ABSTRACT
Consider the situation in which k studies have been conducted to investigate the differences between two treatments, and denote by Yi the summary statistic of study i
(i=1,…, k). For example, Yi may may denote the treatment difference for normal data, the log odds ratio for binary data, the log hazard ratio in a proportional-hazards regression model for survival data, or the odds ratio in a proportional odds model for ordinal categorical data; see for example, Refs 2 and 3. If the precision associated with each individual study estimate is high, or the effect of interest is normally distributed, we may assume:
where wi denotes the (known) precision of the observed estimate, 0,- the treatment effect in study i, and N(a, b) the normal density with mean a and variance b. If we further assume the individual study effects are exchangeable and that there is no systematic bias in the selection of studies, as might arise due to publication bias (Smith et al., Chap. 13), we may model the distribution of θis as a normal density with mean µ, representing the population treatment effect, and variance τ2, representing the extent of between-study heterogeneity:
Model (1) combined with (2) implies that marginally the Yi follow independent
distributions. The inclusion of study-level covariates that contribute to heterogeneity can make the
exchangeability assumption more appropriate. Covariates are formally introduced into the model by replacing (2) with
where is a p-dimensional vector of covariates (with p<k−1) for study i(i=1,…, k), and β is a p-dimensional vector of population regression parameters. We assume that the xij have zero mean so that µ still measures the average treatment effect. Often the effects of covariates are of interest in their own right. For example, in their meta-analysis of nicotine-replacement therapy, DuMouchel and Normand (Chap. 6) analyze the effect of two covariates on smoking cessation: type of therapy and intensity of support.
