State estimates of infants and children income eligible for the WIC Program in 1992 /

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Contract No.: 53-3198-4-031
MPR Reference No.: 8207-050
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
\qs-0wf67 <*/
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 state-level 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 0-4 in each state in households with incomes below 185% of poverty. These
proportions were then applied to 1992 state population estimates of infants (0-1) and children
(1-4) 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 state-level economic data. The direct sample
estimates used were the March 1993 Current Population Survey (CPS) estimates of the
proportion of children 0-4 below 185% of poverty in each state in 1992. The model
estimates were developed using a regression model which estimated the proportion of
children 0-4 below 185% of poverty using the following state-level 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 (0-4) 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 STEP-BY-STEP 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
Pa|c
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 90-PERCENT 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 90-PERCENT CONFIDENCE INTERVALS
FOR ESTIMATES OF NUMBERS OF ELIGIBLE INFANTS 53
A12 APPROXIMATE 90-PERCENT 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 WIC-eligible
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 confidence-according to widely accepted statistical standards-only 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 one-third 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
method-shrinkage estimation-to 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
four-person 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 STEP-BY-STEP 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 income-eligible 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 variable-the change in FSP participation-that 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 out-migration) and adding persons "entering" the population (due to birth or net in-migration).
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 90-percent confidence
intervals showing the uncertainty remaining after using shrinkage estimation. One interpretation of
a 90-percent confidence interval is that there is a 90 percent chance that the true value-that is, the
true number of eligibles-lies 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 income-eligible 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, Delaware-which
has about 0.2 percent ((3,761 + 15,092) + (1,717,746 + 6,925,819)) of all eligible infants and
children-has 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 90-PERCENT 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 report--the 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. 293-315.
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 Small-Places: An Application of
James-Stein Procedures to Census Data." Journal ofthe American StatisticalAssociation, vol. 74,
no. 366, June 1979, pp. 269-277.
Fay, Robert E, Charles T. Nelson, and Leon Litow. "Estimation of Median Income for 4-Person
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. 43-46.
Ghosh, M. and J.N.K. Rao. "Small Area Estimation: An Appraisal" (with comments). Statistical
Science, vol. 9, no. 1, February 1994, pp. 55-93.
Hogan, Howard. The 1990 Post-Enumeration Survey: Operations and Results." Journal of the
American Statistical Association, vol. 88, no. 423, September 1993, pp. 1047-1060.
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. 209-217.
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
One-person family 8^90 8,500 8395
Each extra person 2,820 2,980 2,900
Hawaii
One-person family 7,610 7,830 7,720
Each extra person 2,600 2,740 2,670
Other States and DC
One-person 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 one-eighth
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 reduced-price meals relative to the size of the school-age
population-ages 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 preschool-age
population-ages 0 through 4)
• Chapter 1 (Compensatory Education) Program funding (basic grant, in dollars,
relative to the size of the school-age 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 one-person family
Total personal income data are from Table 1,
Total and Per Capita Personal Income by
State and Region, 1985-90," in U.S.
Department of Commerce (1991b, p. 30) and
Table 1, "Total and Per Capita Personal
Income by State and Region, 1987-92," in
US Department of Commerce (1993b, p.
74).
■Data on the resident population as of Jury 1 are from Table 26, "Resident Population-States and Puerto Rico: 1960 to 1990," in U.S. Department of
Commerce (1991a, pp. 20-21) and Table 31, "Resident Population-States: 1970 to 1992," in U.S. Department of Commerce (1993a, pp. 28-29).
*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 one-person 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 one-person
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 variance-covariance 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 coefficients--and
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 closed-form 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 variance-covariancc matrix:
(") Vcj D + 1M
-i
These expressions for the shrinkage estimates and the variance-covariance 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^-
y-i
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 variance-covariance 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 variance-covariance
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 90-percent 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 90-PERCENT 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 90-PERCENT 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