Measuring the Number of People Without Health Insurance:

A Test of a Synthetic Estimates Approach for Small Areas

Using SIPP Microdata

By

Carole Popoff

Branch Chief, Modeling & Outreach Branch

Housing and Household Economic Statistics Division

U.S. Census Bureau

D. H. Judson

Mathematical Statistician and Group Leader

Administrative Records Evaluation and Linkage Group

Planning, Research and Evaluation Division

U.S. Census Bureau

Betsy Fadali

Research Associate

University of Nevada, Reno

Paper presented at:

2001 Federal Committee on Statistical Methodology Conference

Fall, 2001

This paper reports the results of research and analysis undertaken by Census Bureau staff. It

has undergone a Census Bureau review more limited in scope than that given to official

Census Bureau publications. This report is released to inform interested parties of ongoing

research and to encourage discussion of work in progress.

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Abstract

The need for detailed small area estimates of policy-relevant population characteristics has

increased dramatically in recent years. In particular, the management and assessment of new

health care initiatives such as the State Child Health Insurance Program (SCHIP) has created

the need for estimates of the uninsured at the county and subcounty levels. Our ultimate goal

is to develop a methodology that satisfies state and local needs and can be implemented

within their level of expertise, budget, and other constraints. The goal of this paper is to test

the validity of the specific estimation system we developed to address these needs.

In prior work we presented a mixed-method approach to estimating the number and percent

uninsured based on a synthetic estimation technique that would satisfy the goal stated above.

For this system we used: 1) a state representative population survey that captures uninsured

status; and 2) county level estimates of age, race, sex and Hispanic origin (ARSH) that are

constructed from state-specific fertility, mortality and migration experience. Using a

synthetic technique, we thus make the link from the state level estimates to derive the county

level estimates by specific ARSH categories.

This approach relies on the assumption that ARSH characteristics are adequate predictors of

uninsured status. This paper tests that assumption. We use a logistic regression model to

predict uninsured. We limit ourselves to ARSH variables in order to use the logistic

regression results in our synthetic estimation system. Thus, strictly speaking, we are not

developing a causal model, but instead are using maximum likelihood logistic regression

techniques for “curve fitting.” Though we fit the model at the individual level, we are really

interested in the group-level proportions, not the individual outcomes. Finally, we generate

the predicted proportions uninsured and regress the actual population proportions uninsured

on the predicted proportions uninsured without an intercept to test for goodness of fit of the

aggregate predicted proportion uninsured. We use the Survey of Income and Program

Participation (SIPP).

The use of synthetic estimation techniques in conjunction with detailed program data

illustrates the potential of such a system for performing policy-relevant demographic

analysis. In this evaluation, we found that, at the aggregate level, ARSH characteristics

generally predict the proportion of uninsured very well. This gives us substantial confidence

in using such characteristics for a county level synthetic estimates system.

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Introduction: The Need for Small Area Estimates

The National Research Council’s Modernizing the U.S. Census (Edmonston and Schultze,

1995:16), a panel recommended that the “...Census Bureau work with state and local

governments to enhance the quantity and frequency of small-area data.” States need small

area estimates (see, for example, Schirm, Zaslavsky, and Czajka, 2000), because they are

mandated to serve new and ongoing demands for services while budgetary constraints limit

the resources and expertise for producing reliable local information. For example, recent

requests from such sources as State Health Departments and Community Development Block

Grant writers highlight the continuing need for detailed, up-to-date demographic estimates.

Targeting needs at this level has become even more critical as block grants have replaced

earlier federally supported open-ended programs. For example, the 1996 Personal

Responsibility and Work Opportunity Reconciliation Act (PRWORA), which replaced the

Aid to Families With Dependent Children Act (AFDC) allocates block grants to states rather

than funding based on needs.

In this paper we examine the usefulness of a synthetic estimation methodology that will not

only allow states to produce local-area estimates of the population without health care

coverage, but which may also be useable to assess many policy issues. We assert that states

and local entities can implement this technique within constrained budgets, timeframes and

expertise in statistics or estimation techniques.

State Health Department needs

Because of rapid changes in the nature of the health care system at the state and national

levels, researchers and policy makers need access reliable current information about the

access to health insurance coverage and health care over time. Under Healthy People 2000

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and its successor project, Healthy People 2010, the federal government outlined a series of

measures for states to examine when developing health policy or public health strategies. In

addition, the Centers for Medicare and Medicaid Services (CMS) new program called State

Children's Health Insurance Program (SCHIP), established by the Balanced Budget Act of

1997, has expanded health care coverage for children (U.S. Department of Health and

Human Services, 2001). SCHIP is a capped entitlement program that requires states to

develop plans that include consistent estimates of eligible recipients to determine program

efficacy. Medicaid and other state initiatives have attempted to improve the data; however,

in most cases the need still exists.

1

Focus of the Study

About 16 percent of the U.S. population was uninsured in 1997, 1998 and 1999 (U.S. Census

Bureau, 2001). As stated above, some data collection efforts have been initiated. The State

Systems Development Initiatives sponsored by the Health Resources and Services

Administration (HRSA, 1999) targeted data collection and inventories for needs assessments,

but good quality, periodic, county-level data are not available in most states. In this study,

we continue prior work on the development of reliable estimation methods to estimate the

number and percentage of uninsured people.

A Review of Some Techniques for Small Area Estimation

Sophisticated Techniques

Private vendors have responded to meet the need for estimates of basic demographic

characteristics at low levels of geography. The American Community Survey will provide

some estimates at the county level. Other sources include targeted surveys (either large

surveys representative at the U.S. or state level or locally sponsored surveys); various

1 CMS was formally called the Health Care Financing Association (HCFA).

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administrative records series; and the decennial census long form (which may or may not

include the variables of interest). For many of the small areas, the sample size will be small

or non-existent, and using these sparse data for small area estimates results in large variances

that make the estimates unreliable or simply not usable. For example, the U.S. Census

Bureau’s Current Population Survey is often used for state level estimates because the

sampling scheme is a state-based design; however, it is not designed to be representative at

lower levels of geography. County-level specialty surveys that actually are able to obtain

reliable or unbiased results are prohibitively expensive. Administrative records series often

have very limited detail on personal and household characteristics and are typically not

designed for such work. Thus there is no simple solution to obtain the relevant information

at small geographies. Often, the solution is to estimate the variable(s) of interest.

Several sophisticated estimation techniques have been tested and employed by federal

agencies or consulting firms. In general, these methods are highly technical in nature; they

require a sophisticated knowledge of modeling and statistical inference in addition to the

construction, implementation and maintenance of elaborate estimation systems. These

techniques "borrow strength" by employing either a variance components or an empirical

Bayes (EB) approach to derive an indirect estimator (Harville, 1988, 1990, in Datta and

Ghosh, 1991). Variations that have been tested include hierarchical Bayes (HB) or nested

error regression models (see, for example, Datta and Ghosh, 1991 for a review and test of

these techniques; and comments by Cressie and Kaiser, 1991; and Holt, 1991). These

methods have been used, for example, to allocate federal funds for the Special Supplemental

Nutrition Program for Women, Infants, and Children (WIC) (Schirm, et. al., 2001) and for

allocating federal Title I funds for compensatory education in secondary schools (National

Research Council, 1998, in Schirm, et. al., 2001).

