455 5430 •0.012 5430 Arizona 45.474 42402 5473 1672 5473 Arkansas 51206 53.929 7.405 1.723 7.405 California 37.760 46473 1487 8413 1487 Colorado 36.057 36.754 6431 0497 6431 Connecticut 21.200 34.766 9440 13466 9440 Delaware 28.543 31065 5414 3422 5414 District of Columbia 46.241 67.187 13.154 20446 13.154 Florida 40.021 49.457 1446 9.436 1446 Georgia 40615 40.193 6.443 •0422 6.443 Hawaii 36.821 46420 6.187 9.799 6.187 Idaho 46.808 46422 5448 0414 5448 Dlinoii 33.183 40496 3.736 7413 3.736 Indiana 35.470 47.068 5.924 11418 5.924 Iowa 36446 41.081 3407 4435 3407 Kansas 36.760 34428 5489 •1732 5489. Kentucky 48.123 53468 8.401 5.445 8.401 Louisiana 51651 55499 11.137 3448 11.137 Maine 34.528 47.455 11415 11927 11415 Maryland 24.246 37.928 3410 13482 3410 Massachusetts 25.087 27420 3.700 1533 3.700 Michigan 37.172 39426 4413 1454 4413 Minnesota 29.362 41.236 9409 11474 9409 Mississippi 57.544 59.956 8470 1412 8470 Missouri 38.929 59.946 7.180 21417 7.180 Montana 46439 53.704 14.153 7465 14.153 Nebraska 38.100 31261 4424 •5439 4424 Nevada 34.353 38.135 7.787 3.782 7.787 New Hampshire 20.531 30.094 4.184 9463 4.184 New Jersey 21446 31436 1461 8490 1461 New Mexico 53.995 54.170 8.942 0175 8.942 New York 35.136 44464 1461 9428 1461 Norm Carolina 39.911 45.196 3412 5485 3.412 Worth Dakota 41554 41977 9417 0423 9417 Ohio 37.048 39.171 3.114 1123 3.114 Oklahoma 47438 51280 7444 4442 7444 Oregon 39479 44.718 7.123 4439 7.123 33.428 33.162 1952 0.266 1952 Rhode Island 29423 38419 6.903 8496 6403 South Carolina 43447 51130 5499 8483 5499 South Dakota 47.214 40461 8.148 6453 8.148 Tennessee 44404 60452 4492 16448 4492 Taxat 4S435 51.141 4481 5406 4481 Utah 39.999 31884 7.756 7.115 7.756 Vermont 31.164 29.180 11009 1.984 11009 Virginia 31469 33435 4442 1266 4442 Washington 34.764 34.242 4.757 •0522 4.757 Watt Virginia 51403 60489 9473 8486 9473 Wnconsin 34494 29470 3430 4434 5430 Wyoming 41.211 36473 11.290 •4438 11490 United States 37.789 43480 0.795 5.791 0795 32 TABLEAU WIC POVERTY GUIDELINES FOR 1992 (Dollars) HHS Poverty Guidelines WIC Poverty Guidelines State and Family Size 1991 1992 1992 Alaska Oneperson family 8^90 8,500 8395 Each extra person 2,820 2,980 2,900 Hawaii Oneperson family 7,610 7,830 7,720 Each extra person 2,600 2,740 2,670 Other States and DC Oneperson family 6,620 6,810 6,715 Each extra person 2,260 2380 2,320 NOTE: The WIC Dovertv guidelines are simnle arithm ctic averasei of the*IHS novertv guidelines 33 year. Therefore, eligibility workers determined a family's eligibility for WIC using the 1991 HHS poverty guidelines during the first six months of 1992 and the 1992 HHS poverty guidelines during the last six months of 1992. The Office of the Secretary, Department of Health and Human Services, is responsible for developing the HHS poverty guidelines. The HHS poverty guidelines are derived from the Census Bureau poverty thresholds (Fisher 1992). We estimated standard errors for our sample estimates using the jackknife estimator proposed by Rao, Wu, and Yue (1992), treating CPS rotation groups as clusters. A rotation group, about oneeighth of a monthly CPS sample, consists of a group of households that begin the CPS at the same time. They are in the CPS for four months, rotate out for eight months, and rotate back in for four months, after which they are dropped from the CPS. To obtain jackknife standard errors, we let Z, equal the CPS sample estimate of the number of eligible infants and children in state i (i = 1,2, .... 51) and Z^ equal the contribution of rotation group r (r = 1,2,..., 8) to that estimate. In other words: 0) Zi  E zi, ■ r  1 If we were to exclude the observations in rotation group r, we could estimate the number of poor persons in state / by: (2) Zl(f)  • (Z,  Zf>) . The "(/■)" subscript indicates that rotation group r has been excluded. The factor 8/7 enters the expression because when (approximately) 1/8 of the sample is removed, an estimate from the remaining 7/8 of the sample needs to be inflated to get an estimate for the whole. By excluding each of the eight rotation groups in turn, we can get eight alternative estimates for the number of poor 34 persons in state i. Then, we can assess the degree of sampling variability (estimate the variance of Z,) by measuring the variability among the eight estimates according to: (3) var(Z,)  1 £ (Zi(r)  Zf . 