Incorporating historical control information in ANCOVA models using the meta‐analytic‐predictive approach

Abstract The meta‐analytic‐predictive (MAP) approach is a Bayesian meta‐analytic method to synthesize and incorporate information from historical controls in the analysis of a new trial. Classically, only a single parameter, typically the intercept or rate, is assumed to vary across studies, which may not be realistic in more complex models. Analysis of covariance (ANCOVA) is often used to analyze trials with a pretest–posttest design, where both the intercept and the baseline effect (coefficient of the outcome at baseline) affect the estimated treatment effect. We extended the MAP approach to ANCOVA, to allow for variation in the intercept and the baseline effect across studies, and possibly also correlation between these parameters. The method was illustrated using data from the Alzheimer's Disease Cooperative Study (ADCS) and assessed with a simulation study. In the ADCS data, the proposed multivariate MAP approach yielded a prior effective sample size of 79 and 58 for the intercept and the baseline effect respectively and reduced the posterior standard deviation of the treatment effect by 12.6%. The result was robust to the choice of prior for the between‐study variation. In the simulations, the proposed approach yielded power gains with a good control of the type I error rate. Ignoring the between‐study correlation of the parameters or assuming no variation in the baseline effect generally led to less power gain. In conclusion, the MAP approach can be extended to a multivariate version for ANCOVA, which may improve the estimation of the treatment effect.

2 Results of the sensitivity analysis for the motivating data analysis In the sensitivity analysis, the proposed MAP approach was implemented with different priors for the between-study standard deviations of β 0 and β 1 to assess the method's robustness to different priors of between-study heterogeneity, and the results with different model specifications were also derived.

Different priors for the between-study standard deviations
In this part of the sensitivity analysis, exponential and uniform priors that assign 5% probability for large between-study heterogeneity were considered. Namely, the exponential priors for τ 0 and τ 1 were Exp(0.15) and Exp (3), and the uniform priors for τ 0 and τ 1 were Uniform(0, 21.05) and Uniform(0, 1.05), respectively. The above priors along with the half-normal priors used in the main analysis are visualized in Figure S1.
Despite the same probability assigned to large between-study heterogeneity, the exponential prior assigns more probability to small between-study heterogeneity than the half-normal prior, while the uniform prior provides less probability to small betweenstudy heterogeneity. Hence, the exponential prior express more confidence for a small between-study heterogeneity and the uniform prior is the least informative among the three priors.
6 Figure S1: Different priors for between-study standard deviations of (A) the intercept (τ 0 ) and (B) the baseline effect (τ 1 ) The prior ESS for the intercept was 108, and the prior ESS for the baseline effect was 75 with the exponential priors. While the prior ESS for the intercept and the baseline effect were 72 and 48 respectively based on the uniform priors. The prior ESS for the two parameters based on half-normal priors (79 and 58) were between those with uniform priors and exponential priors. The results were reasonable in that the exponential prior is more informative while the uniform prior is less informative than the half-normal prior.
The MAP priors and the prior ESS are shown in Table S5.
7 The parameter estimates based on the exponential and uniform priors are presented in Table S6. The treatment effect estimate with the exponential priors was -1.27 (SD: 1.16), and the treatment effect estimate based on the uniform priors was -1.28 (SD: 1.18).
The estimates were similar to that based on the half-normal priors, which indicated the inference of the parameter of interest was robust to differently shaped priors (but same tail probability) for the between-study standard deviations.

Different MAP approaches
In the simulation study, there were three other MAP approaches, namely MMAP+IND, UMAP+COM, and UMAP+SEP considered. The results of the abovementioned models are also presented in Table S7 and Table S8.
The prior ESS of the intercept and the baseline effect with the MMAP+IND were larger than those in the MMAP+COR. The UMAP+COM yielded the largest prior ESS for the intercept, while the UMAP+SEP led to the smallest. The results were in line 8 with those in the simulation study. The inference for the treatment effect in the MMAP+IND was similar to that in the MMAP+COR, while the treatment effect estimates of the UMAP+COM and the UMAP+SEP were more deviated from that in the MMAP+COR.

JAGS scripts for the motivating data analysis
In this section, JAGS scripts for 'No borrowing", "Pooling", and 'MMAP+COR" approaches used in the motivating data analysis of Section 5 are presented. "No borrowing" and "Pooling" approaches share the same JAGS scripts, while the "MMAP+COR" approach has its unique JAGS scripts. 9 3.1 JAGS scripts for "No borrowing" and "Pooling" approaches In this subsection, JAG scripts for "No borrowing" and "Pooling" approaches are presented. To implement the two approaches, a single data set is needed (either the new trial data only or the pooled data without study identifier).