Formulas for School District Estimators and Evaluation Statistics
The generic form of the school district estimators is as follows Let be the quantity of interest (poor 5-17, population 5-17, or total population) for school district j in county i. Also define
For a school district whose boundaries cross county lines, the subscript j corresponding to that district for county i will indicate the piece of the school district in county i. The ij thus can denote either pieces of school districts or complete school districts that are contained in county in county i. When a school district is split across two or more counties, the estimate for that district is obtained by summing estimates for all of its pieces obtained from (1) across all the counties i that it intersects. For simplicity of discussion, this complication will be be ignored in what follows.
For the 1995 school district estimates, come from the 1990 census, and the other quantities refer to income year 1995 (for 5-17 poverty) or to calendar year 1996 (for population). For the evaluation of 80-based school district estimates against the 1990 census results, in (1) come from the 1980 census (and will be written to distinguish this), and the other quantities refer to income year 1989 (for 5-17 poverty) or the calendar year 1990 (for population), as do the 1990 census results. In fact, for purposes of evaluation we make the assumption that
Of course, this assumption is untrue as the census poverty figures are themselves estimates containing nontrivial amounts of sampling error (for small to moderate size school districts), and both poverty and population figures can contain varying amounts of nonsampling error.
Alternative Estimates for Comparison in Evaluations
Two alternative estimates are considered for purposes of evaluation against 1990 census results. The census county-based estimate replaces in (1) by the 90 census estimate for county i, This yields
This "estimate" in not usable for 1995, of course. It is included in the evaluations merely for purposes of comparison of its performance to that of the model-based estimator (1). Note i.e., the estimates from (3) aggregated to counties agree with the 90 census results, which are being used as the standard of comparison. Thus, evaluating the Census county-based estimator (2) shows, to the extent these evaluations can examine this, the conributions to error of change from 1980 to 1990 in the school-district-to-county ratios Note that if these ratios don't change from 1980 to 1990, comes out to be the standard.
The naive estimate replaces the county i census growth ratio in (3), with the corresponding ratio for the total U.S.:
This estimate changes the previous census school district estimates only by the crude measure of growth at the national level . This growth ration refers to total population for evaluating estimates of same, or to 5-17 population for evaluating estimates of 5-17 population or poor children 5-17.
The same evaluation statistics are computed for all three estimates (model-based, Census county-based, naive). Explicit formulas will be given here only for the model-based estimates
The mean algebraic percent error is
where sums over all districts involved in the particular comparison, and n (=) is total number of such school districts. The number n varies over the various evaluations shown, which include groupings of school districts into categories according to various characteristics (e.g., size). The MALP is a measure of bias (relative to the 90 census).
The mean absolute percent error is
where and n are defined as for (5). The MAPE is a measure of accuracy (with respect to the 90 census results).
The above two measures are unweighted statistics. Corresponding weighted statistics are
where for all ij that The following weights were used:
children 5-17 in the 1990 census for school district in j in county i.
The same weights are used for all three sets of evaluations (poor 5-17, 5-17 population, and total population). Later evaluations will examine use of different sets of weights, e.g., weights proportional to 5-17 poor for evaluating the 5-17 school district poverty estimates.