8,^ I The factor 7/8 enters this expression because the Zi(r) are obtained from samples that are only 7/8 the size of the full CPS sample for state i and, hence, are expected to be more variable than Z, (by a factor of 8/7). If Y, equals the CPS sample estimate of the percentage of infants and children eligible in state /: (4) y.  100 % , where N, is the CPS sample estimate of the population of infants and children in state i. We estimate the variance of Yt by. , var(Z) (5) var(y,)  lOO2 _1£ where var(Z,) is calculated according to Equation (3). Our jackknife estimate of the standard error of yj is obtained by taking the square root of var(yi). Estimated jackknife standard errors for the CPS sample estimates for 1992 are presented in Table A.1. 3. CoBStroct sample estimates of the esuutge ia the pcrceatafe eligible betwcea 1989 aad 1992. A state's sample estimate of the change between 1989 and 1992 in the percentage of infants and children who were income eligible was obtained by subtracting the census estimate for 1989 from the CPS estimate for 1992. Sample estimates of change and their standard errors are presented in Table A. 1. We assumed that the sampling error associated with a census estimate is negligible. Therefore, the standard errors for the estimates of change in the percentage eligible equal the standard errors for the 1992 estimates of the percentage eligible. 4. Using a regression model, predict the change in the percentage eligible for each state based on observed changes in (i) Food Stamp Program (FSP) participation, (ii) Unemployment Insurance (UI) Program participation, and (ill) per capita income. Our "best'' regression model has three predictors that measure the changes between 1989 and 1992 in: • FSP participation • UI Program participation • Per capita income These three predictors were selected from a list that included variables measuring the changes in: • National School Lunch Program (NSLP) participation (number of students approved for free or reducedprice meals relative to the size of the schoolage populationages 5 through 17) • Supplemental Security Income (SSI) Program participation (number of recipients relative to the size of the population) • Aid to Families with Dependent Children (AFDC) Program participation (number of recipients relative to the size of the population) • Head Start Program participation (enrollment relative to the size of the preschoolage populationages 0 through 4) • Chapter 1 (Compensatory Education) Program funding (basic grant, in dollars, relative to the size of the schoolage population) • Per capita residential construction (in dollars) • Per capita nonresidential construction (in dollars) • Crime rate • Population density 36 We considered these variables because (1) we believed that they might indicate differences among states in the incidence of poverty (especially child poverty), socioeconomic conditions related to poverty, or the health of the state economy and (2) they could be measured uniformly across states for 1992 from nonsample or highly precise sample data. Variables measuring vital events (e.g., infant deaths), WIC participation, and Medicaid participation were rejected as potential predictors because they are often used as outcome measures in analyses of the effectiveness of the WIC Program.2 We selected our best regression model on the basis of its consistently strong relative performance in predicting changes in WIC eligibles for three time periods: 1989 to 1990, 1989 to 1991, and 1989 to 1992. We judged performance by examining numerous functions of the regression residuals, including R2 as well as measures that adjust for the loss in degrees of freedom from adding predictor variables.3 Definitions and data sources for the three predictor variables in our best regression model are given in Table A.3. Tables A.4 and A.5 provide the raw data for 1989 and 1992, respectively, used to calculate the predictor variables, and Table A.6 displays the calculated predictor variables for each state. Following the estimation procedure described in Step 5, we obtained the estimated regression equation shown below: Change in percentage eligible =  1.899 + 1.617 x Change in FSP participation + 4.644 x Change in UI Program participation  6.498 x Change in per capita income ^Estimating the numbers of WIC eligibles and, implicitly, WIC participation rates using the infant mortality rate (IMR), for example, as a predictor would have "built in" a relationship between WIC participation and the IMR, therefore biasing analyses of the effectiveness of WIC in reducing infant deaths. 3The residual for a state is the difference between the sample estimate and the regression prediction. Our best model tended to produce smaller residuals than did alternative models. 