# 95%CI for the difference in risk (Cox regression)

I have using following cox model to investigate the association between two exposures and a time to event outcome (0=no event, 1=event).
Model= coxph(Surv(time, as.numeric(event))~V1+V2, data=data) Where v1=exposure1, v2=exposure2

I would like to predict the difference in risk due to difference in these two exposures. Thus I created a new dataset with the new values of exposures and used that dataset to predict a diference in rsik due to difference in exposures. This new dataset with new values looks like this

newdata= data.frame(V1=c(-2.17, -1.99), V2=c(.44, .43))


For this, I used the following function on this newdata to predict risk associated with new exposure values.

Prediction=predict(Model, newdata=newdata)


this gives me this output (hazards)

-0.66 , -0.60

Based on this output I can calculate the difference in risk due to difference in exposure like this

Prediction-Prediction


But I am struggling to calculate the 95% confidence interval of this difference in risk. Any suggestions on how I can calculate this?

Somebody in this post (Cox regression. Find 95% confidence interval for comparison of two groups) suggested to use ‘contrast’ function from rms package but I am not sure how to use this function since I have never used that package. Any suggestions on how to calculate such 95%CI?

• Just to clarify, 1) are the exposures 1 or 0? + 2) on what scale do you want the answer? Do you want a (log-)hazard ratio (if your exposures are just 0 and 1, that's an easy answer to get)? Do you want a hazard difference? Do you want the difference (or ratio or odds ratio) in the probability of an event by a specific time (e.g. after 1 year)? When you say difference in risk, I would normally interpret that as a difference in such probabilities, but that would require specifying a time-horizon and is not entirely consistent with other things you mentioned. Thus, it would be good to clarify. Jun 17, 2021 at 8:35
• hi Thanks. i would like to get log(hazard ratio). so i indicated the values from 'predict' function and those i got from type='lp'. those are actually log(HR). later i would like calculate the difference of these two log(HR) and find out the 95%CI of this difference. Jun 17, 2021 at 12:18

The advantage of working in the linear-predictor scale is that a simple difference between two cases is easily written in terms of a single case. (V1, V2) for your two cases are (-2.17, 0.44) and (-1.99, .43) respectively. Thus, with $$\text{lp}_i$$ the linear predictor for case i and coefficient estimates $$\beta_{V1}$$, $$\beta_{V2}$$:
$$\text{lp}_1 = -2.17 \beta_{V1} + 0.44 \beta_{V2}$$ $$\text{lp}_2 = -1.99 \beta_{V1} + 0.43 \beta_{V2}$$ $$\text{lp}_1 - \text{lp}_2 = (-2.17+1.99) \beta_{V1} + (0.44 - 0.43) \beta_{V2}= -0.18 \beta_{V1} + 0.01 \beta_{V2}$$
So a test of a significant difference between these two cases, or the CI for the difference, resolves to calculations on this difference in linear-predictor values, equivalent to a single case with (V1, V2) of (-0.18, 0.01). Most prediction routines provide the option to calculate confidence intervals in addition to point estimates; for predict.coxph() you have to specify se.fit=TRUE, while that's the default in survfit.coxph().