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The variations of EB or HB techniques are appropriate to obtain estimates of a single

characteristic or a small number of closely related characteristics for each geographical area.

These techniques may require the analyst to formulate and test the appropriate models for

each of the different characteristics of interest. Schirm, Zaslavsky and Czajka, (2000)

developed a method to produce estimates of large numbers of characteristics for each area

using an elaborate re-weighting scheme. They fit a Poisson regression model to obtain an

estimated prevalence for each small area for every household type in the database, the

household types being defined by household characteristics. Thus a matrix of weights is

produced with each household having a weight for each state in a state-based survey, for

example. Since each household has a weight for every state, the entire database can be used

to estimate variables of interest at the state level that increases the sample size and reduces

the problem of small cell sizes. While this could theoretically be applied at the county level,

no such scheme exists and to develop such a system would require enormous effort.

While all these techniques have much to recommend them, we reiterate that they require the

resources and the sophisticated efforts of highly trained personnel and large organizations.

The thrust of this paper is to expand on a system we developed to answer the needs of states

and localities that have limited resources. For example, it is highly unlikely that state and

local jurisdictions would have the expertise required to develop and test extremely

sophisticated techniques to address their needs.

Drawbacks of Typical Estimation Techniques Used by States

States have typically used one or more of three techniques: state-specific or locality-specific

surveys, state level data from the Current Population Survey (CPS) as an estimate for

substate areas, or administrative data sets maintained at the state level. Each of these has

features that make it useful for certain types of estimates, but all have significant weaknesses

that preclude their use on a broader scale.

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A Locally-Sponsored Special Survey Designed to Estimate the Topic-Specific Characteristics

Some states and counties have implemented topic-specific surveys to collect the relevant

data. For example, the Nevada Center for Business and Economic Research conducted a

statewide demographic survey to assess health insurance coverage. At a cost of

approximately $93,000 the initial response rate of 10.1 percent was increased to 18.2 percent

with a later telephone follow-up (Pencek, Daneshvary, and Schwer, 1998:244). Because

health insurance status is highly correlated with social and economic characteristics that also

correlate with survey non-response, insurance estimates based on surveys with this level of

non-response are subject to great potential bias (Dillman, 1978; Tanur, 1981).

CPS-Based Estimates of Topic-Specific Characteristics

The US Census Bureau's Current Population Survey (CPS), March Supplement is widely

used for comparative state-to-state characteristics (U.S. Census Bureau, 2001), and, in the

mid-eighties it began being used to estimate the uninsured population (Nelson and Mills,

2001). The survey asks a series of questions as to whether each household member had

insurance coverage at any time during the prior year by naming a fairly exhaustive list of

types of coverage (e.g., Medicaid, private coverage, CHAMPUS, etc.). If the respondent

provides negative responses to all of these questions, he/she is considered to be uninsured for

all of the prior year. However, it is inadequate to accurately assess health insurance coverage

at the sub-state level because the sampling scheme is a state-based design with the sample in

each state being independent of the others (U.S. Census Bureau, 2001).

Pure Administrative Record Estimates of Topic-Specific Characteristics

Administrative data estimates are even more difficult to use as the basis of estimates for a

characteristic like health insurance coverage. First, administrative data cannot provide direct

information on people who are not covered, but only provide information on the number of

people who accessed the various programs; therefore, presumably people covered subtracted

from the total population should equal the number of people without coverage. Second, for

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this type of calculation to be accurate, in most cases, it is necessary to combine, sort and

unduplicate administrative records from a variety of sources. However, technically, these

records are not complete counts of the insured population making it difficult to accurately

calculate an unduplicated count (for a list of potential errors, biases, and assorted limitations

of administrative records data, see Judson and Popoff, 1998). Finally, while Medicaid,

Medicare and insurance department enrollment records could potentially help in estimating

the population covered by insurance, many legal, proprietary, and political barriers may

prevent the establishment of the necessary linkages for unduplication operations. For

example, privacy and confidentiality legislation often prevents this type of access. Given

these conditions, administrative data are often used only to verify survey coverage (for

example, see Card, Hildreth, and Shore-Sheppard, 2001).

Goals for a Small Area Estimating System To Be Useful to States

for Small Area Estimates

More effective small areas estimates that can be easily and cost-effectively implemented

require:

1. Developing an estimating system for health-related characteristics that can be replicated

on a periodic basis to evaluate trends rather than single “points in time”;

2. Creating a system that provides estimates useable (and reasonable) at the county (or subcounty) level of geography;

3. Developing a system that will serve intercensal estimation needs;

4. Designing the system to be portable, i.e., useable by different states;

5. Designing the system to be expandable, i.e., useable to estimate different characteristics

of interest, not tied to a particular data source or content;

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6. Designing a system that can be implemented within budget and the technical constraints

of state and local entities; and finally,

7. Verifying that the system is capable of providing reasonable estimates of the

characteristic(s) of policy interest.

Appraisal of Goals

Each of the above data collection and estimation options (national or special surveys,

administrative data, or elaborate estimation schemes) fails on at least one of the seven criteria

above. For example, a single state-specific survey, representing only a point in time

estimate, is not sufficient for developing a true estimating system nor does it offer trend

analysis if not repeated on a regular basis, thus it fails goal #1 (periodic implementation,

trend analysis). Similarly, using the CPS as the state-level data collection method suffices

for goals #1 (estimating system), #4 (portable), and #5 (useful for many characteristics

because of its broad coverage of many issues), but is not sufficient for goal #2 (county level

of geography), due primarily to its cluster sampling strategy. Using a pure administrative

records approach cannot address estimating people who are not in the records (such as the

uninsured) but probably suffices for goal #1 (estimating system), goal #2 (county level of

geography). It also fails on goals #4 and #5 (each administrative data set is specific to the

program thus limiting the expandability and portability). A complex and sophisticated

estimating system requires using one or several data sets as well as costly design and

implementation. This type of system may be out of reach for most state and local entities

thus goal #6 (ability of state and local jurisdictions to implement) is not satisfied.

Given this evaluation, we chose to develop a system based on synthetic estimation. For this

system, we used two data sources; 1) the CPS, March Supplement as the state-level survey

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from which to derive estimates of uninsured status by ARSH characteristics; and, 2) statespecific ARSH estimates derived from administrative records and produced in the usual

manner. This system is replicable, portable, and relatively straightforward and cost-effective

to implement within a reasonable level of expertise. It depends on an annually administered

survey, can be expanded to other characteristics of interest because of the variety of survey

questions, and can be localized by using specific ARSH estimates.