37 BLANK PAGE TABLE A.3 DEFINITIONS AND DATA SOURCES FOR PREDICTOR VARIABLES 00 Predictor Variable Definition: Change between 1989 and 1992 in Principal Data Sources' FSP participation* trw v Number of participants during August Resident population FSP participation data are population counts of participants from state program operations data and were obtained electronically from the Food and Consumer Service, U.S. Department of Agriculture. UI Program participation0 im v Number of first payment beneficiaries during year Resident population UI data for 1992 were obtained electronically from the Unemployment Insurance Service, U.S. Department of Labor. Data for 1989 are from Table 603, "State Unemployment Insurance, by State and Other Areas: 1989,' in U.S. Department of Commerce (1991a, p. 367). Per capita income4 (Total personal income + Resident population) WIC poverty guideline for oneperson family Total personal income data are from Table 1, Total and Per Capita Personal Income by State and Region, 198590," in U.S. Department of Commerce (1991b, p. 30) and Table 1, "Total and Per Capita Personal Income by State and Region, 198792," in US Department of Commerce (1993b, p. 74). ■Data on the resident population as of Jury 1 are from Table 26, "Resident PopulationStates and Puerto Rico: 1960 to 1990," in U.S. Department of Commerce (1991a, pp. 2021) and Table 31, "Resident PopulationStates: 1970 to 1992," in U.S. Department of Commerce (1993a, pp. 2829). *Data for August are often used to measure FSP participation. See, for example, Schirm, Swearingen, and Hendricks (1992). *A first payment beneficiary is a person receiving a UI payment for the first time in more than a year. *We measure per capita income relative to the WIC poverty guideline for a oneperson family to account for inflation. Poverty guidelines are adjusted annually based on the Consumer Price Index (CPI). The 1992 WIC poverty guidelines are displayed in Table A.2. The 1989 guidelines for a oneperson family are $7345, $6760, and $5875 for Alaska, Hawaii, and the rest of the U.S., respectively. >/ TABLE A.4 1989 DATA FOR CALCULATING PREDICTOR VARIABLES UI First Total Personal Resident Population FSP Recipients Payment Income on Jury 1 State in August Beneficiaries ($1*00.000) (1.000) Alabama 428*80 152,000 56*98 4.118 Alaska 23.766 33*00 11*76 527 Arizona 276*62 74*00 55352 3356 Arkansas 226,262 83*00 31*90 2,406 California 1327,414 1*24,000 576,489 29*63 Colorado 206384 74*00 58*15 3317 Connecticut 117396 119,000 80*09 3*39 Delaware 30286 22*00 12*93 673 District of Columbia 58.903 19*00 13*00 604 Florida 691.285 187,000 225*61 12371 Georgia 486.762 210*00 104,107 6*36 Hawaii 79.135 19*00 20,417 1,112 Idaho 57378 37,000 14.153 1*14 Dlinois 973*76 303,000 220*89 11358 Indiana 282.643 116.000 88308 5393 Iowa 163*10 73*00 44356 2*40 Kansas 132.794 69*00 41,916 2313 Kentucky 446.171 112,000 51396 3,727 Louisiana 725332 99*00 56320 4*82 Maine 84.185 44*00 20*81 1*22 Maryland 248.688 89*00 98*31 4*94 Massachusetts 319341 261,000 131.403 5*13 Michigan 875,425 393,000 163*69 9*73 Minnesota 248354 123,000 77334 4*53 Mississippi 483.489 72*00 31*89 2*21 Missouri 402*92 161.000 85,163 5,159 Montana 52313 22*00 11*48 806 Nebraska 91*87 27*00 25.772 1*11 Nevada 44*04 36*00 20.919 1.111 New Hampshire 23*22 32,000 22*46 1.107 New Jersey 357,935 268,000 182*82 7,736 New Mexico 150*28 28.000 20*40 1328 New York 1,409,738 544,000 374*92 17*50 North Carolina 381*99 211.000 101.440 6371 North Dakota 37*36 15*00 9*47 660 Ohio 1,072380 305.000 180,197 10*07 Oklahoma 253,921 50*00 43*91 3*24 Oregon 208*95 106,000 45.409 2320 Pennsylvania 901.156 406,000 209*00 12*40 Rhode Island 57*80 46*00 18*92 998 South Carolina 249*51 97*00 48*44 3*12 South Dakota 48*00 8*00 10*22 715 Tennessee 500,159 164,000 72*12 4*40 Tax* 1*81*21 340,000 263*58 16*91 Utah 93.793 31*00 22*87 L707 Vermont 34*92 19*00 9*34 367 Virginia 325,167 131*00 113*46 6*98 Washington 319*47 169*00 84*08 4,761 West Virginia 257,470 33*00 23*41 1*37 WsKonsin 280*11 172*00 80*79 4*67 Wyoming 26*19 10*00 6*44 475 TABLE AJ 1992 DATA FOR CALCULATING PREDICTOR VARIABLES UI First Total Personal Resident Population FSP Recipients Payment Income on July 1 State in August Beneficiaries ($1,000,000) (1.000) Alabama 555.232 157,084 68321 4.136 Alaska 40,477 44394 13.157 587 Arizona 475,882 90.486 66386 3332 Aifcaiuat 278,876 99322 37317 2399 California 2358340 1,443,782 662,786 30367 Colorado 264,118 79360 71354 3.470 Connecticut 207380 157319 89336 3381 Delaware 54360 28,787 15301 689 District of Columbia 86,135 26331 15390 589 Florida 1.398.057 339388 262,929 13,488 Georgia 777,194 231.957 124303 6.751 Hawaii 95.484 39381 25355 1.160 Idaho 71321 46.156 