Time Dependent Covariates Overview
In Cox proportional hazards models, a covariate can be a one-time static measurement or it can vary across time. When the predictor changes value over the observed time frame, the time dependent covariate model should be used.
A good example of time dependent covariates is vital health statistics including patient weight, blood pressure, and cholesterol level. These are measures that can certainly vary over time and also have an influence on the hazards model in medical studies.
For some analyses, the covariate changes in a systematic way that can be expressed as a function of the observed covariate value and time. For other situations, the covariate is measured at various time points. In this case, multiple variables are used to express the change related to time. For either of these cases, the time dependent covariate is written as a function of time and the variable(s).
One Covariate Measure
The covariate can change across time in some well-defined, systematic way. This change can be represented with a simple mathematical function. For example, this might be the sum or product of time and the baseline covariate. Time is represented in the expression by T_. A systematic time dependent covariate might look like this:
“Covariate1” * log(T_)
This time dependent covariate is the product of the covariate and the log of time.
Multiple Covariate Measures
When the variation across time is not systematic, multiple measurements across time are needed. Then the expression of the time dependent covariate is a function of time, T_ and the set of covariate measurements. For example, say a patient is weighed every 90 days as part of the study. The weight measurements are listed as Weight1, Weight2 and Weight3. Assume time is measured in days. The time dependent covariate function would be:
(T_ < 90) * "Weight1" + (T_ >= 90 and T_ < 180) * "Weight2" + (T_ >= 180 and T_ < 270) * "Weight3"
The logical expressions evaluate to 1 if true and 0 if false. So if time is less than 90 days, (T_ < 90)=1 and the initial weight measurement is used as the covariate. All other logical expressions would evaluate to 0, effectively adding 0 to the function. Only one of the time expressions can be true at once.