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Small Area Income & Poverty EstimatesModel-based Estimates for States, Counties, & School Districts |
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The 1993 state and county estimates were originally released in April of 1997. Revisions in the 1993 estimates were made in January of 1998. The following documentation outlines our estimation procedure for the modified estimates.
There are several points to consider about our 1993 estimates of poverty for counties:Using counties in the CPS sample. Our use of the CPS implicitly assumes that the counties in the survey sample are representative of those not selected, but this need not be the case. The CPS sample is designed to represent the population, and only incidentally represents counties. The characteristics of some counties guarantee that they are included, e.g., most counties in large metropolitan areas and counties with large populations. More generally, while all counties have a nonzero probability of being included in the sample, some have higher probabilities than others. Further, the probability of selection of a county may be related to its income and poverty level. On the other hand, comparison of regression equations based on 1990 census data for counties in the CPS sample and equations based on all counties indicate remarkably similar results, providing some assurance that the CPS counties are largely representative of all counties.
The survey weights used in estimation at the national level are not appropriate for county-level estimates. The CPS sample design selects some primary sampling units (usually a county or group of counties) to represent a set of counties in the same stratum. The sum of the weights for sample households from such a county estimates the total population of the entire set of counties it represents. Because we want each county in the CPS sample to stand for itself, we have adjusted the weights to make each county self-representing.
Estimation of the model equation. CPS sampling variances are not constant over all counties. We avoid giving observations with larger variances (a great deal of uncertainty) the same influence on the regression as observations with smaller variances (less uncertainty) by, in effect, weighting each observation by the inverse of its uncertainty. Representing this uncertainty requires recognizing that it arises from two sources:
To estimate the lack of fit component, we estimate our model using the 1990 census data and assume that the lack of fit component of residual variance is the same when the same model is fit to the CPS and to the census. Since we do have separate estimates of sampling variance for each observation in the 1990 Census, we use them to estimate the unknown lack of fit component with a maximum likelihood procedure. (See "Appendix C: Accuracy of the Data" in 1990 Census, STF3 documentation.)
Next we fit a regression equation to the CPS data. We assume the sampling variance of the log of the number of poor is inversely proportional to the sample size (in households) and the lack of fit variance is the same as that estimated in the Census regression. We estimate the CPS regression parameters and the two components of CPS variance with a maximum likelihood procedure.
Combining model and direct survey estimates. The final estimates are weighted averages of the model predictions and the direct CPS estimates, where they exist. The two weights for each county add to 1.0 and the weight on the model prediction is computed as the sampling variance divided by the total variance (sampling plus lack of fit) of the direct estimate. Using this technique, the larger the sampling variance of the survey estimate, the smaller its contribution and the larger the contribution from the prediction model. These weights are commonly referred to as "shrinkage weights," and the final estimates as "shrinkage" or "Empirical Bayes" estimates. For counties which are not in the CPS sample, the weight on the model's predictions is one and the weight on the direct survey estimate is zero.
Controlling to State Estimates. Completion of the shrinkage estimates does not produce the final county estimates of the number of poor. The last step in the process is to transform the county estimates from the log scale to estimates of numbers and control them to the state estimates that have been derived independently. In our current approach, a simple ratio adjustment is made to the county level estimates to assure that they add to the state totals. Model-based estimates at the state level are controlled to the national level direct estimates provided by the March 1994 CPS. The estimated standard errors of the county estimates are adjusted to reflect this additional level of control.
Standard Errors and Confidence Intervals. One of the goals of our small area estimation work is to provide estimates of the uncertainty surrounding the estimates of the numbers of poor. The census and model-based estimates shown in the tables are accompanied by their 90 percent confidence intervals. These intervals were constructed from estimated standard errors.
For the model-based estimates, the standard error depends mainly on the uncertainty about the model and the CPS sampling variance. While the variance of the shrinkage weights could also be a significant component of uncertainty about our estimates (if sizeable and ignored we would be underestimating the standard errors), our research indicates that its contribution is negligible. For the census, the standard errors were derived from a set of generalized variance functions that reflect the nature of the census sample design for the long form questionnaire. (For further information, see Quantifying Uncertainty in the Estimates.)
The Model for Total Number of Poor PeopleThe model is multiplicative, that is, the number of poor is modeled as the product of a series of predictors which are numbers (not rates), and unknown errors. To estimate the coefficients in the model, we take logarithms of the dependent and all independent variables. Our choice of a multiplicative model is motivated in part by the fact that the distribution of county numbers of poor has a huge range -- the 1990 Census distribution ranges from zero in some counties to more than a million in the largest county, with a mean of 10,000 -- and it is highly skewed. Taking the logarithm of all variables makes their distributions more centered and symmetrical. It has the effect of diminishing the otherwise inordinate influence of large counties on the coefficient estimates. Another advantage of a multiplicative model is that it makes it plausible to maintain that the (unobserved) errors for every county no matter how large or small, are drawn from the same distribution. All variables in the county regression models for numbers of poor are logarithmic.
The predictor variables in the income year 1993 regression model for the total number of poor people by county are:
(For further information on these variables see Information about Data Inputs.)
The dependent variable is the log of the total number of poor in each county as measured by the three-year average of values from the March 1993-1995 CPS's. The regression predictions, in the log scale, are combined with the logs of the direct CPS sample estimates and then transformed to estimates of the numbers of poor, which are finally controlled to the independent estimates of state totals.
The Model for the Number of Related Children Ages 5 to 17 in Families in Poverty
The estimation model for related children age 5 to 17 in poverty parallels that for all people in poverty in structure. There are five predictor variables:
(For further information on these variables see Information about Data Inputs.)
The dependent variable is the log of the number of poor related children age 5 to 17 in each county as measured by the three-year weighted average of the March 1993-1995 CPS's. The regression predictions, in the log scale, are combined with the logs of the direct CPS sample estimates and then transformed to estimates of the numbers of poor, which are finally controlled to the independent estimates of state totals.
The Model for the Number of Poor People Under Age 18
The estimation model for poor people under age 18 in poverty is quite similar. There are five predictor variables:
(For further information on these variables see Information about Data Inputs.)
The dependent variable is the log of the number of poor persons under age 18 in each county as measured by the three-year weighted average of the March 1993-1995 CPS's. The regression predictions, in the log scale, are combined with the logs of the direct CPS sample estimates and then transformed to estimates of the numbers of poor, which are finally controlled to the independent estimates of state totals.
The Model for Median Household Income
The predictor variables in the regression model to generate the estimates for median 1993 household income by county are:
(For further information on these variables see Information about Data Inputs.)
The dependent variable is the county median household income as measured by the three-year average of the March 1993-1995 CPS's (income for years 1992-1994). Adjustments were made to the March 1993 and 1995 CPS to express incomes in 1993 dollars before the median incomes were computed.