17334 1367 Dlinois 1.158311 390,904 255351 11331 Indiana 464394 149343 104304 5362 Iowa 191,727 88304 52,103 2312 Kansas 179,183 70323 48307 2323 Kentucky 527,608 127,034 63361 3,755 Louisiana 773335 109,968 68355 4387 Maine 133330 58340 22360 1335 Maryland 355,947 144326 114,115 4,908 Massachusetts 430,034 249341 142328 5.998 Michigan 1,002,451 487346 185,713 9,437 Minnesota 317332 133306 91312 4,480 Mississippi 540,061 79,145 36336 2314 Missouri 558.861 184.467 98363 5.193 Montana 66.965 25,147 13397 824 Nebraska 109353 33,436 30,438 1306 Nevada 83.417 60368 28354 1327 New Hampshire 57302 39315 25,100 1.111 New Jersey 510,070 339337 210,059 7,789 New Mexico 233334 31,702 24309 1381 New York 1,921386 673398 432,001 18,119 North Carolina 608,734 243,700 123.074 6343 North Dakota 47324 14336 10334 636 Ohio 1347.751 357397 207,769 11316 Oklahoma 352,129 65369 52347 3312 Oregon 258,457 141,756 54340 2377 Pennsylvania 1,157341 517,8i0 244314 12309 Rhode Island 88.795 60,746 19396 1305 South Carolina 380309 125.030 58362 3303 South Dakota 55300 8368 12,147 711 Tennessee 725.074 189367 88384 5324 Taut 2305.165 429,726 323387 17356 Utah 122358 37385 28328 1313 Vermont 53326 26377 10,732 570 Virginia 518397 137398 135.003 6377 Washington 439,451 219317 108301 5,136 West Virginia 310,970 60322 27.784 1312 Wisconsin 339.986 215369 95336 5307 Wyoming 33317 12322 8345 466 40 TABLEAU VALUES FOR PREDICTOR VARIABLES IN REGRESSION MODEL Change Between 1989 and 1992 in FSP UI Per Capita State Participation Participation Income Alabama 3.024 0.107 0.112 Alack* 2386 1350 •0.243 Arizona 4.633 0380 0.084 Arkansas 2.221 0.715 0.134 California 2325 1.154 •0.178 Colorado 1389 0.056 0.083 Connecticut 2.681 1.121 ■0.179 Delaware 3390 0.909 0.097 District of Columbia 4.872 1.409 0.109 Florida 4.909 1.039 •a 124 Georgia 3.949 0.173 0.000 Hawaii 1.115 1.686 0.104 Idaho 0.967 0.677 0.085 Dliuois 1.610 0.762 0.055 Indiana 3.151 0373 0.038 Iowa 1.071 0381 0.071 Kansas L818 0.061 0.042 Kentucky 2.080 0378 0.153 Louisiana 1.480 0306 0.157 Maine 3.923 L147 a 101 Maryland 1.954 1.051 •0.099 Massachusetts 1.761 0.257 0.237 Michigan 1.182 0.925 0.066 Minnesota 1364 0.154 0.018 Mississippi 1213 0381 0.085 Missouri 2.962 0.431 0.028 Montana 1375 0322 0.018 Nebraska 1.118 0.406 0.099 Nevada 1280 1302 •0.034 New Hampshire 3.060 0.702 0.103 New Jersey 1.922 0399 •0.008 New Mexico 4.914 0.173 0.063 New York 2.752 0.686 0.002 North Carolina 3.093 0350 0.050 North Dakota 1.738 0.075 0327 Ohio 1.492 0.448 0.003 Oklahoma 3.087 0303 0.038 Oregon 1181 1.003 0X02 Pennsylvania 1152 0.940 0.078 Rhode Island 3.055 1.435 ■a123 South Carolina 3.467 0.708 O069 South Dakota 0.939 a128 0.158 Tennessee 4307 0.455 0.114 Texas 4295 0.433 0.090 Utah. 1387 0363 0.105 Vermont 3395 1394 0.028 Virginia 1796 0.010 0.072 Washington 1344 0.718 0.122 West Virginia 3397 0308 0.171 Wisconsin 1.026 0.773 0021 Wyoming 1.672 0318 0379 41 As expected, the signs of the regression coefficients imply that, all else equal, states with (1) larger increases in FSP participation, (2) larger increases in UI Program participation, or (3) larger decreases in per capita income tend to have larger increases in the percentage of infants and children eligible for WIG4 Table A. 7 presents regression estimates and their standard errors for each state.' 5. Using "shrinkage" methods, average the sample estimates of change and the predictions of change. We have used a shrinkage estimator based on the Empirical Bayes estimator proposed by DuMouchel and Harris (1983). Their estimator was used by Ericksen and Kadane (1985) to estimate population undercounts in the 1980 census for 66 areas covering the entire U.S. and by Schirm, Swearingen, and Hendricks (1992) to estimate state poverty rates and FSP participation rates. The Empirical Bayes shrinkage estimator proposed by DuMouchel and Harris (1983) is: (6) Y,c.EB 1 D ♦ 1M u2 DYS, where K,ff is a (51 x 1) vector of Empirical Bayes shrinkage estimates, and 7, is a (51 x 1) vector of direct sample estimates. D is a (51 x 51) diagonal matrix with diagonal element (i,i) equal to one divided by the variance (standard error squared) of the direct sample estimate for state i.* M = I  X(X'X)~XX', where / is a (51 x 51) identity matrix and A' is a (51 x K) matrix containing data for 4This equsjon does not express a causal relationship. It does not imply that more FSP participants cause more WIC eligibles. Rather the equation implies only a statistical association: states with more FSP participants typically have more WIC eligibles than states with fewer FSP participants. For this reason, predictors are often called "symptomatic indicators." They are symptomatic of differences among states in conditions associated with having more or fewer WIC eligibles. 