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For the 2010 estimates, the only change in methodology was the incorporation of population estimates based on the decennial 2010 counts. Control populations used for the 2010 release of the American Community Survey (ACS), all population-based inputs, and the denominator for the published SAIPE poverty rates are derived from these decennial 2010 counts.
Estimation of the Model Equation
ACS sampling variances are not constant over all counties. When calculating model parameters, we avoid giving observations with larger variances (a great deal of uncertainty) the same influence on the parameter estimates 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 the residual variance by a maximum likelihood procedure. Next we estimate the ACS regression parameters using the variance components as observational weights with a maximum likelihood procedure.
Combining Model and Direct Survey Estimates
Final estimates are weighted averages of direct ACS estimates, where they exist, and the model predictions. The two weights for each county add to 1.0, and we compute the weight on the model prediction as the sampling variance divided by the total variance (sampling plus lack-of-fit) of the direct estimate. With this technique, the larger the sampling variance of the direct 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 that have zero poor children in sample, the weight on the model's predictions is 1.0 and the weight on the direct survey estimate is zero.
Controlling to State Estimates
The last steps in the production process are transforming the county estimates from the log scale to estimates of numbers and controlling them to the independently derived state estimates. We make a simple ratio adjustment to the county-level estimates to ensure that they sum to the state totals. We control model-based estimates at the state level to the national level direct estimates derived from the ACS. We adjust the estimated standard errors of the county estimates to reflect this additional level of control. We do not control estimates of county median household income to the state medians. This would require that the estimation model produce the entire household income distribution, rather than just the median as it does now.
The estimates for the number of school-aged children in poverty are handled slightly differently. The Department of Education, a major sponsor of the SAIPE program, requires that the estimated number of school-aged children in poverty be integers. We use an algorithm to round the counties' estimates in a way that forces the sum of the estimates of school-aged children in poverty for the counties to sum to the estimate for the state. Note that this algorithm is first applied to the states' estimates, so they are integers and add to the integer-valued national estimate.
Standard Errors and Confidence Intervals
One goal of our small area estimation work is to provide estimates of the uncertainty surrounding the estimates of the numbers of people in poverty. The 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 ACS 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.
Predictor and Dependent Variables
For 2010, the dependent variable is based on the single-year 2010 ACS sample. The predictor variables described below use aggregated 2009 federal tax filings, SNAP benefits, Bureau of Economic Analysis (BEA) data and 2010 population estimates. SNAP stands for Supplemental Nutrition Assistance Program, which was known as the federal Food Stamp Program prior to October 1, 2008. For further information on these variables see Information About Data Inputs.
The Model for Total Number of People in Poverty
The model is multiplicative; that is, we model the number of people in poverty as the product of a series of predictors that are numbers (not rates), and we model the unknown errors. To estimate the coefficients in the model, we take logarithms (logs) of the dependent and all independent variables. Our choice of a multiplicative model is motivated, in part, by the fact that the distribution of the number in poverty has a huge range -- from zero in some counties to more than a million in the largest county (with a mean of 10,000), based on Census 2000 -- and the distribution is highly skewed. Taking the log of all variables makes their distributions more centered and symmetrical and 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.
The predictor variables in the regression model used to estimate the total number of people in poverty are:
The dependent variable is the log of the total number of people in poverty in each county as measured by the ACS. We combine the regression predictions, in the log scale, with the logs of the direct ACS sample estimates, and then transform the results into estimates of the numbers of people in poverty. Finally, we control the estimates 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 ages 5 to 17 in poverty parallels that for all people in poverty in structure. There are five predictor variables:
The dependent variable is the log of the number of related children in poverty ages 5 to 17 in each county as measured by the ACS. We combine the regression predictions, in the log scale, with the logs of the direct ACS sample estimates, and then transform the results into estimates of the numbers in poverty. Finally, we control the estimates to the independent estimates of state totals.
The Model for the Number of People Under Age 18 in Poverty
The estimation model for people under age 18 in poverty is quite similar. There are five predictor variables:
The dependent variable is the log of the number of people in poverty under age 18 in each county as measured by ACS. We combine the regression predictions, in the log scale, with the logs of the direct ACS sample estimates, and then transform the results into estimates of the numbers in poverty. Finally, we control the estimates to the independent estimates of state totals.
The Model for Median Household Income
Like the models for the number of people in poverty, the model for median household income is multiplicative. A consequence of the multiplicative form, when the model performs well relative to the direct ACS estimates of median household income, is that the standard errors of the estimates are proportional to the point estimates. In other words, the unobserved errors associated with high-income counties are larger than the unobserved errors in counties with high proportions of people in poverty. To estimate the model, we take logs of the dependent and all independent variables; i.e., the model is linear in logs. However, we report median household income in the linear scale and, as a result, the confidence intervals are asymmetric. The predictor variables are:
We define the nonfiler rate as the ratio of estimated total population minus total exemptions claimed on IRS tax returns to estimated total population.
The dependent variable is the log of county median household income interpolated with the ACS survey.