15.2 Convergence Problems in Fitting Linear Mixed-Effects Models . 260

15.2.1 A Simulation Study ..................................... 261

15.3 Model Convergence Issues and Multiple Imputation ............ 263

15.3.1 A Simulation Study ..................................... 265

15.3.2 Case Studies ............................................ 266

15.3.2.1 The Age-Related Macular Degeneration Trial 266

15.3.2.2 Five Clinical Trials in Schizophrenia ......... 271

15.4 A Formal Basis for the Two-Stage Approach .................... 272

Introduction

Evaluating a surrogate endpoint typically requires a large amount of data, and this is particularly so when the contemporary multiple-trial surrogate endpoint evaluation methods are used. The need for a large sample size is an issue in all clinical trials (e.g., an increased study duration and cost, a higher probability of missing values due to study drop-out, and so on), and it is particularly problematic in clinical trials in rare diseases (or in clinical trials in small patient subgroups that are identified within the context of a common disease). Indeed, in rare diseases the number of patients that are available for study participation is typically substantially smaller compared to what is the case in non-rare disease clinical trials. Yet, surrogate endpoints may be particularly useful in rare disease clinical trials (Korn et al., 2013). Indeed, the use of a surrogate endpoint may result in a smaller sample size that is needed to show the effectiveness of a new treatment, because (i) the surrogate endpoint may be an event that occurs more frequently than the true endpoint, and/or because (ii) the treatment effect on the surrogate endpoint may be larger compared to the treatment effect on the true endpoint (in particular when the surrogate endpoint is closer to the treatment in terms of time and the biological mechanism that is being targeted (Korn et al., 2013).

As detailed in Part II, the contemporary multiple-trial surrogate evaluation methods usually involve the fitting of complex hierarchical models. For example, when both S and T are normally distributed endpoints, a linear mixed- effects model (see Eq. (4.1)) is fitted to estimate the trial- and individual-level surrogacy coefficients. When the number of available patients is small, fitting such linear mixed-effects models is often problematic. Indeed, when the data are sparse, the Newton-Raphson (or quasi-Newton)-based iterative process that is typically used to fit the model (i) may not converge at all, or (ii) may converge to values that are close to or outside the boundary of the parameter space. In the latter case, the estimated variance components are close to zero or negative, which may lead to a non-positive-definite variance-covariance matrix of the random effects D (see (4.2)). A non-positive-definite D matrix is not an issue when one is merely interested in the fixed-effect components, because the marginal model can still be used to make valid inferences regarding the fixed-effect parameters as long as the overall V matrix is positive-definite (with V = ZiDZi + Ej, where Z_{i} are matrices of known covariates associated with the random effects (Verbeke and Molenberghs, 2000; West et al., 2007)). However, in a surrogate evaluation context, interest is in the random effects rather than in the fixed effects. In the latter setting, it is paramount that D is positive-definite, as this condition will guarantee that R_{rial} is within the unit interval.

The remainder of this chapter is organized in the following way. In Section 15.2, a simulation study is conducted to examine which factors affect convergence of linear mixed-effects models. Based on the results of these simulations, a multiple imputation (MI)-based strategy that can be used to overcome model convergence problems is detailed in Section 15.3. As an alternative to overcome model convergence issues, a two-stage model-fitting approach can be used (for details, see Section 4.3). Section 15.4 provides a formal basis for the latter approach. Notice that the current chapter focuses on the setting where both S and T are normally distributed endpoints, but similar strategies are conceivable in other settings as well.