5As shown in the next step, we do not have to calculate regression estimates as a separate step, although we do have to select a best regression model before we can calculate shrinkage estimates. 6The fourth column of numbers in Table A1 is Yr while D can be obtained from the last column in that table. 42 TABLE A.7 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE REGRESSION ESTIMATES Stale Estimate Standard Error Alabama Alaaka Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida 2.760 9.343 7.439 4.142 8377 0.068 8.805 7.174 1L814 11470 2427 1868 1888 2.478 1618 2449 2389 1573 3.450 1898 Georgia Hawaii Idaho Elinois Indiana Iowa Kansas Kentucky Louisiana Maine 5290 7.058 1256 3.886 5.611 1070 L051 1226 (X895 10.428 1685 3.350 1525 1362 1337 1476 1575 1531 1607 1607 Maryland Massachusetts Michigan Minnesota Mississippi Missouri Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island 6.785 1295 4.737 0.905 1432 4.710 1260 1.151 8.055 6.979 5.436 6.441 5.750 4.403 0.215 1614 5.182 4417 5.439 10.505 1437 3.693 1505 1577 1421 1336 1450 1527 1540 1362 1314 3.072 1254 1406 1891 1390 1339 1477 1422 1702 South Carolina South Dakota Tennessee Utah Vermont Virginia Washington Wast Virginia Wisconsin Wyoming United States 6347 0L813 6.438 6.472 0721 9.782 3.137 3425 4481 3213 L397 5303 1454 1785 1822 1765 2354 1607 1728 1459 1699 1468 3.028 0499 43 each state on a set of k = K  1 symptomatic indicators. (The other column of A" consists of all ones and allows for an intercept in the regression model.)7 u2, a scalar reflecting the lack of fit of the regression model, is estimated by maximizing the likelihood function: (7) L  \W\W \X>WX\xri exp V\fr*\ , where W = (D~x + u2/)1 and S = W  WX(X'WX)XX'W. The variancecovariance matrix of the Empirical Bayes shrinkage estimator is: l (8) K. c.EB D * ±M u2 This estimator treats the maximum likelihood estimate of u2. once it is calculated, as known. We have taken a more fully Bayesian approach, treating u2 as estimated. Ifwe specify flat prior distributions for both B~the (K x 1) vector of regression coefficientsand U, that is, distributions proportional to one, the posterior density of u, evaluated at Uj, is proportional to: (9) P;  n)iw i*'»;.*r,/2exp( J(y,  tyywp,  &,)), where Wj^ (Z)'1 ♦ ufl)'1 and 6..  (X'WjX)1XtW.Yt. Under this formulation treating u as unknown but following a particular distribution, there is no closedform expression for our shrinkage estimator. Instead, we must numerically integrate over u. 7Except for a column of ones to allow for an intercept in the regression model, Table A.6 is the X matrix To perform the numerical integration, we selected a grid of 701 equally spaced values of u, starting with 0.00 and incrementing by 0.01. For each value u; * 0.00, 0.01 7.00 of u, we calculated a vector of shrinkage estimates: (10) n, D ♦ 1M i DYt, and a variancecovariancc matrix: (") Vcj D + 1M i These expressions for the shrinkage estimates and the variancecovariance matrix are the same as when u is treated as known.8 For each u;, we also calculated p* according to Equation (9). After calculating Yc , Vc , and p* 701 times (once for each value of u;), we calculated the probability of Uf. <>2) n ■ T^ yi which is also an estimate of the probability that the shrinkage estimates V are the true values. As Equation (12) suggests, the/>; are obtained by normalizing the p* to sum to one.9 •For a,  0, we set K,  XQ(fDK)'xX!B¥t and VeJ  XtfDX)^, the limiting values derived by DuMouchel and" Harris (1983). *The pi should approach 0 as u approaches the upper limit of the grid over which we integrate. If that does not occur, the grid should be extended, and the calculations repeated. 45 To complete the numerical integration over u and obtain a single set of shrinkage estimates, we calculated a weighted sum of the 701 sets of shrinkage estimates, weighting each set YCJ by its associated probability/^. Thus, our shrinkage estimates are: 701 j • i The variancecovariance matrix is: 701 701 (14) Ve  V PjVc. ♦ £ Pj(Ycj  Yc)(YcJ  Yc)' > 1 / • l The first term on the right side of this expression reflects the error from sampling variability and the lack of fit of the regression model. The second term captures how the shrinkage estimates vary as our estimate of u varies. Thus, the second term accounts for the variability from not being able to estimate u very well. Our shrinkage estimates and their standard errors are displayed in Table A.8.10 Our regression