# Multivariable survival analysis: adding another variable lowers the p value?

When I was performing the Cox survival analysis on my data, I tried to look at the predictive value of different variables to survival. For example, here I have two variables: 'size' and 'surface'. When I tested the 'size' in a uni-variable model, I got

Call:
coxph(formula = Surv(time_to_therapy, therapy) ~ size)

n= 174, number of events= 54

coef exp(coef) se(coef)     z Pr(>|z|)
size 0.004399  1.004409 0.004814 0.914    0.361


The second variable itself is not a significant predictor, either:

Call:
coxph(formula = Surv(time_to_therapy, therapy) ~ surface)

n= 174, number of events= 54

coef exp(coef)  se(coef)     z Pr(>|z|)
surface 3.553e-06 1.000e+00 1.359e-05 0.261    0.794


The two variables are not independent.

However, when I put the two variables together in a multi-variable Cox model, I got

Call:
coxph(formula = Surv(time_to_therapy, therapy) ~ size + surface)

n= 174, number of events= 54

coef  exp(coef)   se(coef)      z Pr(>|z|)
size     2.884e-02  1.029e+00  1.480e-02  1.949   0.0513 .
surface -7.058e-05  9.999e-01  4.158e-05 -1.697   0.0896 .


which shows that the p value decreased for both variables: size - from 0.36 to 0.05, surface - from 0.79 to 0.09.

I had the impression that when you add more variables, the p values usually become higher (when they are dependent, as they often are). Does my example imply that the two variables have some consequence together to the survival? Can I make a composite parameter out of them which is significant?

• Does the proportional hazards assumption hold in the first model? How about the second model? – Theodor Apr 13 '16 at 7:30
• I assume the assumption of proportional hazards holds. The second is the same model. – alize Apr 13 '16 at 9:30

First check for proportional hazards, for example see ?cox.zph. The hazard ratios (or covariate effects) do not have a real interpretation if this does not hold.

Second, there is no a priori reason why p-values should decrease if you add another covariate. If the two are strongly correlated, then it is more likely to happen, because the effects are confounded in this case.

Third, what happens when you add a second covariate is not too surprising. The second covariate lowers the hazard, and the first raises the hazard. The effect of the first is then more pronounced (goes up from 0.004 to 0.2).

Lastly, the hazard ratios have a relative interpretation in the Cox model. Namely, a baseline case in the first model is an individual with size = 0, while in the second model a baseline case has both size = 0 and surface = 0. Thus, the hazard ratio in the first model (1.004409) is the ratio between an individual with size = 1 and an average value of surface for such individuals, and one with size = 0 and an average value of surface for such individuals. In other words, in the first model you estimate a marginal effect.

In the second model the hazard ratio for size (1.029) is the hazard ratio between an individual with size=1 and surface = 0 and an individual with size = 0 and surface = 0. This is a conditional effect. Basically you adjusted for a confounder.

In linear regression this would happen when adding the second covariate reduces the sum of squares for the error (which is the basis of testing significance). It's true that you also lose 1 degree of freedom in this case, but if you have sufficient data (so that the loss of 1 df is not a big deal) and the decrease in the sum of squares for the error is large enough compared to the loss of 1 df, then this will happen.

In conclusion,

I had the impression that when you add more variables, the p values usually become higher (when they are dependent, as they often are).

This is in general a wrong impression.

• Thank you. So it does not make sense at all to feed two dependent variables into a Cox model? For example, if I have to identify a predictor from a number of variables (they are all dependent), what can I do? – alize Apr 19 '16 at 13:50
• Identify in what way? If you are after model selection, then p-value based selection is arguably one of the worse ideas. Then it really depends what you are after: a model which explains the mechanism of the data or a prediction model. Anyway, there is a huge amount of literature, course materials and tutorials on the internet about variable selection. – Theodor Apr 21 '16 at 8:50