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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*
Nick Cox
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user29689
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2 Answers2

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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.

Marc Claesen
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  • 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..? – user29689 Feb 20 '14 at 12:31
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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.

Chamberlain Mbah
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  • 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? – user29689 Feb 20 '14 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. – Chamberlain Mbah Feb 20 '14 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 – user29689 Feb 21 '14 at 17:42
  • I thought a 3-way interaction indicated that a 2-way interaction was significantly different in the presence or absence of at third variable... for example a:b may be significant when c = 0 but not when c = 1. I don't see how W being different from everything else tells us that? – user29689 Feb 21 '14 at 17:49
  • I think you should read your question once small – Chamberlain Mbah Feb 21 '14 at 21:09
  • sorry I'm not sure what that means? – user29689 Feb 21 '14 at 22:05
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    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 c1`vs`l1 c0` and `l2 c1`vs`l2 c0`.In your case only `l2 c1`vs`l2 c0` was significant which was `a1:b1:c1` vs `a1:b1:c0` which is 0.18.I have used `c` as the third variable. – Chamberlain Mbah Feb 22 '14 at 12:54
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    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\ – Chamberlain Mbah Feb 22 '14 at 13:03
  • Thanks for taking the time. It appears that to fully explain the interaction I do have to look at the data rather than woking it out from the coefficients. The cox ph section of the linked item describes how to attain the HR of each treatment compared to the baseline (treatment 1) http://cran.r-project.org/web/packages/survival/vignettes/tests.pdf – user29689 Feb 22 '14 at 15:59