estimates, which were presented in the previous step, were similarly obtained. They are: 701 (i5) r,  EPjYrJ, where Y ■ X& is the vector of regression estimates obtained when u * u, The variancecovariance matrix is: 10The standard errors were calculated by taking the square roots of the diagonal elements of Vc. 46 TABLE A3 CHANGES BETWEEN 1989 AND 1992 IN PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE. SHRINKAGE ESTIMATES Sute Ffimur Standard Error Alaska Arizona Arkansas California Colorado Connecticut Delaware District of Columbia Florida 2306 8.238 6336 3.977 8.619 0.133 9.044 6.7S8 11032 1O907 2.484 2.724 2.724 2302 1.403 1407 2.445 2308 3.372 1365 Georgia Hawaii Idaho Dlinoa Indiana Iowa Kansas Kentucky Louisiana Maine 4.785 7325 2.020 4.638 6.221 2.634 0.620 2.409 a977 10310 2.474 3.006 1249 1.936 1157 1.877 1310 1381 1510 1512 Maryland Massachusetts Michigan Minnesota Mississippi Missoun Montana Nebraska Nevada New Hampshire New Jersey New Mexico New York North Carolina North Dakota Ohio Oklahoma Oregon Pennsylvania Rhode Island 8.268 1396 4331 1.454 1436 5.911 2366 0.191 7.788 7.436 6309 6.123 6317 4.608 0.184 2322 5.145 4.830 3.904 10390 1161 1891 1165 1460 1284 1417 2388 2361 2384 1.959 1.670 1907 1.684 1J59 1750 1.783 1161 1278 1114 1469 South Carolina South Dakota Te Ta Utah Ve Virginia Washington West Virginia Wisconsin Wyoming United States 6.717 •L148 7373 6306 0227 9.442 3.033 3.041 4368 1413 1.212 5390 1203 1643 1725 2375 1451 2345 2347 1169 2379 2347 1925 0383 47 701 701 (16) vr  £ PjvrJ ♦ £ j^c*^i  w„  n>'. i 1 > 1 wbcrc Kf « XiX'WXj'W * up. We can estimate the regression coefficient vector by: 701 (17) 6  £ />,*, . i 1 Estimated values for the regression coefficients were displayed in the previous step. 6. Add the shrinkage estimate of the change between 1989 and 1992 to the census estimate of the percentage eligible in 1989 to get a shrinkage estimate of the percentage eligible in 1992. To facilitate a comparison of the alternative estimates, we have displayed in Table A. 9 not only shrinkage estimates but also sample and regression estimates of the percentage eligible in 1992. These estimates were obtained by adding the census estimates for 1989 from Step 1 to the estimates of change derived in Steps 3, 4, and 5. The estimates of change were displayed together in Chapter II, Table II.2 The sample estimates in Table A.9 are, of course, equal to the sample estimates obtained in Step 2 because we have just added and then subtracted the census estimates. 7. Multiply the shrinkage estimate of the perceatage eligible by the state population of infants and the state population of children to get preliminary shrinkage estimates of the numbers of eligible infants and children. As we stated in Chapter II, we assumed in this step that the percentage of infants who were income eligible equals the percentage of children who were income eligible. Census estimates show that this assumption is reasonable. Nationwide, 37.982 percent of infants and 37.739 percent of children were income eligible in 1989. TABLE A.9 PERCENTAGES OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 Sample Regression Shrinkage Slate Estimate Estimate Estimate Alabama 44.476 49.062 48308 Alaska 41.355 50.710 49305 Arizona 42J0S 51913 51310 Arkansas 53.929 56348 56.183 California 46.573 46.137 46379 Colorado 36.754 36.125 36.190 Connecticut 34.766 30,005 30344 Delaware 32.065 35.717 35331 District of Columbia 67.187 58355 58373 Florida 49.457 51391 50.928 Georgia 40.193 45.905 45.400 Hawaii 46.620 43379 44.146 Idaho 46.822 49.064 48328 Dlkiois 40.696 37369 37321 Indiana 47.088 41381 41391 Iowa 41.081 38.916 39.480 Kansas 34.028 37311 37380 Kentucky 53.568 50349 50332 Louisiana 55.899 53346 53328 Maine 47.455 44.956 45.038 Maryland 37.928 31331 31514 Massachusetts 27.620 26382 26383 Michigan 39.026 41.909 41303 Minnesota 41236 30367 30316 Mississippi 59.956 59.976 59.980 Missouri 59.946 43339 44340 Montana 53.704 48399 49.005 Nebraska 31261 39351 38391 Nevada 38.135 41408 41141 New Hampshire 30.094 27310 27.967 New Jersey 31.036 27382 28.955 New Mexico 54.170 60.436 60.118 New York 44364 40386 41053 North Carolina 45.196 44314 44319 North Dakota 42.977 42339 42370 Ohio 39.171 39362 39370 Oklahoma 52380 51820 51783 Oregon 44.718 44396 44.709 Pennsylvania 33.162 38367 37332 Rhode bland 38319 40328 40313 South Carolina 51130 50394 50364 South Dakota 40361 46.401 46366 Tennessee 60352 50342 51377 Texas 51.141 52307 51141 Utah 32384 40.720 40326 Vermont 29.180 40.946 40306 Virginia 33335 34306 34.402 Washington 34342 38389 37305 Watt Virginia 60389 56384 56.471 Wisconsin 29370 37307 36307 Wyoming 36373 42308 41423 United States 43380 43392 43379 The independent population estimates used in this step and displayed in Chapter II, Table 11.4 were obtained electronically from the U.S. Bureau of the Census. The release date for the estimtics was March IS, 1994. 