This document presents a study that uses Bayesian statistical methods to generate more stable estimates of poverty rates for US counties using data from Census 2000 and the American Community Survey (ACS). The authors develop 6 Bayesian hierarchical models to estimate county-level poverty rates. The models incorporate spatial, temporal, and space-time random effects. The results show that the Bayesian models produce poverty rate estimates similar to traditional methods, with average errors between 10-11% compared to accepted estimates. The authors conclude the Bayesian approach provides reliable small area estimates by borrowing information across time and neighboring counties.

1. AN APPLICATION OF BAYESIAN METHODS TO
SMALL AREA ESTIMATES OF POVERTY RATES
Joey Campbell
Corey Sparks
The University of Texas at San Antonio
Department of Demography

2. 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.

3. 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.

4. 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.

5.  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

6. ESTIMATING COUNTY-LEVEL RATES


7. 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

10. THE MODELS
 yi~ bin(pi, ni)
 logit(pi) = μ0+Ai+Bj+ Cij

11. THE MODELS
 yi~ bin(pi, ni)
 logit(pi) = μ0+Ai+Bj+ Cij
Overal
l
rate

12. THE MODELS
 yi~ bin(pi, ni)
 logit(pi) = μ0+Ai+Bj+ Cij
Overal
l
rate
The
spatial
group

13. THE MODELS
 yi~ bin(pi, ni)
 logit(pi) = μ0+Ai+Bj+ Cij
Overal
l
rate
The
spatial
group
The
time
group

14. THE MODELS
 yi~ bin(pi, ni)
 logit(pi) = μ0+Ai+Bj+ Cij
Overal
l
rate
The
spatial
group
The
time
group The
space
-time
group

15. THE MODELS
 yi~ bin(pi, ni)
 logit(pi) = μ0+Ai+Bj+ Cij
Summary of Model Specification
Spatial Temporal Space-time
Terms Terms 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 ψij
Each model was evaluated with respect to how it
recreated the overall poverty rate, the known time trend,
and the known spatial distribution

16. 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.

20. 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
10.6
1 11% 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%

21. 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

22. 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)