American Community Survey (ACS)

Small Area Estimation with Administrative Records and Continuous Measurement

Nanak Chand, Charles H. Alexander

U.S. Census Bureau
Washington, DC 20233

Presented to the Annual Meeting of the American Statistical Association (ASA), Chicago, Illinois, August 1996.

This paper reports the results of research and analysis undertaken by Census Bureau staff. It has undergone a more limited review than official Census Bureau publications. This report is released to inform interested parties of research and to encourage discussion.

I. INTRODUCTION

The American Community Survey (ACS) component of the Continuous Measurement program  is designed to provide reliable direct estimates of the various population characteristics for substate areas. For small areas, such as census tracts, it is desirable to improve the ACS estimates by borrowing strength from other areas and other sources of data. In this project, we will develop procedures to derive indirect estimates of characteristics of interest by integrating ACS data with administrative records and the previous census data.

Synthetic estimators which borrow strength from similar areas may be sensitive to the similarity assumption. Regression synthetic estimators based on auxiliary data taken from other sources for the same and similar areas will be less sensitive to this assumption. The composite estimation (Singh, Gambino and Mantel (1994)) combines direct and synthetic estimators, and thus balances the potential bias of synthetic estimators against the instability of the direct estimators. In addition, the procedure may provide estimators with between area variation much smaller than the prior known variance (Spjotvoll and Thomsen (1987)).

However, composite estimators under fixed effect models provide best linear unbiased estimators which reduce to synthetic estimators for areas with small sampling fractions, irrespective of the size of between area variance relative to the within area variance. This limitation is avoided by using models which take into account random area effects (Chand and Alexander (1995), Cressie (1989, 1990, 1992), Datta et al (1992), Ericksen and Kadane (1985, 1987, 1992), Fay (1987), Fay and Herriot (1979), Ghosh and Rao (1994), and Prasad and Rao (1990)).

The paper adapts the small area methods for application to the ACS variables of interest such as proportion of population below poverty level. The applications pertain to developing estimates and their mean squared errors of such proportions for census tracts.

II. ASSUMPTIONS

A large area A is composed of m small areas, i = 1, ..., m. The parameter of interest foris the true population proportion.

A direct estimatorofis available from the ACS. The auxiliary data are available from administrative records and from previous censuses for each. These data are related to.

The transformation g is a function of a single variable and has a nonzero and continuous first derivative. Let

.

We consider the small area model,

,

whereandare mx1 vectors,represents random area effects,represents random sampling errors, andhas a multivariate normal distribution. X is a mxs design matrix andis a sx1 vector of unknown parameters. and are statistically independent. Letandbe mxm diagonal matrices with the (i, i)th elements respectively equal to and . We also assume that

, and

In this paper, we consider two transformations. The first is the variance stabilization function given by

and the second is the logistic function given by

i = 1,..., m. (Cox and Snell (1989)).

III. EMPIRICAL BEST LINEAR UNBIASED PREDICTORS (EBLUP) AND THEIR MEAN SQUARE ERRORS (MSE)

We consider four estimators of the variance componentunder the model of the previous section. These are the maximum likelihood (ML) estimator, the restricted maximum likelihood (RML) estimator (Cressie (1989, 1992)), the Fay and Herriot (FH) estimator (Fay and Herriot (1979)), and a quadratic moment (QM) estimator (Prasad and Rao (1990) and Ghosh and Rao (1994)).

While the calculations of the first three estimators require iterative solutions, the last one has an explicit solution.

The ML estimators ofandminimize the expression

where V is a mxm diagonal matrix with the (i, i)th element equal to.

The RML estimators of andminimize

The FH estimator ofis obtained by simultaneously solving

, and

The QM estimator ofis given by

where is the ordinary least square estimator ofgiven by

,

andis the ith row of the design matrix X. Withestimated by one of the above four methods, let be the best linear unbiased estimator ofgiven by

,

where U is the mxm matrix obtained from V by replacingby its estimator.

The measure of uncertainty in the model relative to the total variance is defined as the ratio of the variance component of the random area effects to the total variance, and is given by

, i = 1, ..., m.