2

Attributes of a Synthetic Estimation Technique

A synthetic estimation system that ties age/race/sex/Hispanic origin (ARSH) estimates at the

substate level to a household survey representative at the state level and conducted annually

could potentially satisfy all seven objectives. For example, because the system updates on a

yearly basis (based on estimated changes in the ARSH characteristics of the population and

the use of an annually administered survey), it can be used for intercensal estimation

(satisfying goal #3). The system has the potential to be portable (partially satisfying goal #4)

because we use ARSH estimates that can be constructed by any state and a state-based

survey that covers all states; i.e., the CPS. Also, the CPS contains information on many

variables that are of policy interest (satisfying goal #5). Finally, it is cost-effective and

relatively straightforward to implement and test satisfying goal #6.3

Figure 1 illustrates the advantage of making estimates by individual ARSH cells and is based

on the following assumptions. First, break the population into two age groups, 0 to 40 and

40+; then into five age groups, 0-19, 20-39, 40-64, 64-79, 80+. Second, assume that 12

2

The CPS is the basis of the official state-level health insurance estimates produced annually by the U.S.

Census Bureau (Nelson and Mills, 2001; U.S. Census Bureau, 1996, 1997. 1998, 1999; see also, Czajka and

Lewis, 1999 and Bennefield, 1996 for a comparison of health insurance status estimates using various surveys).

3

Though, in this paper, insurance status is used as a test prototype, there is nothing specific to insurance status

in our synthetic estimation techniques. The techniques could apply to any characteristic of interest for which an

estimate is needed, provided that the assumption that ARSH characteristics are good predictors of the policy

variable of interest is correct.

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percent of the total population is uninsured, but that the proportion is not distributed

uniformly across the age groups. This figure demonstrates why breaking the population into

finer groups must necessarily generate more correct estimates. The dotted line represents the

“true” proportion uninsured by single year of age. As can be seen in this example, using 12

percent across all age categories is dramatically incorrect for certain age groups that deviate

from the average. A similar argument applies to the estimate using only two age groups, and,

while using five (or more) age groups improves the estimate, without the full original age

detail the estimate will not be completely correct. (See Appendix A for a more complete

explanation of the synthetic technique used in previous work.)

--Insert figure 1 about here --

To summarize, the unique portions of this methodology include:

• Tying two independent sources of information together (ARSH estimates at the

county level and survey estimates at a higher level of geography), and using the

ARSH estimates in an attempt to take advantage of the porposed correlation between

individuals’ ARSH characteristics and other characteristics of interest; and

• Constructing detailed estimates at a level of geography much smaller than attempted

before.

An Evaluation of the Synthetic Estimates Approach

Potential Difficulties Using Synthetic Estimation

Under certain conditions the underlying assumptions of a synthetic estimates system may be

incorrect and thus might fail to provide reasonable estimates. The following is a list of the

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major assumptions that need further elaboration and testing to verify the validity of this

method.

1. Representative surveys: A key component of this method is the estimate of mˆ

a ,r ,s ,h

,

4

obtained from the population survey. To the extent that the survey fails to represent the

population as a whole, this estimate may fail to represent the insurance experience of the

respondents. Similarly, a survey with low response rates has the potential to generate

very small cell sizes if respondents are differentially underrepresented in the survey itself.

In addition, imprecise information will be generated from a survey with imprecise or

untested questions.

2. Uniformity within ARSH cells: As illustrated in Figure 1, assuming that mˆ

a,r,s,h

 is the

same for every member of the particular a,r,s,h cell could cause the estimates to be

biased. Breaking out mˆ

a,r,s,h

into ARSH-specific calculations reduces this bias. However,

if the population is heterogeneous relative to the characteristic of interest (insurance

status) within ARSH-specific cells, we would incur biases.

3. ARSH characteristics as useful predictors: The correlation between ARSH

characteristics and certain policy-relevant characteristics may not be high enough to use

to estimate the characteristics. For example, if there were no particular relationship

between a person’s ARSH characteristics and insurance status, using ARSH

characteristics as a tool for estimating uninsured status provides no more information

than multiplying an overall percentage of uninsured people times the total population

number for the area. In multi-way crosstabulation terms, the table could be “collapsed”

(Bishop, Feinberg, and Holland, 1975) across the irrelevant dimensions.

4

See Appendix A for definition.

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Given that we propose to use the CPS March Supplement in our estimation system

proposition 1 is not of great concern in this paper. Propositions 2 and 3 are also of concern,

and are analysed in this paper.

General Evaluation Approach

Synthetic methods are non-parametric methods using multiplication on a cell-by-cell basis.

Our general purpose is to parameterize the predictor variables (ARSH characteristics). In

this case, the synthetic method depends heavily on the ARSH characteristics to customize the

estimates of the characteristic of interest to each local area. For example, suppose there are

differential proportions of the population without health insurance among different the

ARSH characteristics. If so, using ARSH to "customize" uninsured estimates for localities

that have varying population ARSH mixes has the potential to produce customized estimates

of the uninsured that reflect each area's unique population characteristics.

First, we fit an individual level logistic regression model to the data, using various

combinations of ARSH characteristics as predictor variables. We also test the ranges upon

which it is safe to collapse given that disaggregating survey responses into ARSH

characteristics generates some unacceptably small cell sizes. Finally, we test the final

estimated model for goodness of fit both at the individual level and the aggregate level.

Results of this exercise formalize the notion that ARSH characteristics can be used as

predictors of the odds of being uninsured, and hence can be used in a synthetic estimation

system.

Indicators that ARSH characteristics correlate with uninsured status

Some evidence suggests there are differences in health care coverage among subsets of the

population. For example, V. Wilcox-Gok (1989), in an assessment to determine the causal

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factors for the use of private health care coverage by the elderly, found that the presence of

having private insurance was systematically related to race, sex, age and education.

The U.S. Census Bureau's 1996, 1997, 1998 and 1999 reports all show that people18 to 24

years old are less likely than any other age group to be insured for all of the preceding year.

Examining race and ethnicity, the U.S. Census Bureau also reports that Hispanics are

typically three times more likely and blacks are approximately twice as likely as white nonHispanics to be without health insurance coverage. Bennefield (1998) used the U.S. Census

Bureau's Survey of Income and Program Participation (SIPP), a longitudinal survey, to

determine characteristics of people with and without continuous coverage and found that

women were more likely to report continuous coverage.

Looking at changes in the number of uninsured over time, Holahan and Kim (2000) found

that all race and ethnic groups experienced a significant increase in the percentage uninsured

from 1994 to 1998. However, there were differences among the groups. Specifically, white

non-Hispanics experienced an increase of 3.4 percent while Hispanics (of any race)

experienced an increase of 21.5 percent and black non-Hispanics increased by 17.4 percent.

