The trick here is to include interaction terms that cover each combination of treatment and temperature. You have to be careful in interpreting the coefficients returned by the analysis, but you will generally do much better with a combined analysis, taking advantage of all your data, than with separate analyses.
In R, you could simply write the model as:
coxph(Surv(time, status) ~ treatment * temperature
and the software will automatically generate all the implied coefficients. Specify the
control group as the reference for the
treatment categorical predictor. For
temperature, code it as a 3-level categorical predictor, or keep it as numeric if you think that the log-hazard is linearly related to temperature.
If you code
temperature as categorical with
10C as the reference, you will get the following coefficients:
treatment B are their log-hazard differences from
temperature35C are the log-hazard differences from
The interaction coefficients
treatmentB:temperature35C represent the further log-hazard differences under those conditions from what you would get by simply adding the corresponding
So that in principle gives you all 8 differences in log-hazard (or hazard ratios, after exponentiation) from the baseline situation (
10C), by combining the coefficient values and their estimated errors appropriately. The
rms packages in R can help with converting the regression coefficients into estimates for specific conditions.
If you can treat
temperature as a continuous numeric predictor linearly related to log-hazard things might be a bit simpler. To simplify interpreting the coefficients, subtract 10 from all of your temperatures, because by default the baseline conditions are reported with continuous predictors set at values of 0.
With a linear treatment of
temperature, the coefficients for
treatment B would be interpreted as above (at 10C). There would be a single
temperature coefficient representing the change in log-hazard per degree under
control conditions, and two interaction coefficients,
treatmentB:temperature, representing the differences from the
temperature coefficient under those treatments.
Including interactions should work well if your study has enough events. You can evaluate the significance of the interaction terms; if they aren't significant then there's no evidence that the effect of
treatment differs with
temperature (or vice-versa) and you might simplify the model with simple additive
There's nothing specific to Cox regression modeling here except for the baseline hazard conceptually replacing the intercept of a linear regression model and the requirement to check that the proportional hazards assumption is met. So if you are confused by how to interpret coefficients when there are interaction terms, any reputable statistics text or many questions taged
interaction on this site should help.