3.2. Basics of the MAP approach

The MAP approach is based on the assumption of exchangeable parameters in the new trial and historical trials, in which the effects in different studies are different but from the same normal distribution. ²

Suppose that we only account for variation between trials with respect to the intercept β0 of the ANCOVA model. The parameter estimates of the intercept within each historical trial should be approximately normally distributed, so that

where β0j and β^0j are the study‐specific intercept for the jth historical trial and its estimate, and σ0j2 is the variance of β^0j. Then if there are J historical control trials with study‐specific intercepts β01,…,β0J and one new trial with intercept β0*, the MAP approach assumes that these parameters are distributed as

where μ0 is the grand mean, and τ02 is the between‐study variance of the study‐specific means.

For known τ02, the grand mean μ0 is distributed as

where w0j=1σ0j2+τ02 is the inverse‐variance weight for the jth historical trial. For meta‐analysis, inference for μ0 is of primary interest, whereas the MAP approach aims to derive an informative prior for β0* in the new trial, which is given by

The variance of the above MAP prior is the combination of the variance of the grand mean and the between‐study variance. However, the between‐study variance (or heterogeneity) is not known in practice. The estimation of the between‐study heterogeneity can be based on either frequentist (e.g., DerSimonian‐Laird's method, ML/REML) or Bayesian methods. However, it cannot be inferred with great precision if the number of historical trials is limited. ²⁶

In the above paragraphs, the MAP approach is described based on the two‐stage meta‐analytic approach, where the study‐specific estimates are derived from each study in advance, and the MAP prior is then obtained based on the study‐specific estimates. This was done to illustrate the effects of the within‐study variation and the between‐study variation for the parameter of interest in the MAP approach. ²⁷ If the IPD are available, a one‐stage approach that simultaneously analyzes the historical control data can be used to derive the MAP prior.

In addition to the proposal of the MAP prior in the design phase of a new trial, the meta‐analytic methodology can be directly implemented in the analysis of new trial data, which is called the meta‐analytic‐combined (MAC) approach. ²⁸