A Categorical (Ordinal) 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 categorical ordinal (or binary) endpoint and the true endpoint is a failure-time endpoint. Note that the described approach is applicable also in the reverse case, i.e., with a failure-time surrogate and an ordinal true endpoint. We mainly follow the two-stage approach proposed by Burzykowski, Molenberghs, and Buyse (2004).

Theoretical Background

Assume that the true endpoint T is a failure-time random variable and the surrogate S is a categorical variable with K ordered categories, i.e., an ordinal variable. For each of j = 1,... ,n_{i} patients from trial i (i = 1,..., N) we thus have quadruplets (X_{i}j, , S_{i}j, Z_{i}j), where X_{i}j is a possibly censored version

of survival time Tj and Д j is the censoring indicator assuming a value of 1 for observed failures and 0 otherwise.

Model (4.10) is replaced by a copula-based model (see, e.g., Section 5.2) for the true endpoint Tj and a latent continuous variable S_{i}j underlying the surrogate endpoint S_{i}j (Burzykowski, Molenberghs, and Buyse, 2004). In particular, Burzykowski, Molenberghs, and Buyse (2004) used the Plackett copula (Hougaard, 1986) (see also (5.6)-(5.7)). The marginal model for S_{i}j is the proportional odds model:

The model can be interpreted as assuming a logistic distribution for the latent variable S_{i}j. Note that, in the case of a binary surrogate S_{i}j, model (6.1) is equivalent to logistic regression.

It is worth re-parameterizing model (6.1) as follows:

where, for identifiability purposes, one might specify that, for example,

If, for a particular trial, i_{0} say, not all levels of S are observed, one can use model (6.2) with some of the terms щ_{0},1,... ,Цг_{0},к-1, corresponding to the unobserved levels, constrained to 0. As a special case, the following model might be considered:

The model assumes a fixed set of cut-points n^{0},..., nK_{-}i, but allows for trial- specific shifts ni of the set.

For Tj, the proportional hazard model is used:

where в are trial-specific effects of treatment Z and A_{i}(t) is a trial-specific baseline hazard function. Burzykowski, Molenberghs, and Buyse (2004) used a Weibull-distribution-based baseline hazard.

Using the joint distribution function defined by the copula and the marginal models (6.1) and (6.4), it is possible to construct the likelihood function for the observed data (Xj = Xj, Aj = , Sj = Sj, Zj = Zj) and

obtain estimates of the treatment effects а_{г} and вг-

If, as proposed by Burzykowski, Molenberghs, and Buyse (2004), the Plack- ett copula is used in the construction of the joint model for Tj and Sj, the quality of the surrogate at the individual level can be evaluated by using the copula parameter 9. This is because, for the Plackett copula, 9 takes the form of a (constant) global odds ratio. Specifically, in the current setting (for k = 1,..., K — I and t > 0):

Thus, 9 is naturally interpreted as the (constant) ratio of the odds of surviving beyond time t given response higher than k to the odds of surviving beyond time t given response at most k. For a binary surrogate, it is just the odds ratio for responders versus non-responders (assuming that k = 2 indicates response).

The quality of the surrogate at the trial level can be evaluated by considering the correlation coefficient between the estimated treatment effects а_{г} and вг- Note that, in this step, the adjustment for the estimation error, present in а_{г} and /З_{г}, should be made. Toward this aim, model (5.13)-(5.14) (see Section 5.2) can be used.

If the individual-level association is not of immediate interest, one may base analysis on the marginal models (6.1) and (6.4), without specifying the baseline hazards in the latter. When fitting the models, it is worth estimating the variance-covariance matrix of the estimated treatment effects а_{г} and вг while taking into account the association between S and T. Toward this aim, an estimator based on a combination of the results obtained by Liang and Zeger (1986), Lin and Wei (1989), Lin (1994), and Lipsitz, Kim, and Zhao (1994) can be used (see Appendix 6.5).