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I'm new to survival analysis, but I've been reading some papers and books and I got a nice model.

However, one of the variables (Sit) does not met the proportionality assumption for the Cox model. Nevertheless, this got me thinking and it makes sense, since that variable should have a time-varying effect. For instance, in this case, I expect that shortly after treatment, the risk for relapse is higher for some individuals while as time passes, the time for all individuals is similar. Note (to myself) that time-varying effect is different than time-varying covariate. It got me quite some time to fully understand this.

Now, my question is:

I know how to include an interaction between Sit and time (I'm using R). But when I add the interaction some of the variables that were previously significant are not anymore. And it got me thinking, should I repeat the process to select variables? Perhaps, some of the variables previously removed are now significant. Or should I interpret the model as it is, and ignore the variables that are no longer significant?

Another question is:

The proportionality of the Cox model is violated. I have decided to introduce an interaction. So, now I should not do the residuals analysis? Since the proportionality is no longer in order. How should I test for the fitness of the model, then?

I'm really confused about this and it would be great to have some guidance and tips about the best procedure, from now on.

Thank you very much!

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  • $\begingroup$ Could you not stratify your model based on your time interaction? i.e. basically split the data at the time where the nonproportionality is observed (I assume you're using a Schoenfeld residual to look at this)? $\endgroup$ – Corey Sparks Jun 1 '14 at 3:51
  • $\begingroup$ Yes @CoreySparks. I'm using Schoenfeld residual to check the proportionality. The idea to include the interaction is because the variable Sit is interesting and I would like to look at the effect. Using strata I would not have a chance to look at the effect. And an interaction make sense. What will your idea bring to the model? I'm sorry, I'm new to survival analysis. I've been self studying through books. Thank you for your comment! $\endgroup$ – psoares Jun 1 '14 at 14:19
  • $\begingroup$ By stratifying, you can assume proportionality of hazards in each of the strata, so you can use the cox model with the main assumption being met. I think you can still easily visualize the effect through plots of the estimated survival or hazard functions, specified with the covariate in question at different values, i.e. hold all the other predictors at their means, then vary the value of your primary predictor. You can see the effect that way $\endgroup$ – Corey Sparks Jun 1 '14 at 23:53
  • $\begingroup$ Thank you! I will also try that approach. I will first try with the interaction and check the model diagnostics. If no changes occurs, I will then stratify and use plots $\endgroup$ – psoares Jun 2 '14 at 8:03
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Though I'm not an expert in survival analysis, I put here my suggestions and hope they will be helpful.

First of all, selection of variables looking at their p-values is a wrong way, especially when the model is aimed to make statistical inferences. You can read about that in multiple sources searching for "stepwise regression drawbacks". The selection of variables should be based on your domain-specific knowledge. All variables which are relevant (on your opinion) should be present, no matter whether their influence is significant or not. In such way you will report the effect of Sit adjusted for the list of used variables, and that is right. It seems that your research is exploratory but not confirmatory. In such case while interpreting the results, you'd better make emphasis on the sizes of effect (model coefficients, odds ratios or risk ratios) rather then on p-values.

As for violation of proportionality assumption: taking into consideration the interaction between Sit and time, you are incorporating linear dependence of Sit on time into the model. So if the true relationship between Sit and time is really close to linear, then proportionality assumption will be held. Thus all model diagnostics methods remains relevant.

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  • $\begingroup$ Thank you for your answer. First of all, I do no use the p-values for selection variables. I agree with you that p-values are not the best way. I use a method describe in Collett's book "Modelling survival data in medical research". Basically looking more at the value of -2 log likelihood. For the results, I'm interested in the Hazard Ratio. Your last paragraph is interesting, I will look at other model diagnostic methods to check the validity of this new model. If the proportionality is still violated, this would mean that the interaction is not enough, right? $\endgroup$ – psoares Jun 1 '14 at 19:41
  • $\begingroup$ I think yes. Then incorporating spline function of time into Cox model may help, however I'd prefer searching for some prior knowledge or considerations about the dependence of Sit effect on time. $\endgroup$ – O_Devinyak Jun 1 '14 at 20:01

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