Again, these and other studies suggest that there are differential proportions of people who

have health insurance among the ARSH characteristics.

Methodology

The method chosen for this study is to estimate the probability of being without insurance

coverage given individual ARSH characteristics using logistic regression . The response

variable, uninsured, is defined as people who had no health insurance coverage for an entire

year or insured if they have had insurance coverage for at least some part of the year. This

5

5 We limit ourselves to ARSH variables in order to use the logistic regression results in our synthetic estimation

system. Thus, strictly speaking, we are not developing a causal model, but instead are using maximum

likelihood logistic regression techniques for “curve fitting.” Furthermore, though we fit the model at the

individual level, we are really interested in the group-level proportions, not the individual outcomes.

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definition allows us to replicate the way in which the state of being uninsured is defined by

the U.S. Census Bureau's uninsured estimates from the CPS (Mills, 1999; Bennefield, 1996,

1997, 1998). If we were to fit an "intercept only" model, the mean response would simply be

the overall odds of being uninsured. By fitting group- and age-specific dummy variables, the

resulting odds ratios represent the odds of a particular population subgroup being uninsured

relative to the reference group. (For example, an odds ratio of three for males of Hispanic

origin would mean that people in this group would be three times as likely to be uninsured

than people in the reference group.) Finally, we generate the predicted proportions uninsured

and regress the actual population proportions uninsured on the predicted proportions

uninsured without an intercept to test for goodness of fit of the aggregate predicted

proportion uninsured.

Data

While we have designed our system to use the CPS which is representative at the state level,

we have chosen to use the Survey of Income and Program Participation for this test. The first

four waves of the 1996 panel are combined to represent an entire year's observations for each

respondent who remained in sample for the entire year; this simulates the reference period for

the CPS, March Supplement. The 1996 panel had an increase in the sample size from prior

panels. There are 80,923 (unweighted) cases or individuals included in the study.

6

Choosing the SIPP for this evaluation rather than CPS requires some explanation. It should

be clarified that the CPS is the official source of estimates of the uninsured at the state level

and that the SIPP is a nationally based survey, thus not appropriate for the system we have

designed. However it is appropriate for this study. It was crucial that the survey minimize

the inaccuracies in individual responses, recognizing that all surveys have accuracy

6 Although, in theory, one year's responses are captured in three waves, we used portions of the forth wave to

cover an entire calendar year.

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deficiencies (see, for example, evaluations by Nelson and Mills, 2001; ASPE, 2001; Czajka

and Lewis, 2000; Pascale, 2000; Lewis, Ellwood, and Czajka, 1998; Bennefield, 1996).

Other large scale surveys also capture health insurance characteristics: The Medical

Expenditures Survey (MEPS) administered by the Agency for Healthcare Research and

Quality (AHRQ); the National Health Interview Survey (NHIS) administered by the National

Center for Health Statistics (NCHS); the National Survey of American Families (NSAF)

administered by the Urban Institute; and the Community Tracking Survey (CTS)

administered by the Center for Studying Health System Change (Office of the Assistant

Secretary for Planning and Evaluation [ASPE], 2001). All produce different estimates.

There are several factors that are responsible for differing estimates:

• Reporting current status (estimates of the proportion of uninsured people would be

inflated if people who are currently uninsured, but who were insured for some duration

during the year, report their current status);

• The recall or reference period (the longer the time frame a person has to remember, the

more likely he/she will misreport);

• How "insurance" is defined and how the questions are framed (whether or not a

comprehensive list with explanations is given or whether single service versus

comprehensive coverage is counted);

• The focus or detail of the questionnaire (whether or not health insurance status is a

primary focus or simply adds to other information on well-being, for example); and,

• Misunderstanding of Medicaid coverage (recognized as least likely to be accurately

reported in any survey due to lack of information on program eligibility [Czajka and

Lewis, 2001; Halahan and Kim, 2000; Pascale, 2001]).

Given the list above of possible data accuracy problems SIPP ranks as a good choice. The

recall period is four months instead of a prior year, thus the problem of point-in-time

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reporting instead of reference period reporting is minimized (Bennefield, 1996). In fact, one

study concludes that MEPS and SIPP are the best sources to examine changes in status

during a certain time period (ASPE, 2001), thus we are able to capture people who were

insured for some part of the year using SIPP. The SIPP insurance questions are given in

sufficient detail to cover the various types of insurance available and the focus is on

comprehensive coverage. Also, SIPP is a person-based rather than household-based survey

thus may more accurately reflect the status of each person (Bennefield, 1996).

Addressing the underreporting of Medicaid, a study by Card, Heldreth and Shore-Sheppard

(2001) found that the SIPP adequately captured people who are covered by Medicaid. Their

study matched Medicaid administrative records to SIPP respondents in California and found

that the probability that the respondent reported Medicaid coverage when an administrative

record existed indicating coverage was about 85 percent, (92 percent for children). They also

report problems with the administrative records suggesting that the practice of using

administrative records as the benchmark may not always be the best practice .

7

Model Specification and Results

First, we test the notion that ARSH characteristics are indicative of the proportions of the

population within these categories having no health care coverage by tabulating simple

descriptive statistics from our data set (unweighted). They strongly suggest that rates of

uninsured people vary by different combinations of ARSH characteristics. However, there

remains some question as to whether weighted or unweighted data more accurately support

the synthetic estimation system. We therefore assessed the impact of different weighting

schemes and within household, within family, and within subfamily clustering. We

7 Given the underreporting of Medicaid coverage, we suggest that the number of people who reported no

coverage may overstate the number of uninsured; however, all estimates using survey data have a similar

problem.

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performed the same regression analyses in this study with weighted data using two different

weights, the sample design weights (alone) and the weights that take into account sample

design, post-stratification, and attrition. Further, to account for within household, within

family, and within subfamily covariance, we performed the same regression analyses with

weights and these three kinds of nonindependent clustering, using robust methods developed

by Huber, (1967) and White (1980, 1982). Though the clustering inflated the robust standard

errors by a nominal amount, we found no substantive differences in the results. Therefore,

for simplicity we present unweighted results for the remainder.

The unweighted proportion uninsured for Whites was lowest of the four race categories at

about 8 percent; the percent of uninsured Blacks was 9.8 percent; it was 10.4 percent for

American Indian / Alaskan Native (AEAN) and 11.2 percent for Asian / Pacific Islander

(API). The uninsured percentage for males was 9.5 percent versus only 7.3 percent for

females. Further cross-tabulations and collapsing of age and racial groups suggested some

models to be tested. For example, females exhibited the same pattern of lower rates of being

uninsured than males for any racial category. Hispanics of any race exhibited the highest

percentage of being uninsured at 21.2 percent; the percentage for white-Hispanic, blackHispanic and AEAN-Hispanic were 22.5 percent and 17.1 percent respectively. Asian /

Pacific Islander-Hispanics' uninsured rate was 13.2 percent. These and other tabulations

suggested the several models tested.

