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AN APPLICATION OF BAYESIAN METHODS TO SMALL AREA ESTIMATES OF POVERTY RATES
Joey CampbellCorey SparksThe University of Texas at San AntonioDepartment of Demography
INTRODUCTION
Estimates of various socio-demographic variables for small geographical areas are proving difficult with the replacement of the Census long form with the American Community Survey (ACS).
Sub-national demographic processes have generally relied on Census 2000 long form data products in order to answer research questions.
INTRODUCTION
ACS data products promise to begin providing up-to-date profiles of the nation's population and economy
Unit and item level non-response in the ACS have left gaps in sub-national coverage
The result is unstable estimates for basic demographic measures.
PURPOSE
Borrowing information from neighboring areas with a spatial smoothing process based on Bayesian statistical methods
Generate more stable estimates of rates for geographic areas not initially represented in the ACS.
A spatial smoothing process grounded in Bayesian statistics, is used to derive estimates of poverty rates at the county level for the United States.
Data come from two sources US Census 2000 Summary File 3 American Community Survey
2001 – 2005 1-year estimates 2005 – 2007, 2006 – 2008 3-year estimates 2005 – 2009 5-year estimates
U.S. Counties N=3,141 (Continental) 2000 Census is missing poverty rates for 0 counties ACS is missing poverty rates for up to 3,123 counties for
some years Primarily due to small population sizes of counties
ESTIMATING COUNTY-LEVEL RATES
Bayesian Statistics Combines observed data with prior
information to “strengthen” estimates for parameters of interest
Allows posterior estimation of these parameters using likelihood and prior information
METHODS: BAYESIAN HIERARCHICAL MODEL
Bayesian Statistics Uses Prior information for estimation of parameters of
interest Allows for posterior estimation of these parameters using the
combination of the information in the likelihood and the prior
Hierarchical Modeling Bayesian Hierarchical Model Allows for a spatially and temporally smoothed estimate of
rates Draws “strength” from neighboring observations Estimated with WinBUGS via Markov–Chain Monte Carlo
methods 100,000 simulations with 20,000 burn in period
THE MODELS yi ~ bin(pi, ni) logit(pi) = μ0 + Ai + Bj + Cij
THE MODELS yi ~ bin(pi, ni) logit(pi) = μ0 + Ai + Bj + Cij
Overall rate
The spati
al grou
p
THE MODELS yi ~ bin(pi, ni) logit(pi) = μ0 + Ai + Bj + Cij
Overall rate
The time grou
p
The spati
al grou
p
THE MODELS yi ~ bin(pi, ni) logit(pi) = μ0 + Ai + Bj + Cij
Overall rate
The spati
al grou
p
The space -
time grou
p
The time grou
p
THE MODELS yi ~ bin(pi, ni) logit(pi) = μ0 + Ai + Bj + Cij
Overall rate
THE MODELS
yi ~ bin(pi, ni) logit(pi) = μ0 + Ai + Bj + Cij
Summary of Model Specification
Spatial Terms
Temporal Terms
Space-time
Terms
Model Ai Bj Cij
1 vi + ui βtj 0
2 vi + ui tj 0
3 vi + ui tj + ξj 0
4 vi + ui tj ψij
5 vi + ui tj + ξj ψij
6 vi + ui tj ψijEach model was evaluated with respect to how it recreated the overall poverty rate, the known time trend, and the known spatial distribution
RESULTS: OVERALL POVERTY RATE
The overall estimate of U.S. poverty in 2001 according to SAIPE = 13.74 percent. Model 1 = 13.97 percent Model 2 = Model 3 =13.96 percent Model 4 = Model 5 = 14.15 percent, and Model 6 = 14.17 percent.
Overall, the Bayesian models produce similar rates of those estimated by more traditional methods.
RESULTS: ERROR RATES
Mean Absolute Percent Error (MAPE) Rates for Bayesian Estimates of US County Poverty Rates compared to SAIPE
Model 2001 2002 2003 2004 2005 2006 2007 2008 2009 Total
1 11% 10.6%
10.8%
11.2% 8.8% 9.3% 9.8% 10.9
%13.1%
11.5%
2 10.5% 10% 10.4
%11.1% 8.2% 9.1% 9.8% 11% 13% 10.3
%
3 10.5% 10.4% 10.4% 11.1% 8.2% 9.1% 9.8% 11.0% 13.0% 10.3%
4 10.6% 11.8% 12.0% 13.1% 10.3% 11.0% 10.6% 11.1% 11.7% 11.3%
5 10.6% 11.8% 12.0% 13.1% 10.3% 11.0% 10.6% 11.1% 11.7% 11.3%
6 10.7% 11.9% 12.2% 13.2% 10.3% 11.0% 10.5% 10.8% 11.8% 11.3%
DISCUSSION
Although the estimates of various socio-demographic variables in the ACS have improved over time, progress is not as fast as expected
Local level efforts have been advocated to help combat various outcomes associated with poverty.
Consequently, reliable estimates for small areas are necessary for these efforts to move forward
DISCUSSION
The Bayesian approach has been demonstrated to produce reliable and dependable estimates by borrowing information both across time and from neighboring counties
Hopefully these estimates (and this method) can be employed to effectively understand how socio-demographic variables vary at the local level
Additionally, models may be formulated that incorporate ACS errors directly (Bayesian SEM)
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