Sampling Errors for SESTAT

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Understanding Sampling Errors

Generalized Variance Functions

Calculating Standard Errors

Generalized Variance Functions: A Methodology for Estimating Standard Errors

A generalized variance function (GVF) is a mathematical model that describes the relationship between a statistic (such as a population total) and its corresponding variance. GVF models are used to approximate standard errors for a wide variety of estimates of characteristics of the target population.

GVF Modeling

GVF modeling consisted of two steps:

(a) calculating population totals and their variances directly for a small subset of the survey items, and

(b) modeling the relationship between the survey-derived totals and their associated variances.

Step 1 - Calculating population totals and their variances

For direct calculation of the variance (Step 1), a successive differences method or a resampling method such as random groups, balanced repeated replication, or jackknife replication might be used. Direct variance estimation techniques used in each survey are described in Calculating Standard Errors.

Step 2 - Modeling relationships between survey-derived totals and sampling errors

GVF models (Step 2) use regression modeling techniques and hence are subject to the same limitations of model specification, fit, and estimation as any other model. The principal advantage of the GVF method is that approximations of sampling errors are simplified for the large amount of estimates that are normally generated from a demographic survey with a large number of variables. For SESTAT, GVF models are available for the total population and for selected domains of interest. Analysts can use these models to predict the variance for a particular statistic by inserting the value of the statistic into the model for the appropriate domain and survey component. The models developed for SESTAT are described in Calculating Standard Errors.

A Methodology Overview

Let denote an estimator of the population total Y. GVF models are usually created for the relative variance of the estimated total, or

where Var( ) is the variance of . The modeling typically begins by assuming that the relative variance of an estimated total is a linear function of the inverse of the total Y being estimated, or

The parameters of the GVF model, and , are unknown and estimated from a subset of all possible survey-derived totals and their variances by some form of least squares regression estimation.

The relative variance of an estimated total can be predicted by evaluating the appropriate GVF model using the estimated values for Y, and . Thus, using the GVF model, the standard error of a specific estimated total can be predicted by inserting the value of the estimated total into the following computational equivalent:

where is the predicted standard error of the estimated total , and and are estimated parameters of and .

The GVF model can also be adapted to estimate the standard error of a percentage. Using the same parameters, the standard error for a percentage can be predicted with this formula:

where is the predicted standard error for a specific estimated percentage , and
is the estimated number of persons in the base of the percentage.

For a useful text with more information on GVFs, see Chapter 5 of Introduction to Variance Estimation, by Kirk Wolter (New York: Springer-Verlag, 1985).