We begin by graphically describing patterns of the uninsured population. These figures

represent unweighted data, and cubic spline interpolations have been overlaid on the data

points. They illustrate differences in proportions uninsured by age, disaggregated by the four

race categories, by Hispanic and non-Hispanic ethnicity, and male and female.

--Insert figure 2 about here.--

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As can be seen in figure 2 (race), the general age pattern of proportions of uninsured people

is similar among the four races, although the level and variation around the pattern differ.

For example:

• The age pattern across race groups is very similar: The proportion of uninsured people

generally increases to peak at about age 25, then drops off relatively slowly until age 64,

when a steep drop occurs—after 64, almost no one is uninsured.

• The level of uninsured people varies slightly among the four race groups, although whites

in general are less likely to be uninsured at every age. And,

• The group-specific proportions uninsured vary notably for AEAN and API groups—this

is a reflection of the relatively small numbers of people in each of those groups. The

white and black groups exhibit less variation around the “typical” age pattern.

This pattern suggests that it is possible to collapse black, AEAN, and API groups into an

overall “other ” category and compare that category to whites.

-- Insert figure 3 about here.--

As can be seen in figure 3, both groups follow the typical age pattern, with people other than

white having a higher level after about age 12 or 13, and more variation in the ages greater

than about 35.

--Insert figure 4 about here.--

As can be seen, figure 4 shows that Hispanics and non-Hispanics exhibit very similar, and

typical, age patterns of uninsured persons. However, the Hispanic level is substantially

higher at every age, and has much more variation after age 35 than the comparable nonHispanics.

--Insert figure 5 about here.--

H:\Final FCSM Papers\II-A Popoff.doc 19
Finally, we can see in figure 5 an interesting interaction between age and sex. The age

pattern for males and females is typical, and the levels are almost exactly the same before age

18 and after age 65. However the levels diverge notably after age 18 and before age 50,

with males showing much higher proportions at all ages within this category. For ages 50 to

64, the proportion uninsured appears to be converging with increasing age.

All of these separate graphs lead to a common supposition: The age pattern of the proportion

of uninsured people is relatively stable across different race, Hispanic, and gender groups.

Thus, a model that uses age information can explain much of the group-level variation in

levels of proportions uninsured. A final question remains: How should age information best

be used? Figures 6 and 7 illustrate different options.

--Insert figure 6 about here.--

In figure 6 above, we have illustrated the population proportion uninsured by age, a cubic

spline interpolation on the population, a “categorical” age model (in which we merely

collapse age groups and fit an age-group-specific mean), and a “simple” age model. In the

“simple” age model, we fit a cubic polynomial on age between ages 0 and 64, and a single

dummy variable for ages 65 and older. After fitting parameters, we obtained the predicted

values seen on the graph: An increasing probability of being without insurance coverage,

peaking at about age 27, curving downward to age 64, with a barely recognizable upturn

prior to age 65. The simple model does not fit the level of uninsured people between ages 0-

64 very well, particularly between the ages of 15 and 40. The “categorical” model fits levels

reasonably well, but at the cost of severely distorting the predicted age pattern, because there

is no within-group slope reflected in the model.

In order to incorporate the best features of both models, we fit an “elaborated” age model.

The goal of the “elaborated” age model is to provide a close fit for each single year of age,

H:\Final FCSM Papers\II-A Popoff.doc 20
while keeping the number of estimated parameters to a minimum. The “elaborated” age

model has the following features:

• An age-specific slope increasing from ages 0 to 15;

• A polynomial curve fit between ages 16 to 30;

• An age-specific slope decreasing from age 31 to 64; and

• A single parameter governing proportion uninsured from ages 65 on (captured in the

intercept in the model fitting process).

The specific form of the regression model fit is as follows:

H:\Final FCSM Papers\II-A Popoff.doc 21

p(Uninsured)

ln = b o + b 1 2 3 Age0 _15 + b Age16 _ 30 + b Agex16 _ 30 + b 4Age216 _ 30 +

1− p(Uninsured)

b 5Age31_ 64 + B7 7 Agex015 + b Agex31_ 64

Where:

Uninsured = person with no insurance coverage for the entire year (1) or having

insurance coverage for some time during the year (0);

Age0_15 = people zero to 15 years old as of December (1) or not (0);

Age16_30 = people 16 to 30 years old as of December (1) or not (0);

Agex1630 = an interaction term, taking the value 0 if the person is outside of ages

16 to 30, and taking their age if they are inside the age group 16 to 30

(age * age16_30);

Age21630 = agex1630 squared, that is, taking the value 0 if the person is outside of

ages 16 to 30, and taking their age squared if they are inside the age

group 16 to 30 (age

2

* age16_30);

Age31_64 = people 31 to 64 as of December (1) or not (0);
Agex015 = an interaction term, taking the value 0 if the person is outside of ages 0

to 15, and taking their age if they are inside the age group 0 to 15 (age

* age0_15); and

Agex31_64 = an interaction term, taking the value 0 if the person is outside of ages

31 to 64, and taking their age if they are inside the age group 31 to 64

(age * age31_64).

The following figure shows the population proportion uninsured, the cubic spline

interpolation, and the predicted values of the fitted model, by age.

--Insert figure 7 about here.--

As can be seen, the fitted values of the elaborated model fit the age pattern of proportions of

uninsured people much better than either the categorical model or the simple model, at a cost

of estimating seven parameters.

Model 1 (“Elaborated age model”)

We shall call the “elaborated age model” Model 1. Model 1 includes no race, ethnicity, or

gender specific effects—it is equivalent to assuming that only age makes a difference in a

person’s probability of being uninsured. As can be seen in figure 7, at the aggregate level

model 1 generates predicted proportions uninsured that are consistent with the population as

a whole. However, model 1 is fitted at the individual level; we present individual level

logistic regression results below.

