A Bayesian Approach to Account for Misclassification and Overdispersion in Count Data

Abstract

Count data are subject to considerable sources of what is often referred to as non-sampling error. Errors such as misclassification, measurement error and unmeasured confounding can lead to substantially biased estimators. It is strongly recommended that epidemiologists not only acknowledge these sorts of errors in data, but incorporate sensitivity analyses into part of the total data analysis. We extend previous work on Poisson regression models that allow for misclassification by thoroughly discussing the basis for the models and allowing for extra-Poisson variability in the form of random effects. Via simulation we show the improvements in inference that are brought about by accounting for both the misclassification and the overdispersion.

Keywords: count data; misclassification; overdispersion.

Figure 1

The naive baseline model: the number of deaths due to cancer ($y_{1i}$) and non-cancer ($y_{2i}$) follow a Poisson distribution with constant parameters.

Figure 2

The no-misclassification Poisson regression model.

Figure 3

The Poisson regression model with misclassification; (a) The graphical model representation of the model. $se$ denotes the sensitivity of the classifier, and $sp$ denotes its specificity; (b) The contingency table representation of the data. $y_{1i}$ ($y_{2i}$) is the true number of deaths due to lung cancer (non-lung cancer). $u_{1i}$ ($u_{2i}$) is the number of true number of lung cancer (non-lung cancer) deaths misclassified. $w_{1i}$ ($w_{2i}$) is the observed number of deaths due to lung cancer (non-lung cancer). Note that C and $\overline{C}$ denote correctly classified and misclassified, respectively.

Figure 4

The random-intercept Poisson regression model with misclassification.

Figure 5

Posterior means and 95% credible sets for sensitivity analysis of ${\beta }_1$ (with true value 0.5).

Figure 6

Posterior means (a) and coverage rates (b) for γ; $se=0.75.$

Figure 7

Posterior means (a) and coverage rates (b) for β; $se=0.75.$