The relationship between a longitudinal covariate and a failure time process can be assessed using the Cox proportional hazards regression model. We consider the problem of estimating the parameters in the Cox model when the longitudinal covariate is measured infrequently and with measurement error. We assume a repeated measures random effects model for the covariate process. Estimates of the parameters are obtained by maximizing the joint likelihood for the covariate process and the failure time process. This approach uses the available information optimally because we use both the covariate and survival data simultaneously. Parameters are estimated using the expectation-maximization algorithm. We argue that such a method is superior to naive methods where one maximizes the partial likelihood of the Cox model using the observed covariate values. It also improves on two-stage methods where, in the first stage, empirical Bayes estimates of the covariate process are computed and then used as time-dependent covariates in a second stage to find the parameters in the Cox model that maximize the partial likelihood.

Summary A treatment regime is a rule that assigns a treatment, among a set of possible treatments, to a patient as a function of his/her observed characteristics, hence “personalizing” treatment to the patient. The goal is to identify the optimal treatment regime that, if followed by the entire population of patients, would lead to the best outcome on average. Given data from a clinical trial or observational study, for a single treatment decision, the optimal regime can be found by assuming a regression model for the expected outcome conditional on treatment and covariates, where, for a given set of covariates, the optimal treatment is the one that yields the most favorable expected outcome. However, treatment assignment via such a regime is suspect if the regression model is incorrectly specified. Recognizing that, even if misspecified, such a regression model defines a class of regimes, we instead consider finding the optimal regime within such a class by finding the regime the optimizes an estimator of overall population mean outcome. To take into account possible confounding in an observational study and to increase precision, we use a doubly robust augmented inverse probability weighted estimator for this purpose. Simulations and application to data from a breast cancer clinical trial demonstrate the performance of the method.

For an experimental animal exposed to k > 1 possible risks of death R1, R2, -*, Rk, the term i-th potential survival time designates a random variable Yi supposed to represent the age at death of the animal in hypothetical conditions in which Ri is the only possible risk. The probability that Yi will exceed a preassigned t is called the i-th net survival probability. The results of a survival experiment are represented by kI "crude" survival functions, empirical counterparts of the probabilities Qi(t) that an animal will survive at least up to the age t and eventually die from Ri. The analysis of a survival experiment aims at estimating the k net survival probabilities using the empirical data on those termed crude. Proof. Because the numbering of the k competing risks is arbitrary, it will be sufficient to prove the theorem assuming i = 1, which will simplify the notation somewhat. Let t and h* be arbitrary positive numbers and 0 < h < h*. The definition of Qi(t) implies that the difference Q,(t) -QI(t + h) = Pt (t < Y1 < t + h) n (Yi > Y1)} [3]

SUMMARYThere is considerable debate regarding whether and how covariate adjusted analyses should be used in the comparison of treatments in randomized clinical trials. Substantial baseline covariate information is routinely collected in such trials, and one goal of adjustment is to exploit covariates associated with outcome to increase precision of estimation of the treatment effect. However, concerns are routinely raised over the potential for bias when the covariates used are selected post hoc; and the potential for adjustment based on a model of the relationship between outcome, covariates, and treatment to invite a "fishing expedition" for that leading to the most dramatic effect estimate. By appealing to the theory of semiparametrics, we are led naturally to a characterization of all treatment effect estimators and to principled, practically-feasible methods for covariate adjustment that yield the desired gains in efficiency and that allow covariate relationships to be identified and exploited while circumventing the usual concerns. The methods and strategies for their implementation in practice are presented. Simulation studies and an application to data from an HIV clinical trial demonstrate the performance of the techniques relative to existing methods.

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