H:\Final FCSM Papers\II-A Popoff.doc 22
 

 

Number of Observations = 80923

LR chi2(7) = 2403.76

Prob > chi2 = 0

Log Likelihood = -22128.05

Pseudo R2 = 0.0515

Odds Ratio Std. Err. z P>|z| Confidence Interval

Age0_15 14.1306600 2.500667 14.97 0.0000 9.98913 19.98930

Age16_30 0.0266538 0.020274 -4.77 0.0000 0.00600 0.11836

Agex1630 1.9058820 0.125307 9.81 0.0000 1.67545 2.16801

age21630 0.9865926 0.001401 -9.51 0.0000 0.98385 0.98934

Age31_64 60.0980500 11.358960 21.67 0.0000 41.49322 87.04494

Agex015 1.0373600 0.007057 5.39 0.0000 1.02362 1.05128

Agex3164 0.9827579 0.002042 -8.37 0.0000 0.97876 0.98677

Note: The number of observations in 80,923 unweighted people in 1996; Age0_15 is an indicator variable

taking the value one for people 0 to 15 years old as of December and zero otherwise; Age16_30 is an indicator

variable taking the value one for people 16 to 30 years old as of December and zero otherwise; Agex1630 is an

interaction term, taking the value zero if the person is outside of ages 16 to 30, and taking their age if they are

inside the age group 16 to 30 (age * age16_30); Age21630 is Agex1630 squared, that is, taking the value zero

the person is outside of ages 16 to 30, and taking their age squared if they are inside the age group 16 to 30

(age2

* age16_30); Age31_64 is an indicator variable taking the value one for people 31 to 64 as of December

and zero otherwise; Agex015 an interaction term, taking the value zero if the person is outside of ages 0 to 15,

and taking their age if they are inside the age group 0 to 15 (age * age0_15); and Agex31_64 is an interaction

term, taking the value zero if the person is outside of ages 31 to 64, and taking their age if they are inside the

age group 31 to 64 (age * age31_64); Odds ratio is the estimated odds of being uninsured relative to the omitted

group (age 65+); Std. Error is the nonrobust standard error of the estimate; z is the z-score associated with a null

hypothesis that the Odds ratio is one; P>|z| is the probability, under the model, of obtaining a z-score at this

value or higher by chance; confidence interval is the 95% confidence interval around the estimated Odds ratio;

and Pseudo R2

is LL[null]-LL[model])/LL[null], where LL[model] is this model’s log likelihood and LL[null]

is the log-likelihood of the null intercept-only model.

As an individual level predictive model only results in a pseudo R2of .0515 and all estimated

odds ratio coefficients are significantly different from one at the conventional 5 percent level

(two-tailed test). The interpretation of these estimates follows from the curve they are

intended to represent: the odds ratio for the dummy variable Age0_15 implies that people

aged 0 to 15 are about an expected 14 times more likely to be uninsured than the reference

group (those 65 and over), holding other effects constant. The coefficient on the Agex015

interaction term implies that a one year increase in age results in an expected 1.04 times

H:\Final FCSM Papers\II-A Popoff.doc 23
increase in the odds of being uninsured. (Thus, for example, a 10 year old person is 14.13 *

1.0410=20.92 times more likely to be uninsured than a person in the reference group, all other

effects held constant.)

Similarly, the odds ratio for the dummy variable Age16_30 implies that people age 16 to 30

have a baseline odds ratio .027 times that of the reference group. The coefficient on

Agex1630 implies that the expected odds of being uninsured increases 1.91 times for each

year of age, for those aged 16 to 30. The odds ratio for the squared term, Age21630, implies

that the expected odds of being uninsured decrease with the square of each year of age, for

those aged 16 to 30. (Note that this is as we would expect with an order-2 polynomial

model—an increase in the proportion of the population who are uninsured, followed by a

local maximum, followed by decrease.)

The estimated odds ratio for the dummy variable Age31_64 indicates that the expected odds

of being uninsured for a person aged 31 are 60.1 times greater than the reference group. The

estimate odds ratio for the interaction term Agex3164 indicate that the odds of being

uninsured decrease past age 31, at an expected decrease of .983 per year, holding other

effects constant. (Thus, for example, a 61 year old person is 60.1 * .98330=35.93 times more

likely to be uninsured than a person in the reference group, all other effects held constant.)

Model 2 - Fully Specified Model

As can be seen in earlier graphs, while the age pattern may be similar, it is inappropriate to

assume that the levels of uninsured people are equivalent for non-whites versus whites,

Hispanics versus non-Hispanics, and males versus females given our tabular results. In

addition, the graphical analysis suggests an interaction between gender and ages 16 through

30. Model 2 also includes the interaction terms of male with the appropriate age categories

already defined as follows:

H:\Final FCSM Papers\II-A Popoff.doc 24
p(Uninsured)

ln = b + b Age0 _15 + b Age16_ 30 + b Age31_ 64 + b Hisp + b Nonwhite 0 1 2 3 4 5 1− p(Uninsured)

+ b Male + b Male_ age1630 + b Male_ age31_ 64 + b Age21630 + b Agex1630 6 7 8 9 10

+ b Agex015 + b Agex31_ 64 + b Male_ agex1630 + b Male_ age21630 11 12 13 14

Where:

Uninsured = person with no insurance coverage for the entire year (1) or

having coverage for some time during the year (0);

Age0_15 = people 0 to 15 years old as of December (1) or not (0);

Age16_30 = people 16 to 30 years old as of December (1) or not (0);

Age31_64 = people 31 to 64 as of December (1) nor not (0);

Hisp = people of Hispanic origin of all races (1) or not (0);

Other = Blacks, American Indian / Alaskan Natives, and Asian / Pacific

Islander (1) or not (0);

Male = males of any age, race and Hispanic origin (1) or not (0);

Male_age1630 = males of any age, race and Hispanic origin inside the age group

of 16 to 30 (1) or not (0);

Male_age31_64 = males of any age, race and Hispanic origin (1) or not (0) within

the age group of 31 to 64;

Age21630 = agex1630 squared, that is, taking the value 0 if the person is

outside of ages 16 to 30, and taking their age squared if they

are inside the age group 16 to 30 (age 

2

* age16_30);

Agex1630 = an interaction term taking the value 0 if the person is outside of

ages 16 to 30, and taking their age if they are inside the age

group 16 to 30 (age * age16_30);

H:\Final FCSM Papers\II-A Popoff.doc 25
Agex015 = an interaction term, taking the value 0 if the person is outside

of ages 0 to 15, and taking their age if they are inside the age

group 0 to 15 (age * age0_15); and

Agex31_64 = an interaction term, taking the value 0 if the person is outside

of ages 31 to 64, and taking their age if they are inside the age

group 31 to 64 (age * age31_64).

Male_agex1630= an interaction term, taking the value 0 if the person is female

and/or outside ages 16 to 30 and taking their age if they are

male and inside the age group 16 to 30 (age * age16_30);

Male_age21630= an interaction term, that is taking the value 0 if the person is

female and/or outside of ages 16 to 30, and taking their age

squared if they are male and inside the age group 16 to 30 (age

* age16_30);

As with Model 1, we present the logistic regression results below.