8. Control the preliminary state shrinkage estimates of the numbers of eligible infants and children to sum to the national totals for eligible infants and children obtained from the CPS. In Chapter III, we presented approximate 90percent confidence intervals for our final shrinkage estimates. The upper and lower bounds of the confidence intervals were calculated according to: (18) Upper Bound. = Eci * 1.645 eci and (19) Lower Bound.  Eci  1.645 eci , where Eci is the final shrinkage estimate (for infants or children) for state i and eci is the standard error of that estimate. That standard error is: (20) ^.r.H.ljJpL. where rc equals 1,717,743 + 1,737,837 (for infants) or 6,925,815 + 6,721,734 (for children), N„ is the independent population estimate of either infants or children in state i, and Vc(ii) is the (L,i) diagonal element of Ve, which was calculated according to Equation (14). In other words, the square root of Vc(ii) is the standard error of the shrinkage estimate of the percentage of infants and children eligible 50 in state i in 1992.11 We can find values for Eci, Nti, and the square root of Vc(ii) in Tables III. 1, 114, and A.8, respectively. In addition to presenting the confidence intervals for our shrinkage estimates in Chapter III. we discussed the relative precision of sample and shrinkage estimates. To inform that discussion, we derived "final" sample estimates in the same way as we derived our final shrinkage estimates.12 Both sets of final estimates appear in Table A. 10. In Tables A. 11 and A. 12, we present confidence intervals for sample and shrinkage estimates of eligible infants and children, respectively. We calculated bounds for confidence intervals of sample estimates according to Equations (18) and (19), replacing shrinkage estimates by sample estimates. The standard error for a sample estimate is given by: (2.) «,,»,«. where r, equals 1,717,743 + 1,746,319 (for infants) or 6,925,815 + 6,754,737 (for children), and the square root of MD(ii) is in the third column of numbers in Table Al. 11As in Step 3, we assumed that the sampling error associated with a census estimate is negligible. Therefore, the standard error for the shrinkage estimate of the proportion eligible is the same as the standard error for the shrinkage estimate of the change in the proportion eligible. Our estimate of ecl does not take account of the correlation between rc and our estimate of the proportion eligible. Instead, rc is treated as a constant "Beginning with the sample estimates of the percentage eligible in 1992, we used the independent population estimates of infants and children to obtain preliminary sample estimates of the numbers eligible and, then, controlled those preliminary estimates to the national totals. The preliminary estimates summed to 1,746319 infants and 6,754,737 children. 51 TABLE A.10 SAMPLE AND SHRINKAGE ESTIMATES OF THE NUMBERS OF INFANTS AND CHILDREN INCOME ELIGIBLE IN 1992 •Final" Sample Estimates Final.Shrinkage Estimates Sutc Infants Children Infants Children Alabama 26,984 107,747 29,757 118322 Alaska 4,683 19304 5345 23310 Arizona 27,750 111371 33.754 135.715 Arkansas 18,278 75.149 19,135 78374 California 271412 1335354 271301 1336304 Colorado 19374 79304 19,170 78,174 Connecticut 15.929 67388 13325 58311 Delaware 3397 13330 3,761 15392 District of Columbia 5,499 21329 4,793 19326 Florida 92,634 381.175 95355 394,439 Georgia 43.183 174332 49316 197,770 Hawaii 8.992 34369 8356 32,705 Idaho 7361 32.188 8338 33,732 Illinois 75371 297325 70389 277364 Indiana 38,129 156313 33324 139,077 Iowa 15351 65.464 14,729 63322 Kansas 12316 52.495 13396 57349 Kentucky 27374 113362 26,423 107367 Louisiana 38385 154.766 37394 149307 Maine 7380 32.458 6342 30356 Maryland 28391 117,770 24371 101,455 Massachusetts 23.429 96368 22,745 93.750 Michigan 53343 230319 56399 246.140 Minnesota 26366 114.025 19,725 85330 Mississippi 25355 98336 25,188 99361 Missouri 44302 185388 33374 139377 Montana 6305 25.730 5306 23394 Nebraska 7346 31.700 8342 37310 Nevada 8323 33072 9,131 36.725 New Hampshire 4384 20.444 4381 19392 New Jersey 36316 145,098 33,765 136.034 New Mexico 14321 58349 16329 65332 New York 122351 491353 117,020 467,950 North Carolina 44,985 182.485 44328 180335 North Dakota 3387 15309 3353 15.761 Ohio 63347 254,736 64304 258395 