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I am confused about time-varying covariates in Cox models. I believe this refers to the fact that in the normal situation, you can calculate a hazard ratio for $X$, assuming $X$ for an individual is not going to change over time. But with time varying covariates you can also take into account that $X$ may change over time.

I don't understand how you interpret hazard ratios in that case. Do you still interpret this as "An individual A with $X=2$ has 4 times the hazard of an individual B with $X=1$", but knowing that later on those individuals will have changes in $X$? What if individual B has $X=2$ at the next time point and individual $A$ has $X=1$? It seems that you can't make an overall statement about the hazard of one person vs another, but rather just at one time point.

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Fitting a Cox model only uses covariate values that are in place at the event time being evaluated as the modeling proceeds from event time to event time. With time-varying covariates, values prior to the event time being evaluated don't matter any more. Values after the event time being evaluated obviously can't enter into the model without violating causality.

Thus you can say things like "An individual A with $X=2$ has 4 times the hazard of an individual B with $X=1$" at any specific time. You are correct that "you can't make an overall statement about the hazard of one person vs another," just about the hazards associated with sets of covariate values.

Under the proportional hazards assumption, those relative hazards between sets of covariate values hold over all time. But if individuals' covariate values change over time you can't translate that into hazards between individuals.

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