Convergence Problems in Fitting Linear Mixed- Effects Models

In a surrogate evaluation context, the (S, T) endpoints (level 1) are nested within patients (level 2), and the patients are nested within clinical trials

(or other relevant clustering units; level 3). Given the complex hierarchical structure of the data, it is hardly surprising that convergence problems are frequently encountered in a surrogate evaluation context. To gain more insight into the factors that affect the convergence of hierarchical models, simulations can be useful.

A Simulation Study

Simulation scenarios

Consider the following mixed-effects model that is typically fitted to evaluate surrogacy in the setting when both S and T are normally distributed endpoints (see also Chapter 4):

where p_{s}, p_{T} are the fixed intercepts for S and T, m_{Si}, m_{Ti} are the corresponding random intercepts, a, в are the fixed treatment effects for S and T, and a_{i}, h are the corresponding random treatment effects. Further, (msi, m_{Ti}, a_{i}, 6j) ~ N (0, D), with D an unstructured variance-covariance matrix of the random effects, and (e_{Si}j, ?_{Ti}j) ~ N (0, E) with E an unstructured variance-covariance matrix of the residuals.

Using Model (15.1), data were simulated. In all simulations, p_{s} = 450, p_{T} = 500, a = 300, в = 500, and

yielding R2_{ndiv} = corr (e_{Si}j, ?_{Tij} )^{2} =0.5. Three conditions were varied in the simulations. First, the number of clusters N = {5, 10, 20, 50}. Second, the between-cluster variability (D), which is either large (7 = 1) or small (7 = 0.1) relative to the residual variability (E):

yielding R‘^{2}liai = corr(a_{i}, hi)^{2} = 0.5. Third, the level of imbalance in the cluster sizes n (the rationale to consider this factor is described in more detail in the next paragraph). In the balanced scenario, all cluster sizes were equal, i.e., n = n = 20. In the two unbalanced scenarios, n was determined based on a draw from a normal distribution and rounded to the nearest integer (i.e., Ui = round (U*)). In the low-imbalance scenario, n ~ N (20, 2.Б^{2}). In the high-imbalance scenario, n ~ N (20, 5^{2}). In the balanced scenario, treatment (Z) is also balanced within a cluster. In the unbalanced scenarios, treatment allocation is determined based on a binomial distribution with success probability 0.50.