# Interpretation of interaction term in Cox PH model when centering LP on mean values of predictors

Let's say I have a Cox PH model for predicting the risk of dying, that in a simplified form looks something like this:

Absolute risk = 1 – baseline hazard^(exp[LP-LPmean]), where

LP (=linear predictor) = 0.54815*age - 0.06318*age [if on treatment]
- 0.06351 [if using treatment]
baseline hazard = 0.98123
LPmean (=linear predictor for patient with mean values for covariables) = -0.03177


The model thus contains a coefficient for age and treatment and an interaction between age and treatment. The LP was centered at the mean values of the predictors, hence, the LP of the mean predictor values is subtracted from the LP in order to get predictions.

I would like to get an idea about the effect of the interaction in the model and give a hazard ratio (HR) for treatment at a few different ages. If I do that as following:

for age 50: HR treatment = exp(-0.06318*(50) - 0.06351) = 0.04;
for age 60: HR treatment = exp(-0.06318*(60) - 0.06351) = 0.02;
etc..,


.. the effects are rather extreme (due to the centering of the LP). I can account for the centering like this:

for age 50: HR treatment = exp(-0.06318*(50-54.6) - 0.06351) = 1.25;
for age 60: HR treatment = exp(-0.06318*(60-54.6) - 0.06351) = 0.67;
etc..,


where 54.6 is the mean age in the study population. However, I am not sure whether this is correct and how I should interpret this.

Many thanks!

• Are you doing two models, one for the linear predictor and then a Cox, or am I just misunderstanding your simplification? It would help if you wrote out the Cox regression as modeled. Jul 9 '13 at 15:39
• @Azula R. It is just an ordinary Cox model; in R it would look something like this: coxph(Surv(time,status)~age*treatment) or more explicit: coxph(Surv(time,status)~age + treatment + age:treatment).
– Rob
Jul 9 '13 at 16:13
• Okay, and your 'age' variable in the model is actually 'age minus mean age', correct? In that case, you absolutely need to subtract the mean age when calculating the HR because the term 'age' in the model is actually 'years from the mean age' and not the subject's age. Jul 9 '13 at 16:23
• @Azula R. I did not explicitly enter the age variable in my model as 'age minus mean age', but the linear predictor as a whole is centered at the mean values of all predictors (also see link, page 49) when making predictions.
– Rob
Jul 10 '13 at 7:27
• thanks for the link. This is a new usage for me, so I'll be interested to see if anyone else has any thoughts. My feeling is that subtracting the mean when evaluating is still the correct thing to do, but I'm not sure. Jul 10 '13 at 14:13