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Let's say I would like to see which factors affect the survival of patients suffering from cancer. And let's assume I have two variables:

  • chemo ($0$ - if a patient is not treated with chemotherapy or $1$ - otherwise),

  • stage ($0$ - early stage or $1$ - advanced stage).

Now I see that patients treated with chemotherapy are generally those with advanced stage of cancer. The natural idea coming up to my mind is to perform stratified log-rank test. The question is how to define "strata"? Should I stratify by chemo or by stage? Or maybe both and perform two tests? How to generally solve such problems?

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A stratified log rank test is appropriate if you would like to test the association between one variable (chemo or stage) and survival while adjusting for another variable. For example, if you only care about the association between chemotherapy and survival but would still like to adjust for stage, stratify by stage and test for chemotherapy association. Note this won't tell you anything about the relationship between stage and survival, but it will still adjust for stage.

I would like to see which factors affect the survival of patients suffering from cancer.

Your question seems more general though, and it appears you are interested in developing an understanding of the relationship between several factors and survival. There are two key ideas to begin answering this question

  1. Don't use p-values to find associations - P-values don't describe the full relationship between a prognostic factor (chemo or stage) and survival. Estimating a hazard ratio and accompanying confidence interval for the prognostic factor tells you the observed value (based on your data) and gives you a range of plausible values. If there isn't enough data and the test is underpowered, a P-value > .05 will miss prognostic factors that may still be associated with survival. While the confidence intervals will be wider, looking at the intervals and observed values will make you think about precision (how many samples do I need) and give you a better understanding of the observed relationship between each variable, even if it is not significant at p < .05. To estimate hazard ratios and confidence intervals, see idea #2.

  2. Use multivariable models - As mentioned above, stratifying only tells you about the relationship of the non-stratified variable. What if we want to adjust for stage but also know its relationship with survival? Multivariable regression models such as the semi parametric Cox Proportional Hazards (PH) model let you adjust for the effect of several variables while simultaneously estimating the effect (hazard ratio) for each variable. Regression models also estimate confidence intervals for the hazard ratio. It's important to analyze the effect of these variables together, because analyzing variables individually (unadjusted analysis) can give biased results.

The log-rank test is a specialized version of the Cox PH model. If you are considering adjustment for multiple variables, the key question is how do the different variables relate to each other? If the effects of chemo and stage on survival are independent from one another, then simply including both in the model is appropriate. But if the treatment effect of chemotherapy changes depending on the stage of the patient, it's important to use an interaction term (see my answer about subgroup analysis for more info on interaction terms). Note that every statistical model comes with assumptions, and it's important to evaluate the model's assumptions with your data to ensure a good fit. In your case with chemo and stage, it will be important to check for proportional hazards and the potential for an interaction between stage and chemotherapy.

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  • $\begingroup$ Thanks a lot for the reply. You say that with a lot of predictors it is better to use multivariable models because of possible interactions and confounding variables which are hard to detect using just log-rank tests. Moreover, obtained results can be misleading. But what if I want to check relationship between survival and just one specific variable. Let chemo be that variable. Would log-rank test be appropriate for this kind of situation assuming all other predictors have approximately the same number of events in each group and the empirical distribution is also similar between groups? $\endgroup$
    – treskov
    Mar 29, 2020 at 22:07
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    $\begingroup$ For a single variable, log-rank test and Cox model will be equivalent. Deciding whether to adjust for other variables depends on study design. If chemotherapy was randomly assigned, it's fine to only analyze chemotherapy effect. If the data is observational, it's important to adjust for other strong prognostic factors (i.e. stage) that affect survival to make as similar comparisons between groups as possible. So in theory, if the patients are the same except for chemotherapy then log-rank test of chemotherapy would be appropriate, but if there are other differences it would be wise to adjust. $\endgroup$ Mar 29, 2020 at 23:11

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