H:\Final FCSM Papers\II-A Popoff.doc 26
 

 

 

Number of Observations = 80923

LR chi2(14) = 4164.96

Prob > chi2

= 0.0000

Log Likelihood = -21247.45

Pseudo R2

= 0.0893

Odds Ratio Std. Err. z P>|z| Confidence Interval

Age0_15 10.887650 1.933589 13.44 0.0000 7.687148 15.420660

Age16_30 0.431951 0.491948 -0.74 0.4610 0.046345 4.025926

Age31_64 41.087450 7.888657 19.35 0.0000 28.202090 59.860060

Hisp 3.662423 0.114251 41.61 0.0000 3.445205 3.893338

Nonwhite 1.485525 0.047887 12.28 0.0000 1.394571 1.582410

Male 0.952728 0.054953 -0.84 0.4010 0.850888 1.066757

Male_age1630 0.003616 0.005523 -3.68 0.0000 0.000181 0.072175

Male_age31_64 1.324937 0.091133 4.09 0.0000 1.157835 1.516155

Age21630 0.992285 0.002149 -3.58 0.0000 0.988081 0.996506

Agex1630 1.442898 0.144432 3.66 0.0000 1.185855 1.755658

Agex3164 0.986493 0.002070 -6.48 0.0000 0.982444 0.990559

Agex015 1.044741 0.007200 6.35 0.0000 1.030724 1.058947

Male_agex1630 1.689270 0.228495 3.88 0.0000 1.295875 2.202090

Male_age21630 0.989360 0.002891 -3.66 0.0000 0.983710 0.995042

Note: Sample size is 80923 unweighted people in 1996; Age0_15 is an indicator variable taking the value of

one for people 0 to 15 years old as of December and zero otherwise; Age16_30 is an indicator variable taking

the value one for people 16 to 30 years old as of December and zero otherwise; Age31_64 is an indicator

variable taking the value one for people 31 to 64 as of December and zero otherwise; Hisp is an indicator

variable taking the value one for people of Hispanic origin of all races and zero otherwise; Other is an indicator

variable taking the value one for blacks, American Indian / Alaskan Natives, and Asian / Pacific Islanders and

zero otherwise; Male is an indicator variable taking the value one for males of any age, race and Hispanic origin

and zero otherwise; Male_age1630 is an indicator variable taking the value one for males of any age, race and

Hispanic origin inside the age group of 16 to 30 and zero otherwise; Male_age31_64 is an indicator variable

taking the value one for males of any age, race and Hispanic origin (1) within the age group of 31 to 64 or zero

otherwise; Age21630 is Agex1630 squared, that is, taking the value zero if the person is outside of ages 16 to 30,

and taking their age squared if they are inside the age group 16 to 30 (age 2

* age16_30); Agex1630 is an

interaction term taking the value zero if the person is outside of ages 16 to 30, and taking their age if they are

inside the age group 16 to 30 (age * age16_30); Agex015 is an interaction term, taking the value zero if the

person is outside of ages 0 to 15, and taking their age if they are inside the age group 0 to 15 (age * age0_15);

Agex31_64 is an interaction term, taking the value zero if the person is outside of ages 31 to 64, and taking their

age if they are inside the age group 31 to 64 (age * age31_64); Male_agex1630 is an interaction term, taking

the value zero if the person is female and/or outside ages 16 to 30 and taking their age if they are male and

inside the age group 16 to 30 (age * age16_30); and, Male_age21630 is an interaction term, that is taking the

value zero if the person is female and/or outside of ages 16 to 30, and taking their age squared if they are male

and inside the age group 16 to 30 (age * age16_30); Odds ratio is the estimated odds of being uninsured relative

to the omitted group (age 65+); Std. Error is the nonrobust standard error of the estimate; z is the z-score

associated with a null hypothesis that the Odds ratio is one; P>|z| is the probability, under the model, of

obtaining a z-score at this value or higher by chance; confidence interval is the 95% confidence interval around

the estimated Odds ratio; and Pseudo R2

is LL[null]-LL[model])/LL[null], where LL[model] is this model’s log

likelihood and LL[null] is the log-likelihood of the null intercept-only model.

H:\Final FCSM Papers\II-A Popoff.doc 27
 

As with model 1, we can inspect this model at the individual level to examine coefficients

and other regression results. In this regression, the reference group is white, non-Hispanic

females aged 65 and older; the odds ratios represent deviations from the reference group.

Appendix B contains a table that illustrates the contributions of parameter estimates to

particular age and sex groups under this model8.

Coefficients for the age components that are also in model 1 are very similar; thus, we focus

on the new components for model 2. The estimated odds ratio of 3.66 for Hispanics (Hisp)

indicates that Hispanics are 3.66 times more likely to be uninsured at every age group,

holding all other effects constant. Similarly, the estimated odds ratio of 1.49 for other s

(Other ) indicates that other s are 1.49 times more likely to be uninsured at every age group,

holding other effects constant.

The male x age dummies and terms require careful consideration. Relative to the reference

group, males are a fraction less likely to be uninsured, although this coefficient is not

statistically different from one at conventional levels. For the Male_age1630 term, males

begin the age 16 age group very unlikely to be uninsured, holding other effects constant. But

the term on Male_agex16_30 is 1.69, indicating that for every year of age past age 16, a male

is 1.69 times more likely to be uninsured. The coefficient of .989 on the Male_age216_30

term indicates that there is a local maximum to males’ odds of being uninsured, and then the

odds decrease thereafter.

Goodness of fit at the aggregate level

The results of this model can best be seen by comparing aggregate proportions uninsured

with the predicted values for the eight race/Hispanic/sex groups separately. This figure is

presented as figure 8.

--Insert figure 8 about here.--

8

The authors thank Brett O’Hara of the U.S. Census Bureau, who suggested the table in Appendix B.

H:\Final FCSM Papers\II-A Popoff.doc 28
This figure displays the predicted proportion uninsured, in aggregate, for the eight groups

separately, and the obtained proportions uninsured by age. Each group (white and other,

Hispanic and non-Hispanic, males and females) has their own group-specific intercept,

representing differences in level of people uninsured across the age span. For males, we

have fit a male-by-age interaction term, representing the increased likelihood of males aged

16 to 30 to be uninsured, relative to females of the same age.

As can be seen in figure 8, for the first six groups (white non-Hispanic females, white nonHispanic males, white Hispanic females, white Hispanic males, other non-Hispanic females,

and other non-Hispanic males), the aggregate model fit is very reasonable, generating

proportions as they are approximately found in the population. For the last two groups (other

Hispanic females and other Hispanic males), small cell sizes begin to cause great variability:

There are only 368 (unweighted) other Hispanic males in the data set, and only 438

(unweighted) other Hispanic females in the data set. Thus, there are many age cells with

only one or a few observations.

As noted earlier in the paper, the purpose of fitting this regression model is to determine

whether we can use these predicted values to support a county-level synthetic estimates

system. Thus, the estimates will be used at the age/race/sex/Hispanic origin cell level, not

the individual level. Since the estimated model will be used in an aggregate fashion, we

submit that the key question for this method is whether the regression model generates the

correct aggregate proportions uninsured, not whether it fits individual level outcomes.

In order to assess the aggregate question, we calculated the proportion uninsured in the

population for each age, race, Hispanic, and sex cell, and also estimated the proportion

uninsured for that cell under the modeled coefficients. We regressed the population

H:\Final FCSM Papers\II-A Popoff.doc 29
 

proportion uninsured on the modeled proportion uninsured, without an intercept, and report

the regression R2, by group, as a measure of goodness of fit .

