Bayesian analysis on meta-analysis of case-control studies accounting for within-study correlation

Abstract

In retrospective studies, odds ratio is often used as the measure of association. Under independent beta prior assumption, the exact posterior distribution of odds ratio given a single 2 × 2 table has been derived in the literature. However, independence between risks within the same study may be an oversimplified assumption because cases and controls in the same study are likely to share some common factors and thus to be correlated. Furthermore, in a meta-analysis of case-control studies, investigators usually have multiple 2 × 2 tables. In this article, we first extend the published results on a single 2 × 2 table to allow within study prior correlation while retaining the advantage of closed-form posterior formula, and then extend the results to multiple 2 × 2 tables and regression setting. The hyperparameters, including within study correlation, are estimated via an empirical Bayes approach. The overall odds ratio and the exact posterior distribution of the study-specific odds ratio are inferred based on the estimated hyperparameters. We conduct simulation studies to verify our exact posterior distribution formulas and investigate the finite sample properties of the inference for the overall odds ratio. The results are illustrated through a twin study for genetic heritability and a meta-analysis for the association between the N-acetyltransferase 2 (NAT2) acetylation status and colorectal cancer.

Keywords: Sarmanov family; bivariate beta-binomial model; exact method; hypergeometric function; meta-analysis; odds ratio.

Figure 1

Histograms of 5, 000 odds ratio samples overlaid on density functions calculated by formulas (1) and (3) under independent prior (left panel) and correlated prior (right panel)

Figure 2

Variance estimates under different correlation coefficients from Sarmanov beta-binomial model (solid lines) and independent beta-binomial model (dotted lines). True standard error (left panel). Model based standard error (right panel). Number of studies is 40.

Figure 3

Prior and posterior distributions of odds ratio under Jeffreys prior, Laplace prior and informative prior (upper panels) and Sarmanov priors (ρ = −0.5 and ρ = 0.5) and Howard prior (lower panels). Odds ratios are defined as the ratio of odds of conviction comparing dizygotic twins to monozygotic twins.

Figure 4

Scatter plot of risks of exposure among cases and controls for Ye and Parry dataset. The area of each circle is proportional to the total sample size of the study. A regression line is overlaid with coefficients estimated by weighted least squared (weights proportional to the total sample size of the study). Left: scatter plot based on all twenty studies. Right: scatter plot based on dataset with the largest study removed.

Figure 5

Posterior distributions of study-specific odds ratios for four studies: Ladero et al., Chen et al., Yoshioka et al., and Butler et al.. The numbers in the legend are the total sample sizes of the studies. Odds ratios are defined as the ratio of odds of having rapid N-acetyltransferase 2 (NAT2) acetylator status comparing those with colorectal cancer to those without.

Figure 6

Forest plot of 20 study-specific and the overall odds ratios with 95% credible intervals. The numbers on the y-axis are the total sample sizes of the studies. Odds ratios are defined as the ratio of odds of having rapid N-acetyltransferase 2 (NAT2) acetylator status comparing those with colorectal cancer to those without.