# Interaction terms in Cox PH model

I am wondering if there is a way to interpret an interaction term from the coefficients rather than just looking at the survival curves?

The factors involved are A, B, and C, which are all binary coded (0,1). I therefore have 8 treatment groups (and survival curves).

1. a0:b0:c0
2. a1:b0:c0
3. a1:b1:c0
4. a1:b1:c1
5. a1:b0:c1
6. a0:b1:c0
7. a0:b0:c1
8. a0:b1:c1


The significant 3-way interaction tells me survival depends on the combination of A B and C - but how do I tell which combination? For example how is treatment 7 different from 8, or 1 from 3.... etc.? Is it possible to work out the HR for c when both a and b are 1 (comparing groups 3 & 4)?

             exp(coef)      z     p
a1                0.85  -0.46 0.650
b1                1.07   0.19 0.848
c1                0.83  -0.53 0.598
a1:b1             1.96   0.42 0.157
a1:c1             2.39   1.83 0.066
b1:c1             2.89   2.32 0.030*
a1:b1:c1          0.18  -2.67 0.008*


You interpret them in the same way as for a single coefficient: if the hazard ratio (exp(coef)) is larger than 1 it means the hazard increases (lower survival) and vice versa.

The high hazard ratio for a1:b1 basically means that if both a1 and b1 occur (whatever it means) survival degrades significantly, more than is explained by a1 and b1 separately.

• Thanks for the comment - but I think I'm looking for a bit more detail. I've edited my question, hopefully it'll make it clearer..? Commented Feb 20, 2014 at 12:31

Lets go slowly. Consider 3 binary random variables $X$, $Y$ and $Z$ coded as 0 and 1 each. Then the 3 way interaction is a new variable $W$=$XYZ$. What kind of variable is $W$?. $W$ is binary too, $W$=0 when one of $X$, $Y$ and $Z$ is 0 and 1 when all 3 variables $X$, $Y$ and $Z$ are 1. So you have formed two groups by the variable $W$. The group 0 is the combination of the treatments 1,2,3,5,6,7,8 and the group 1 is treatment 4. Interpreting $W$ does not compare the eight treatment groups to each other, it only compare group 4 against any of 1, 2,3,5,6,7,8 and says the effect is thesame. That is if you pick any member from group 0 compared to group 1(treatment 4) the effect(HR) is 0.18.

• Thanks. I followed this up until the last 2 sentences. The survival of treatment 8 (or 4 on my list) is almost identical to 8 and 5 on my list - so "pick any member from group 0 compared to group 1(treatment 8) the effect(HR) is 0.18" doesn't seem right? Commented Feb 20, 2014 at 15:09
• Actually I thought they were well ordered, sorry for the mix-up. I have edited the answer.It is a good indication that you followed. Commented Feb 20, 2014 at 16:07
• I still don't get this part " it only compare group 4 against any of 1, 2,3,5,6,7,8 and says the effect is thesame. That is if you pick any member from group 0 compared to group 1(treatment 4) the effect(HR) is 0.18 Commented Feb 21, 2014 at 17:42
• Alright alternatively. Well 2-way interactions means one variable has different effects on the different levels of the other. 3 way interactions will build on that. In the different levels of the two way interaction, the 3rd variable has different effects. But what does that mean? It means the effect of c {0} vs {1} is different at levels of a:b which are l1=a0:b0,a0:b1,a1:b0, and l2=a1:b1. So we will compare l1 c1vsl1 c0 and l2 c1vsl2 c0.In your case only l2 c1vsl2 c0 was significant which was a1:b1:c1 vs a1:b1:c0 which is 0.18.I have used c as the third variable. Commented Feb 22, 2014 at 12:54
• I have used c as the third level, but your model also has two way interactions ac and bc at the end of the day repeating the above explanation for a and b we dive back to the my initial answer. We lets see how it goes it is interesting\ Commented Feb 22, 2014 at 13:03