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ABSTRACT
The Cox proportional-hazards regression model has achieved widespread use in the analysis of time-to-event data with censoring and covariates. The covariates may change their values over time. This article discusses the use of such time-- dependent covariates, which offer additional opportunities but must be used with caution. The interrelationships between the outcome and variable over time can lead to bias unless the relationships are well understood. The form of a time-- dependent covariate is much more complex than in Cox models with fixed (non-- time--dependent) covariates. It involves constructing a function of time. Further, the model does not have some of the properties of the fixed-covariate model; it cannot usually be used to predict the survival (time-to-event) curve over time. The estimated probability of an event over time is not related to the hazard function in the usual fashion. An appendix summarizes the mathematics of time-dependent covariates.
KEY WORDS: survival analysis, longitudinal analysis, censored data, model checking
INTRODUCTION
One of the areas of great methodologic advance in biostatistics has been the ability to handle censored time-to-event data. "Censored" means that some units of observation are observed for variable lengths of time but do not experience the event (or endpoint) under study. Such data were first studied and analyzed by actuaries. Kaplan & Meier presented the product limit or Kaplan-Meier curve to efficiently use all of the data to estimate the time-to-event curve (6).
Comparison of groups based on this nonparametric estimate is given by the logrank test. Sir David Cox considered the introduction of predictor/explanatory variables or covariates into such models. The hazard may be thought of as proportional to the instantaneous probability of an event at a particular time. Cox (2) proposed a model in which the effect of the covariates is to multiply the hazard function by a function of the explanatory covariates. This means that two units of observation have a ratio of their hazards that is constant and depends on their covariate values. This model is usually called either the Cox regression model or the proportional-hazards regression model. It is important that covariates in this model may also be used in models in which the underlying survival curve has a fully parametric form, such as the...