Oklahoma 24324 99367 24378 101.019 Oregon 18,157 77319 18342 78,185 Pennsylvania 52350 220367 59399 249361 Rhode Island 5367 22328 5.795 23347 South Carolina 28367 116.175 27344 113339 South Dakota 4.400 18339 5322 20317 Tennessee 43394 173.919 37396 150230 Taut 159340 620,788 163.761 636334 Utah 11310 47.704 14318 58341 Vermont 2.162 9384 3323 13342 Virginia 31318 128,169 32,497 131.735 Washington 26389 109.746 29377 121,761 West Virginia 13326 53370 12300 50368 WaMoaan 20366 88391 25313 108307 Wyoming 2337 10445 2317 12376 United States 1.717,740 6325316 1,717,746 6325319 52 TABLE A.11 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE INFANTS Sample Estimates Shrinkage Estimates State Lower Bound Upper Bound Lower Bound Upper Bound Alabami 18,579 35389 27366 32348 AlMfc. 3.690 5376 5,135 6.155 Arizona 21,700 33300 30335 36373 Arkansas 14,149 22,407 17345 20325 California 255.145 287.479 257,990 285312 Colorado 13.624 25124 17373 21367 Connecticut 9.116 22,742 12373 15,777 Delaware 2384 4310 3357 4.165 District of Columbia 3,728 7370 4337 5349 Florida 85,098 100.170 89,771 101.939 Georgia 31,796 54370 44322 53.410 Hawaii 7,029 10355 7398 9314 Idaho 6329 9393 7314 8362 Illinois 63.989 86.753 64362 76316 Indiana 30.238 46320 31337 36311 Iowa 13,292 17310 13377 15381 Kansas 9,107 15325 12314 14378 Kentucky 20.683 35365 24375 28,471 Louisiana 26.006 51364 34,423 40,165 Maine 4.450 10.110 6305 7379 Maryland 23361 32.721 21.706 27336 Massachusetts 18.266 28392 18391 26,799 Michigan 42390 63396 52316 61,782 Minnesota 16326 35.706 17,135 22315 Mississippi 19,095 31315 23310 26,766 Missouri 35332 52372 30.141 36307 Montana 3.402 8308 5365 5347 Nebraska 5.464 9328 7,765 9319 Nevada 5.461 10385 8381 9381 New Hampshire 3336 5332 3.788 4,774 New Jersey 31318 40,714 30361 36369 New Mexico 10,796 18346 15314 17344 New York 111341 134.061 109312 124,728 North Carolina 39398 50372 41.469 47387 North Dakota 2330 5344 3363 4343 Ohio 55.063 71331 59338 69370 Oklahoma 18,780 29368 23316 26340 Oregon 13399 22315 16,713 19,771 Pennsylvania 45.196 60.704 54319 65379 Rhode Island 3339 7,195 5310 6380 South Carolina 23,159 33375 25348 29340 South Dakota 2346 5354 4348 5396 Tennessee 37,757 48331 34,165 40327 Texas 134331 185349 15L491 176331 Utah 7328 16392 13363 15373 Vermont 698 3326 2.711 3335 Vnginia 23376 39360 28350 36,144 Washington 20358 32320 26314 32340 West Virgkua 9341 1M11 11383 13,117 Wkcoosin 14331 26381 22368 27358 Wyoming L210 3364 2397 3,137 53 TABLE A12 APPROXIMATE 90PERCENT CONFIDENCE INTERVALS FOR ESTIMATES OF NUMBERS OF ELIGIBLE CHILDREN Sample Estimates Shrinkage Estimates State Lower Bound Upper Bound Lower Bound Upper Bound Alabama 74,184 141310 108374 128,770 Alaska 15.369 23339 21486 25334 Arizona 87J45 135397 123,977 147.453 Arkansas 58,175 92,123 73471 83477 California 974,131 1397477 985.020 1388.188 Colorado 55,557 102,451 69321 86,727 Connecticut 38463 96413 51377 66,745 Delaware 9^65 17395 13,470 16,714 District of Columbia 14,799 28359 17415 20837 Florida 350,164 412.186 369,404 419,474 Georgia 128,288 220176 180,042 215.498 Hawaii 26366 41372 29342 36468 Idaho 25.914 38,462 31,176 36488 Dlinois 252494 342,456 254.466 301462 Indiana 123,964 188,662 127440 150914 Iowa 57,057 73371 58478 68.166 Kansas 38319 66.171 52358 63340 Kentucky 84,042 142,482 99345 115.689 Louisiana 104,043 205,489 137.719 160695 Maine 19340 45376 28.116 33,796 Maryland 99331 136409 90463 112447 Massachusetts 75,288 117348 77341 110459 Michigan 185335 275.103 225,018 267462 Minnesota 73,046 155.004 74485 96375 Mississippi 75325 122447 93,137 105485 Missouri 148,781 221.795 126,927 151327 Montana 14375 36385 21,703 25.485 Nebraska 23,903 39.497 33475 41345 Nevada 21.963 44,181 33407 40143 New Hampshire 15,768 25.120 16392 21492 New Jersey 126,171 164.025 123.128 148.940 New Mexico 42369 74329 60.411 70353 New York 446.425 536.081 437,125 498.775 North Carolina 159323 205,147 168427 193,043 North Dakota 10053 21.765 14378 17,444 Ohio 221,423 288.049 239,427 277,763 Oklahoma 76372 122462 94416 107322 Oregon 57,428 98410 71332 84,738 Pennsylvania 188325 253409 226486 273,136 Rhode Island 15367 28489 20489 25.705 South Carolina 94.183 138,167 105,123 121455 South Dakota 12412 24466 18352 22,782 Tennessee 151377 196,161 137449 163411 Texas 521326 720450 588477 683391 Utah 29,195 66413 52,763 64419 Vermont 3428 16440 12,146 14438 Virginia 97,190 159.148 116451 146419 Washington 84366 134326 110469 133453 West Virginia 39.721 67319 46,492 54344 Wisconsin 61.123 116,059 974OO 120414 Wyoming 5484 15,706 10706 13446 54 
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