I have a competing risk survival model with a log transformed interaction term and I am not sure how to interpret and report these results.
I'm evaluating the effect of a treatment and the effect modification based of age on cardiovascular disease events as outcome. In this case age is log transformed, because it was not linear.
The model contains the following variables
- Treatment: treatment 1 and treatment 2
- Age as continuous variable, log transformed
- Riskfactor as categorical variable: no hypertension and hypertension
- Interaction term treatment*log(Age)
This is the model:
library(cmprsk)
cov <- model.matrix(~ factor(treatment)*log(Age) + factor(riskfactor), data=dat02) [,-1]
SurvModel <- crr(dat02$FU, dat02$status, cov, cencode="censored", failcode="Event")
summary(SurvModel)
Output:
Competing Risks Regression
Call:
crr(ftime = dat02$FU, fstatus = dat02$status,
cov1 = cov, failcode = "Event", cencode = "censored")
coef exp(coef) se(coef) z p-value
factor(treatment)2 10.204 199569.6284 5.274 2.314 0.021000
log(Age) 7.021 55.7549 0.940 4.279 0.000019
factor(riskfactor)Hypertension 0.900 2.4587 0.266 3.383 0.000720
factor(treatment)2:log(Age) -3.017 0.0489 1.270 -2.375 0.018000
I reported the results as a HR of 10.2 for treatment 2 compared to treatment 1 in developing the disease. And a HR of 7.0 for each unit (year) increase of Age.
Now I'm not sure how to include the interaction term results. Also the HRs are rather extreme, could there be a mistake in the log transformation of age?
Could anyone help me with that?
Thank you in advance!
Best Answer
That's incorrect, in several ways.
HR versus
coef
The value reported for a
coef
is for the association of a predictor with the log-hazard of an event. The baseline HR forfactor(treatment)2
is given byexp(coef)
, or 199569.6284 in your case! Yes, that's troubling, but makes (a bit) more sense when you consider the interaction term involved. See below.log(Age)
The
coef
value of 7.0, again, is for a log-hazard association with outcome. The HR is 55.7549! That's for 1 natural-log-unit increase of age, not for a 1-year increase in age. Also, aslog(Age)
is included in an interaction withtreatment
, the reported value is only for the baseline level oftreatment
, presumablyfactor(treatment)1
.The coefficient for a natural-log-transformed continuous predictor represents the change in outcome per e-fold change in the predictor. For example, that could be approximately between
Age = 20
andAge = 55
(natural logs of about 3 and 4, respectively). As this answer notes, a different choice of logarithm base can be easier to explain in words. For example, if you use base-2 logarithms, the HR would be per doubling of age. You could choose base-1.1 logarithms to give you the HR for a 10% increase in age.Implications of the interaction
As
treatment
is included in an interaction withlog(Age)
, thecoef
and HR (exp(coef)
) reported forfactor(treatment)2
are the values whenlog(Age) = 0
orAge = 1
. Did you have any 1-year-olds in your study?With a significant negative interaction between
treatment
andlog(Age)
, you need to consider specific ages to estimate the association oftreatment
with outcome. These calculations are best done on thecoef
scale, only exponentiating at the end to get the corresponding HR.For example, someone with
Age = 55
haslog(Age) = 4.0
. Multiply that by thecoef
value of -3 for thefactor(treatment)2:log(Age)
interaction, and you get at value of -12. Add that to thecoef
value of 10 forfactor(treatment)2
at the baseline oflog(Age) = 0
and you get a net coefficient of -2 forfactor(treatment)2
at an age of 55, or a hazard ratio of 0.14 forfactor(treatment)2
versusfactor(treatment)1
. The association oftreatment
with outcome switches at aboutAge = 29
in your model.I can't rule out some problems with the data themselves, and I certainly can't help with interpretation of things specific to the Fine-Gray analysis (which I don't understand very well). The problems in what you show here, however, mostly have to do with the interpretation of the results, in a way that often catches people by surprise when they start modeling with interactions in any type of regression.