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S and T Time-to-Event Variables

Prior to describing the assessment of the individual-level surrogacy in this scenario, let us introduce some general notation for time-to-event outcomes. The random variables of interest in this setting are represented by the vector (Y,C, XT), where Y is a time-to-event variable, C the censoring time, and X is a p-dimensional column vector of possibly time-dependent explanatory variables. Time dependency is sometimes indicated through X = X(t).

The response variable for individual j is given by the vector (Tj, Sj), where Tj = min(Yj, Cj) and Sj = I(Yj < Cj), with I(?) an indicator function. Furthermore, let Kj (t) denote a function indicating whether subject j is at risk (Kj (t) = 1) or not (Kj(t) = 0) at time t. The regression model for T, given the observed values of the covariates X = x, tries to describe the conditional density f (t|x), although it is more frequently expressed equivalently in terms of the hazard function, also referred to as the intensity function. The proportional hazard (PH) model specifies the intensity function as (Cox, 1972)

where Ao (t) is a fixed but unknown baseline hazard function, and в is a p- dimensional vector of unknown coefficients. For a binary covariate xq, fiq can be interpreted as the log-relative risk, conditional on the values of all other covariates. The baseline hazard function A0(t) can be specified using a power form or a constant, in which cases the Weibull and exponential models are recovered (Cox and Oakes, 1984; Kalbfleisch and Prentice, 1980). In the following it will be assumed that censoring is non-informative, i.e., the censoring and failure times are statistically independent. The parameters of interest in (10.11) are estimated based on the so-called partial likelihood introduced by Cox (1972).

Given the continuous nature of time-to-event variables, Alonso and Molen- berghs (2008) proposed to quantify the individual-level surrogacy in this scenario using the SICC, i.e., Rflindiv = 1 — e-2I(T,S), where I(T, S) is the mutual information between both endpoints after adjusting by trial and treatment. The estimation of the mutual information I(T, S) is particularly challenging in this setting for two main reasons: i) both endpoints S and T are potentially censored and ii) the estimation of the parameters in (10.11) is carried out using a non-likelihood procedure, i.e., it is based on the partial likelihood function. In the next section two estimation strategies will be discussed.

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