# Cox proportional hazard model and interpretation of coefficients when higher case interaction is involved

Here is the summary-output of the Coxph-model I used (I used R and the output is based on the best final model i.e. all significant explanatory variables and their interactions are included):

 coxph(formula = Y ~ LT + Food + Temp2 + LT:Food + LT:Temp2 +
Food:Temp2 + LT:Food:Temp2) # Y<-Surv(Time,Status==1)


n= 555

               coef         exp(coef)          se(coef)      z           Pr(>|z|)
LT             9.302e+02      Inf             2.822e+02    3.297        0.000979 ***
Food           3.397e+03      Inf             1.023e+03    3.321        0.000896 ***
Temp2          5.016e+03      Inf             1.522e+03    3.296        0.000979 ***
LT:Food        -2.250e+02    1.950e-98        6.807e+01    -3.305       0.000949 ***
LT:Temp2       -3.327e+02    3.352e-145       1.013e+02    -3.284       0.001022 **
Food:Temp2     -1.212e+03    0.000e+00        3.666e+02    -3.307       0.000942 ***
LT:Food:Temp2   8.046e+01    8.815e+34        2.442e+01     3.295       0.000986 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Rsquare= 0.123   (max possible= 0.858 )
Likelihood ratio test= 72.91  on 7 df,   p=3.811e-13
Wald test            = 55.79  on 7 df,   p=1.042e-09
Score (logrank) test = 78.57  on 7 df,   p=2.687e-14


Question is:

How to interpret coefficient and exp(coef) values in this case, as they are very large values? Also 3-case interaction is involved, which confuses the interpretation more.

All the examples concerning Coxph-model I have found so far online have been really simple regarding the interaction terms (which have always turned out to be unsignificant) and also coefficient-values (=hazard rates) and exponentials of these (=hazard ratios) have been pretty small and "easy to handle" numbers, e.g. coefficient = 1.73 -> exp(coef) = 5.64. BUT mine are way bigger numbers as you can see from the summary output (above). And because they are so large vaues, they almost seem to not make any sense.

It seems to be a bit ridiculous to think that survival is e.g. 8.815e+34 (hazard ratio taken from the interaction LT:Food:Temp2) times lower when the interaction increases by one unit (?).

Actually I don't know how to interpret this 3-case interaction either. Does it mean that when all of the variables in interaction increase by one unit, the survival decreases by the certain amount (told by the exp(coef)-value)?

Would be great if somebody can help me out here. :)

Below is the part of my data sheet I used for the cox-analysis. Here you can see, that I have used many times same explanatory variable value (i.e. LT, Food, and Temp2) for several "Time, Status response variable". These explanatory variable values are already the mean values of these variables (due to the field-work setup out in the nature, it was not possible to get individual explanatory variable value for each observed response individual, hence the mean values used already in this phase), and this would answer to suggestion 1 (?) (see the first answer).

Suggestion 2 (see the 1st answer): I am using R, and I am not yet super god in it. :) Thus, if I use function predict(cox.model,type="expected"), I get a huge amount of different values and have no clue to which explanatory variable they are referring to and in which order. Or is it possible to highlight certain interaction term in predict function? I am not sure if I am making myself very clear here.

Suggestion 3 (see 1st answer): in the part of the data sheet below, one can see the units of different explanatory variables. They are all different and include decimals. Can this have something to do with the cox outcome?

Part of the data sheet:

Time (days)     Status      LT(h) Food (portions per day) Temp2 (ºC)
28                0         14.42        4.46             3.049
22                0         14.42        4.46             3.049
9                 1         14.42        4.46             3.049
24                0         15.33        4.45             2.595
24                0         15.33        4.45             2.595
19                1         15.33        4.45             2.595


## 1 Answer

A couple suggestions, not directly related to CoxPH but to interactions and collinearity

1) When you are getting "crazy" values like these, one possiblitiy is collinearity. This is often a problem when you have interactions. Have you centered all your variables (by subtracting the mean from each)?

2) You can't interpret one interaction among many quite so easily. LT, food and temp2 are all involved in many interactions. So, look at predicted values from different combinations.

3) Check the units of the different variables. When you get crazy parameters, sometimes it's a problem of units (e.g. measuring a human height in millimeters or kilometers)

4) Once you've got that stuff straightened out, I find the easiest way to think of the effects of different interactions (esp. higher level ones) is to graph the predicted values with different combinations of the independent values.

• Hei, below is the part of my data sheet I used for the cox-analysis. Here you can see, that I have used many times same explanatory variabe value (i.e. LT, Food, and Temp2) for several Time, Status response variable. These explanatory variable values are already the mean values of these variables (due to the field-work setup out in the nature, it was not possible to get individual explanatory variable value for each observed response individual, hence the mean values used already in this phase), and this would answer to suggestion 1 (?). – Unna Jul 13 '12 at 11:42
• Suggestion 2: I am using R, and I am not yet super god in it. :) Thus, if I use function predict(cox.model,type="expected"), I get a huge amount of different values and have no clue to which explanatory variable they are referring to and in which order. Or is it possible to higlight certain interaction term in predict function? I am not sure if I am making myself very clear here. – Unna Jul 13 '12 at 11:42
• Suggestion 3: in the part of the data sheet below, one can see the units of different explanatory variables. They are all different and include decimals. Can this have something to do with the cox outcome? – Unna Jul 13 '12 at 11:43
• Time (days) Status LT (h) Food (portions per day) Temp2 (ºC) 28 0 14.42 4.46 3.049 22 0 14.42 4.46 3.049 9 1 14.42 4.46 3.049 24 0 15.33 4.45 2.595 24 0 15.33 4.45 2.595 19 1 15.33 4.45 2.595 – Unna Jul 13 '12 at 11:43
• Above comment about the data sheet example I used does not show in the table-shape, but I hope it is possible to make sense out of it. :) – Unna Jul 13 '12 at 11:44