Using informative variance priors in connection with an unrestricted variance structure or borrowing strength by assuming exchangeability or imposing consistency between heterogeneity variances, can all ensure sufficiently reliable and realistic heterogeneity estimation, and thus reliable MTC inferences.
While Trt1 and Trt3 consistently received very low rank probabilities (e.g., 0.5% chance of being the best), the probability of Trt2 versus Trt4 being the best treatment varied from 71.3% versus 28.2% with the homogeneous variance model to informed variance model to 43.2% versus 56.8% with the unrestricted variance model (see Table 5).
The empirically informed variances model had the second highest posterior distribution precision, the consistency variances model third, the exchangeable variances model fourth, and lastly the unrestricted variances model produced the most imprecise posterior distributions.
The empirically informed variances model had the second highest posterior distribution precision, the consistency variances model third, the exchangeable variances model fourth, and lastly the unrestricted variances model fifth.
The unrestricted variances model had the widest credible intervals among the heterogeneous variances models, and the informed variances model based on frequentist approximate distributions had the narrowest credible intervals.
Theoretically, the consistency variances model is optimal in that it largely retains the flexibility of the unrestricted variances model, and additionally restricts variances in alignment with and borrows strength from the seminal assumption of consistency.
The two moderately informative models are structurally based on the unrestricted heterogeneous variance model and the variance priors are either frequentistic distribution approximations from within the MTC data or distributions previously derived from a large external empirical data set.
The below four Bayesian modelling approaches are modifications of the unrestricted heterogeneous variance model that apply different parameter value constraints or moderately informative prior distributions to optimize the estimation of the between-trial variance parameters.
The unrestricted heterogeneous variances model places no structural restrictions on the heterogeneity variances.
These models, which included time, treatment, a time-by-treatment interaction, and the respective baseline value as principal explanatory variables, posited an unrestricted structure on the variance-covariance matrix of the residuals for all 123 participants.
The four weakly informative models include the conventional homogeneous variance model, the unrestricted heterogeneous variance model, the exchangeable variances model, and the consistency variances model.