The regression synthetic estimator of the vector of outcome variables is the product of transpose of the design matrix and the best linear unbiased estimator of the vector of unknown parameters. Thus the regression synthetic estimator ofis .

The EBLUP of the outcome variable is the weighted average of the transformed direct ACS estimate and the regression synthetic estimator, the weight being the estimated measure of uncertainty in the model.

Thus the EBLUP ofis given by

,

whereis the value ofwhenis replaced by its estimator.

The corresponding estimatorofis taken asfor the variance stabilization model and asfor the logistic model.

The MSE of the EBLUP, defined as the expected value of its squared deviation from the true value, consists of three parts. Part one is the sampling error variance times the measure of uncertainty in the model relative to the total variance. The second part is due to estimating the unknown parameters in the model. The third part is due to estimation of the variance component of the random area effects.

The MSE of(Cressie (1992), Kackar and Harville (1984), and Ghosh and Rao (1994)) is given by

whereis the asymptotic variance ofand

An approximately unbiased estimator of(Prasad and Rao (1990)) is given by

This estimator of MSE, using the moment estimators of, is valid under moderate nonnormality of the random effects

IV. ADJUSTMENT OF EBLUP ESTIMATORS

Since ACS is designed to provide unbiased estimates for large areas, we make an adjustment to the EBLUP estimators for eachsuch that an appropriately weighted sum of these adjusted estimators equals the ACS estimate for the large area.

Letbe the ratio of the base population in with respect to, to the total base population in A.

Then the ACS estimate for A is the weighted sum of the ACS estimates forwith weightsi = 1, ..., m.

We define the modified EBLUPofin the following steps:

This modification is similar to the one suggested by Battese, Harter, and Fuller (1988). Their model assumes that element- specific auxiliary data are available for each
Defining for i=1, ..., m,

being MSE of, we have,

If we thus define

it follows that

This derivation ofdoes not require element-specific data.

V. ESTIMATION OF PROPORTION OF PERSONS BELOW POVERTY LEVEL

We illustrate the above estimation procedures by taking {} as the census tracts in Alameda County, California.

The direct estimate of the proportion below poverty level inis calculated as the ratio of weighted number of persons below poverty level to the total weighted ACS population, simulated from the 1990 census long form data. The function g is chosen as described in Section II. The sources of auxiliary data are the simulated administrative records data such as income of tax filers in the tract, and the census data such as number of persons with hispanic origin.

For the logistic model, the design matrix X is defined with s = 4 as

where, for area , is the base population, is the number of persons with a college degree, is number of persons with hispanic origin, and is the simulated median income of tax filers, i = 1, ..., m.

For the variance stabilization model, the design matrix is defined with s = 4 as

The variance componentsare estimated by the Jackknife method using the VPLX program (Fay (1990)).

There are a total of 291 tracts in the above ACS sample for Alameda County, giving m = 291. The suitability of the assumed models is verified by demonstrating that the standardized residuals are approximately normally distributed with mean zero and variance one.

VI. A COMPARISON OF THE VARIANCE COMPONENT ESTIMATION METHODS

The four estimation methods, when applied to the Alameda County data, gave the following estimates of.

Variance Stabilization Model (VSTM)
	RML	ML	FH	QM
	.0688	.0678	.0696	.0710
Logistic Model (LGM)
	RML	ML	FH	QM
	.7416	.7300	.7636	.7960

Tables A1-A2 show the four sets of EBLUP estimators of percent of persons below poverty level along with the weighted ACS estimates, for five of the 291 tracts. The four methods of variance component estimation provide similar results for each of the two models.

Tables B1-B2 show the modified EBLUP estimators of percent below poverty level. An appropriately weighted sum of these estimators equals the ACS estimate of the percent below poverty level for the whole county. This latter percent is equal to 11.01. For comparison, the weighted average of the unadjusted RML for the county is 10.73 under VSTM and is 10.94 under LGM.

Tables C1-C2 give MSE estimates associated with the four EBLUP estimators. The tables show the small levels of MSE of the EBLUP estimators for each of the estimation methods.

SIMULATED ACS SAMPLE

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