A Longitudinal (Normally Distributed) and a Failure-Time Endpoint

Tomasz Burzykowski

Hasselt University, Belgium, and IDDI, Belgium

Introduction

In this chapter, we focus on the case when the surrogate is a continuous, normally-distributed endpoint measured repeatedly over time and the true endpoint is a failure-time endpoint. Technically, the evaluation of the candidate surrogate requires a joint model for longitudinal measurements and failure-time data. Methodology for such joint modeling has been considerably developed in recent years (see, e.g., the recent monograph by Rizopou- los, 2012). The proposed joint models couple the failure-time model, which is usually of primary interest, with a suitable model for the longitudinal measurements of a covariate. We follow the meta-analytic approach developed by Renard et al. (2003), which uses the joint model proposed by Henderson, Diggle, and Dobson (2000).

Theoretical Background

Assume that the true endpoint T is a failure-time random variable and the surrogate S is a normally distributed, continuous variable that is repeatedly measured over time. In particular, for each of j = 1,... ,n_{i} patients from trial i (i = 1,..., N) we have measurements s_{i}j_{k} (k = 1,..., n_{i}j) made at

each patient we have a vector of data (X_{i}j, A_{i}j, sj, тj ,Z_{i}j), where X_{i}j is a possibly censored version of survival time T_{i}j, A_{i}j is the censoring indicator, assuming value of 1 for observed failures and 0 otherwise, and Z_{i}j is the treatment indicator.

To evaluate a longitudinal surrogate S, Renard et al. (2003) applied the joint model proposed by Henderson, Diggle, and Dobson (2000). In what follows, we briefly describe the model.

A central feature of the model is an unobserved (latent) zero-mean bivariate Gaussian process, W_{i}j (t ) = {W_{1},_{i}j (t ),W_{2ji}j (t )} used to induce the association between the longitudinal measurement and event processes. The measurement and intensity models are linked as follows:

(1) The sequence of measurements {s_{i}j_{k} : k = 1,..., n_{i}j} of a subject is modeled using a standard linear mixed-effects model (LMM), possibly allowing for a serially correlated component:

where p_{i}j (T_{ijk}) describes the mean profile and

is a sequence of mutually independent measurement errors. In (8.1) we have explicitly indicated the dependence of the mean response profile p_{i}j on a_{i}, which is a vector of parameters for the trial-specific treatment effects used in modeling the mean response profile. We will discuss the composition of a_{i} in what follows.

(2) The event intensity process is modeled by using a semi-parametric proportional-hazards (PH) model:

where the form of Ao(t) is left unspecified and Yj (t) is a binary indicator of whether a subject is at risk of experiencing an event at time t. The parameters в_{0}у and в represent, respectively, trial- specific overall effects and treatment effects on the hazard function.

W_{1i}j and W_{2ji}j can be specified in various forms. For example, suppressing the indices for notational simplicity, we can assume:

with (Ui,U_{2}) being normally distributed with mean zero and variance- covariance matrix G. This implies the use of a model with random intercepts and slopes for the longitudinal surrogate. Then, W_{2}(t) can be defined by including distinct elements of W_{1}(t) and/or its entire current value. For instance, we can specify that:

The model can be estimated by using the EM algorithm (Henderson, Diggle, and Dobson, 2000; Renard et al., 2003).

In the meta-analytic approach proposed by Renard et al. (2003), model (8.1)-(8.2) is fitted to data from all trials, providing estimates of a; and в_{г}. The quality of the surrogate at the trial level is evaluated by the coefficient Rtriai(r), the reduced value based on not using a trial-specific intercept, obtained from a linear model associating вг and a_{;}. Note that the structure of a_{;} depends on the chosen form of the mean profile over time in (8.1). For practical purposes, the mean trajectory of the surrogate within each treatment group is specified parsimoniously by using, e.g., a low-order polynomial or fractional polynomials. For the sake of illustration, suppose that the profile is quadratic; then pj (т^д,) can be defined as follows:

with a_{;} = (ao,i, a_{1t}i, a_{2},i)'. Subsequently, Rt_{r}i_{a}i(f) can be calculated as the coefficient of determination from the regression model:

Note that, in this step, the adjustment for the estimation error present in вг and a_{г} should ideally be considered (see Section 5.2).

At the individual level, Renard et al. (2003) proposed to consider the strength of the association between W_{1} and W_{2} and estimate R^{2}ndiv as the square of the correlation coefficient of the pair (W_{1}, W_{2}). Note that, in that case, Rtnd_{iv} will not refer to the direct association between the two endpoints, but rather to the association between the two latent processes governing the longitudinal and event processes. Moreover, as W_{1} and W_{2} can be functions of time, it may no longer be possible to summarize the strength of the association by a single number. To illustrate the point, assume that W_{1} and W_{2} are defined by (8.3) and (8.4), respectively. Thus, given that (U_{1}, U_{2})' ~ N(0, G), we have:

where G_{11}, G12, and G_{22} are the elements of the variance-covariance matrix G. From (8.6)-(8.8) it is clear that the correlation between W_{1} and W_{2}, or its squared value R2_{ndiv}, is a function of time. In fact, (8.6)-(8.8) defines a two-dimensional surface, R^{2}ndiv(r_{1},r_{2}), which can be estimated by using the estimates for 71, y_{2}, 73, and G.

Note that the form of the correlation function very much depends on the assumed form of W_{1} and W_{2}. For instance, if we assume that W_{2} (t) = 7W1 (t), then, necessarily, R2_{ndiv} = 1. Hence, Renard et al. (2003) recommended including a sufficiently large number of association parameters {y_{k}} in (8.4) to avoid undue constraints on R^{2}ndiv.