Bayesian testing for independence of two categorical variables under two-stage cluster sampling with covariates

ABSTRACT

We consider Bayesian testing for independence of two categorical variables with covariates for a two-stage cluster sample. This is a difficult problem because we have a complex sample (i.e. cluster sample), not a simple random sample. Our approach is to convert the cluster sample with covariates into an equivalent simple random sample without covariates, which provides a surrogate of the original sample. Then, this surrogate sample is used to compute the Bayes factor to make an inference about independence. We apply our methodology to the data from the Trend in International Mathematics and Science Study [30] for fourth grade US students to assess the association between the mathematics and science scores represented as categorical variables. We show that if there is strong association between two categorical variables, there is no significant difference between the tests with and without the covariates. We also performed a simulation study to further understand the effect of covariates in various situations. We found that for borderline cases (moderate association between the two categorical variables), there are noticeable differences in the test with and without covariates.

KEYWORDS:

• Bayes factor
• hierarchical Bayesian model
• Markov chain
• multinomial logistic regression
• surrogate samples
• TIMSS

Acknowledgments

We thank the referees and the Editor for constructive comments, which have led to an improved presentation.

Disclosure statement

No potential conflict of interest was reported by the authors.