Extending computations for disparity testing when data sources are uncertain

Abstract

The topic of this article is one-sided hypothesis testing on the means of two populations when there is uncertainty as to which population a datum is drawn. Along with each datum, a probability is given as to which of the populations the datum emanated. Such situations arise, for example, in the use of Bayesian imputation methods to assess racial and ethnic disparities with certain survey, health, and financial data. By use of a Bayesian framework and Markov Chain Monte Carlo sampling from the joint posterior distribution of the population means, the probability of a disparity hypothesis is estimated. This approach extends sample size limitations of previous methods given in the literature from a few dozen to well into the thousands. Four methods are developed and compared. Three methods are implemented in R codes and one method in WinBUGS. All the codes are provided in the appendices.

Appendices

Appendix 1: Data for example in Sect. 2, N = 40

This data was created using the following R-Code:

Appendix 2: R code for Laplace calculation in Sect. 2

Appendix 3: R code for MCMC calculations in Sect. 3

Appendix 4: WinBUGS code for MCMC calculations in Sect. 4

Appendix 5: WinBUGS code for MCMC calculations in Sect. 4, common unknown σ

Appendix 6: Some diagnostics from WinBUGS for the model given in “Appendix 5”: Dynamic traces (left), Autocorrelation function (center), and Running Quantiles (right)