HB Methods for Combining Estimates from Multiple Surveys

Transcription

1 Hierarchical Bayesian Methods for Combining Estimates from Multiple Surveys Adrijo Chakraborty NORC at the University of Chicago January 30, 2015 Joint work with Gauri Sankar Datta and Yang Cheng

2 Outline Introduction 1 Introduction 2 3 Model with covariate 4

3 Outline Introduction 1 Introduction 2 3 Model with covariate 4

4 Sometimes multiple surveys are conducted to estimate a characteristic. Estimates of this characteristic from different surveys could be combined to provide an overall, and hopefully better estimate. Estimates obtained from different surveys may not always agree to each other.

5 In order to estimate the number of occupied housing units (households), many surveys are conducted by the United States Census Bureau. Difference among the survey estimates are noticeable in the following table. Table: Estimates of households, obtained in different surveys (numbers in 1000s). Survey CPS/ASEC HVS ACS AHS

6 Estimates from CPS/ASEC are consistently high over the years and estimates from Housing Vacancy Survey and American Housing Survey are typically low Table: Estimates of households, obtained in different surveys (numbers in 1000s). Survey CPS/ASEC HVS ACS AHS

7 Table: Estimates of households (numbers in 1000s). Survey 2007 CPS/ASEC HVS ACS AHS

8 Table: Standard errors of the estimates obtained in different surveys (numbers in 1000s). Survey CPS/ASEC HVS ACS AHS

9 Outline Introduction 1 Introduction 2 3 Model with covariate 4

10 Let, h t = Number of households in year t. y it is the estimate of h t from i th survey, i = 1,..., 4. s it is the estimated standard error of y it. Proposed model M A : y it = h t + e it h t = h t 1 + e t, (1) where, et s are independently distributed with a truncated normal distribution truncated above 0, with variance σe 2 and e it N(0, sit 2 ). We assume uniform priors for the unknown parameters in the model.

11 Table: Posterior standard deviations and the standard errors (numbers in 1000s). Year Proposed method (M A ) CPS/ASEC HVS ACS AHS Posterior sd s.e s.e s.e s.e

12 Estimates CPS/ASEC ACS HB Estimate HB Estimate (log transformation) HVS AHS Estimates CPS/ASEC ACS HB Estimate HB Estimate (log transformation) HVS AHS (a) (b) Figure: (a) Point estimates (b) Uncertainty.

13 Outline Introduction Model with covariate 1 Introduction 2 3 Model with covariate 4

14 Model with covariate Proposed model M B : where, y it = h t + α i + e it h t = h t 1 + e t, 4 α i = 0, α i measures the bias for i th survey. In model i=1 M B, et s are independently distributed with a truncated normal distribution truncated above 0, with variance σe 2 and e it N(0, sit 2). We assume uniform priors for the unknown parameters in the model.

15 Model with covariate Table: Bayesian inference of the bias contrasts based on M B. Posterior Posterior Simulated Quantiles Parameter Mean sd 2.5% Median 97.5% α 1 α α 1 α α 1 α α 2 α α 2 α α 3 α

16 Model with covariate Proposed model M C : y it = h t + α i + e it h t = β 0 + β 1 x t + η t, where η t iid N(0, σ 2 η ), x t is the total population in the United States at year t. As before, We impose an additive constraint model. 4 α i = 0 in the i=1

17 Model with covariate Table: Bayesian inference of the bias contrasts based on M C. Posterior Posterior Simulated Quantiles Parameter Mean sd 2.5% Median 97.5% α 1 α α 1 α α 1 α α 2 α α 2 α α 3 α

18 Model with covariate Table: Bias corrected estimates (rounded) of households from for three different surveys based on model M C (numbers in 1000s). Survey CPS HVS ACS AHS

19 Model with covariate Table: Bias corrected estimates (numbers in 1000s). Survey CPS HVS ACS AHS Table: Original survey estimates (numbers in 1000s). Survey CPS HVS ACS AHS

20 Model with covariate Survey 2007 CPS/ASEC HVS ACS AHS Survey 2007 CPS/ASEC HVS ACS AHS

21 Model with covariate Estimates CPS/ASEC ACS HVS AHS Estimates CPS/ASEC ACS HVS AHS Year Year (a) (b) Figure: (a) Original data (b) After bias correction.

22 Model with covariate Table: HB estimates based on model M B and M C (numbers in 1000s) Estimate Posterior SD Year M B M C M B M C

23 Outline Introduction 1 Introduction 2 3 Model with covariate 4

24 Number of households estimated by different surveys differ considerably, which may create ambiguity among the researchers and impact decisions of the government organizations. We have studied various methods which successfully combine the estimates obtained from different surveys.

25 We have achieved considerable gain in precision using our proposed models. In future, we would like to develop efficient model selection techniques which select the appropriate model for a given data set. These techniques will be helpful for survey researchers particularly when the scope of external evaluations are restrictive.

26 Thank you!

27 Arthur Cresce Jr, Yang Cheng, and Christopher Grieves (2013). Household Estimates Conundrum: Effort to Develop More Consistent Household Estimates Across Current Survey Federal Committee on Statistical Methodology Research Conference. Alan Gelfand and Adrian Smith (1990). Sampling-Based Approaches to Calculating Marginal Densities. Journal of the American Statistical Association, 85, (410) Howard R. Hogan (2012). Reducing User Confusion with Joint Data Releases and User Education Federal Committee on Statistical Methodology Statistical Policy Seminar. Cheng Y., Chakraborty A., Datta G.S. (2014). Hierarchical Bayesian Methods for Combining Surveys Proceedings of the American Statistical Association, Survey Research Methods Section,

28 Collaborators Introduction Yang Cheng, US Census Bureau. Gauri Datta, University of Georgia and US Census Bureau.

29 Collaborators Introduction Yang Cheng, US Census Bureau. Gauri Datta, University of Georgia and US Census Bureau.

30 This presentation is released to inform interested parties of ongoing research and to encourage discussion of work in progress. Any views expressed on statistical, methodological, technical, or operational issues are those of the authors and not necessarily those of the U.S. Census Bureau.