9

Group Regression R2

White non-Hispanic Females .954

White non-Hispanic Males .967

White Hispanic Females .943

White Hispanic Males .956

Other non-Hispanic Females .923

Other non-Hispanic Males .920

Other Hispanic Females .621

Other Hispanic Males .713

As can be seen, at the aggregate level, the model generally predicts age-race-sex-Hispanic

origin specific proportions very well, with R2values above .90 in six cases. In the two

smallest groups, the regression R2indicates that the model is not as predictive for those

groups—there is more variability in aggregate proportions uninsured than can be explained

by our modeling.

Goodness of fit at the aggregate level

Conclusions

To summarize, our goal was to evaluate a synthetic estimation technique for use by state and

local entities for much needed small area estimates of policy relevant characteristics. In

particular, the system would have to satisfy seven criteria; replicable, reasonable at the

9

This regression R2takes the model prediction=0 as the null model when calculating R2, which is standard for

no-intercept models. In this particular case, the proportion uninsured persons is quite low for most age groups,

hence prediction=0 is a reasonable null model.

H:\Final FCSM Papers\II-A Popoff.doc 30
substate level, satisfactory for intercensal needs, portable to other areas, expandable to a

variety of characteristics, able to be implemented within local government constraints, and

verifiable. Because the system we chose uses a well-designed and executed survey

representative at the state level and ARSH estimates generated for each local area using

administrative records and standard demographic techniques, it will be within the local

entities' abilities to implement. Thus, the use of synthetic estimation techniques in

conjunction with detailed program data illustrates the potential of such a system for

performing policy-relevant demographic analysis.

In this evaluation, we found that, at the aggregate level, ARSH characteristics generally

predict the proportion of uninsured people in the population very well. This finding gives us

substantial confidence in using such characteristics for a county level synthetic estimates

system.

H:\Final FCSM Papers\II-A Popoff.doc 31
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U.S. Census Bureau (2001). Current Population Survey Annual Demographic Survey, March

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http://www.bls.census.gov/cps/ads/1995/ssampdes.htm.

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heteroskedasticity. Econometrica, 48:817-830.

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Appendix A: A Brief Description of the Synthetic Estimates

System

The notion of synthetic estimation has been well described by such authors as Gonzalez

(1973), Gonzalez and Waksberg (1973) and Gonzalez and Hoza (1978). In the context of

health-related characteristics, the National Center for Health Statistics has performed a

variety of studies, both theoretical and empirical, on the usefulness of synthetic estimation

techniques (Levy, 1971; National Center for Health Statistics, 1977; 1979; Schaible, Brock

and Schnack, 1977). More recently, Reder (1997) developed methods for estimating literacy

proficiencies for populations using the National Adult Literacy Survey, State Adult Literacy

Surveys, Public Use Microdata Sample data, and 1990 Decennial Census tabulations for

counties10.

The basic synthetic estimation equation used here is:

xˆ = P ×mˆ

a . ,r,s,h a,r,s,h a,r,s,h

Where,

a ∈{0,...,85+}

r ∈{W ,B, API, AI }

s ∈{M ,F}

h ∈{H, ~ H}

and

Pa,r,s,h = the number of people of age a, race r, sex s, and ethnicity h;

10 The approach presented here builds directly on Reder’s work. Though the methods differ in particulars, the

notion of using survey data linked with local area ARSH characteristics is taken directly from Reder (1994;

1997).

H:\Final FCSM Papers\II-A Popoff.doc 35
mˆ

a,r,s,h = the proportion of people of age a, race r, sex s, and ethnicity h that have the

characteristic of interest (in this case, who are uninsured), mˆ

a,r,s,h ∈[0,1];

xˆ

a,r,s,h = the number of people of age a, race r, sex s, and ethnicity h that have the

characteristic of interest (in this case, who are uninsured).

The model applies an estimated group-specific proportion who are uninsured to known

age/race/sex/Hispanic numbers, to estimate the number of uninsured people of the specific

group.

The simplest form of a synthetic estimate is, of course, to simply multiply an overall

population by an overall proportion to get an overall number. “Dot” notation describes the

method within the framework of developing insurance estimates. For any variable ya,r,s,h , we

define: ya,•,s,h = ∑ ya,r,s,h.

r

Other sums are similary defined if a, s, or h are “dotted,” when we “dot” a subscript, it

merely indicates summation over all elements of that subscript, or, using more common

language, “collapsing” that margin.

Using Pa,r,s,h as an example, the total population of an area is equivalent to “collapsing” all

margins, or:

Total Population = P•,•,•,• = ∑∑∑∑Pa,r,s,h.

a r s h

H:\Final FCSM Papers\II-A Popoff.doc 36
Multiplying the total population by some overall proportion of people uninsured in the

population, is expressed in this notation as:

Total Population Uninsured = xˆ = P•,•,•,• ×mˆ

•,•,•,• .

• ,• ,•,• 

However, the ARSH-synthetic method proposed here does make the simple “overall”

calculation above. Instead, we propose making the multiplication cell-by-cell, and then

summing, rather than summing and then multiplying as above. Therefore, our estimating

equation is:

H:\Final FCSM Papers\II-A Popoff.doc 37

Total Population Uninsured = ∑∑∑∑ xˆ = ∑∑∑∑ Pa,r,s,h ×mˆ

, , ,

.

a r s h a,r,s,h

a r s h a r s h
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Appendix B: A Guide to Interpreting the Parameter Estimates of

Model 2

Model 2 contains a complex set of interaction terms intended to represent various effects.

The table below, expressed in terms of the additive parameters of a non-exponentiated logit

model, illustrates the impact of each term on the specified age/sex group. Because race and

Hispanic groups are represented as individual dummy variables and do not interact with age

and sex, we break out only age and sex in this table.

Age 

Sex Parameters associated with that age/sex group (expressed on the additive logit scale)

Age 0-15:

Male ß0 + ß1 + ß6 + ß11*Age

+ ß4

*Hisp

+ ß5

*Nonwhite

Female ß0 + ß1 + ß11*Age

+ ß4*Hisp

+ ß5*Nonwhite

Age 16-30:

Male ß0 + ß2 + ß6 + ß7 + ß10*Age + ß9*Age2 + ß13*Age + ß14*Age2

+ ß4*Hisp

+ ß5*Nonwhite

Female ß0 + ß2 + ß10*Age + ß9*Age2

+ ß4*Hisp

+ ß5*Nonwhite

Age 31-64:

Male ß0 + ß3 + ß6 + ß8 + ß12*Age

+ ß4*Hisp

+ ß5*Nonwhite

Female ß0 + ß3 + ß12*Age

+ ß4*Hisp

+ ß5*Nonwhite

Age 65+:

Male ß0 + ß6

+ ß4

*Hisp

+ ß5*Nonwhite

Female ß0

+ ß4*Hisp

+ ß5

*Nonwhite

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