Solved – Interpretation of Cox regression with time varying covariates

cox-modelsurvivaltime-varying-covariate

After fitting my model to a time varying cox regression, let's say I have a continuos variable $ X $. The hazard ratio of this variable will be higher than 1 (about 2 and it is significantly significant). Is it safe to say that an increase in $ X $ will increase the risk by a factor of two ?

I'm not so sure about this, because this is a time varying covariates.

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There is no problem with that conclusion. In fact, I would say it is especially meaningful for time varying covariates. Think of a time constant covariate like sex: would someone ever experience what happens if she became a he or the reverse (ignoring sex-change operations)? So time-constant covariates are basically group comparisons, but you would not expect someone to experience such a change in group membership. In case of time-varying covariates it is usually realistic for a person to experience such a change (how else could it vary over time?).

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Patients are "taken into account" for their dosage history in 2 distinct ways:

The first is the coding of dose receipt as a time varying covariate. A brief explanation of how they are included in the risk sets for the comparisons in the Cox model: Patient 1 was observed in the risk-set denominators from 0 until 0.034 at which point they were censored because they did not die and their covariates changed. They were then re-entered into the study since they received a second dose at 0.034 and were observed until 12.95 time-units with two cumulative doses at which point they were censored again. They received their 4th dose and were censored for loss-to-follow-up or administratively at time 30.8. The second patient follows the same pattern but experiences the event at time 37.92 with six doses. Since the first patient was censored before then, they are not compared in terms of the study outcome. This is how we account for the amount of time, and the order of dose receipt, in which the patient was at risk for disease/death during the study.

The second way that "dose at n-1" is taken into account is the coding of cumulative dose. The two modeling approaches you propose in fact result in similar predictions and inference. The second formulation, one which adjusts for the "covariate history", is called a "distributed lag". The factor levels are merely the sum of the lags because the time varying covariates follow a specific order (a patient doesn't for instance miss the first dose and then receive their second). You will find, however, that the p-values and global inference from the two formulations are the same.

A hazard to venture here: I don't think this model answers the type of question you believe you are answering. You say: " I am interested in the hazard ratio associated with each level of X, ie. how an additional dose affects the likelihood of recovery." It sounds to me like you should code the dosage variable numerically so that if a patient has 5 doses and dies you know they have had more exposure to the drug than one who has had 4 doses; both formulations you propose make it possible for the cumulative dose effect to vary in unpredictable ways.

Another issue to discuss is the fact that there may not be enough variability to differentiate the person-time-at-risk effect from the effect of cumulative dose. It is assumed that the longer the person is in the study, the more at-risk they generally are for an outcome. This is an issue that is solved by having a lot of data and a lot of variation in the timing of the subsequent doses. Using some parametric models like a log-linear model for the life table may help assess the sensitivity to that.

Solved – Cox regression with time varying covariates

I might be able to give you some tips:

Recommended books...

Kleinbaum, Klein: Survival analysis - A Self-Learning text http://www.springer.com/statistics/life+sciences,+medicine+%26+health/book/978-1-4419-6645-2 In my opinion the best book on this matter and it includes time-varying covariates and also how to program the computations in SAS and R.

Also look at: http://cran.r-project.org/doc/contrib/Fox-Companion/appendix-cox-regression.pdf Which is John Fox explanation and he uses R to calculate, which is great.

Second: what type of time-varying covariates do you have? - Multiple observations per individual? - Multiple endpoints per individual?

In general, if you have 1 endpoint of interest and multiple observations per individual, you usually set up the data frame in a format which means that each observation corresponds to one row (therefore one individual may have several rows of data) and you create a start variable and a stop variable, which is simply the start and stop intervals for each observation.

The usual Cox model:

coxph(Surv(survival, event) ~ predictors, data = df)

The time-dependent Cox model (if data is set up as described above):

coxph(Surv(star, stopp, event) ~ predictors, data = df)

A very well written manual can be found here: http://cran.r-project.org/web/packages/survival/vignettes/timedep.pdf

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