Simpson’s Paradox From Adding Constants in Contingency Tables as an Example of Bayesian Noncollapsibility

Abstract

Although alternatives have long been available, adding a constant to the cells of a contingency table remains common in practice, especially to avoid problems from zero counts or small cells. There is a certain illogic to the practice, however: It injects unfounded prior information about nuisance parameters. As a result, it can induce a form of Simpson’s paradox, producing a point estimate of the target parameter that is outside the interval bounded by the maximum likelihood estimate from the observed table and the null value of the parameter implied by the constants. Furthermore, it can increase apparent evidence against the null hypothesis, even if both the observed data and the added constants fit this null. The paradox can be seen as arising from default independence priors or additive penalty functions. It can be avoided with simple modifications of constants, or by relevant reparameterization. More generally, noncollapsibility over the prior and likelihood may signal a problem with the prior specification when the prior does not penalize the target parameter directly.

Keywords:  :

• Bayesian inference
• Collapsibility
• Conjugate priors
• Penalized likelihood
• Prior distributions
• Shrinkage estimation
• Smoothing