Simplified Model-Fitting Strategies
The mixed-effects (hierarchical) modeling approach of Buyse et al. (2000) poses considerable computational challenges. Indeed, fitting a linear mixed- effects model is typically done using Newton-Raphson or quasi-Newton-based procedures (Lindstrom and Bates, 1988; Verbeke and Molenberghs, 2000). Based on some starting values for the parameters of interest, these procedures iteratively update the parameter estimates until convergence is achieved. Unfortunately, non-converging iteration processes may occur when complex linear mixed-effects models are considered. This means that the iterative process does not converge at all, or that it converges to values that are close to or outside the boundary of the parameter space (i.e., variances that are close to zero or even negative). Simulation studies have shown that such problems mainly occur (i) when the number of clustering (trial-level) units is small, (ii) when the size of the between-cluster variability (i.e., the components in the D matrix (4.2)) is small relative to the size of the residual variability (i.e., the components in the Я matrix (4.3)), and (iii) when the number of patients in each of the trials is strongly unbalanced (Burzykowski, Molenberghs, and Buyse, 2005; Buyse et al., 2000; Renard et al., 2002; Van der Elst et al., 2015).
Unfortunately, the conditions described in (i)—(iii) are often encountered in real-life surrogate evaluation settings and thus convergence problems tend to be prevalent. Buyse et al. (2000) and Tibaldi et al. (2003) proposed a number of simplified model-fitting strategies that can be used when model convergence problems occur. In particular, Model (4.1) can be simplified along four dimensions. The simplified model-fitting strategies are discussed in more detail in the subsequent sections, and an overview that summarizes the different strategies is provided in Table 4.1. Alternatively, the multiple-imputation-based strategy, described in Chapter 15, can be used to improve convergence.