I want to estimate an exponential hazards model with one predictor in R. For some reason, I am getting coefficients with opposite signs when I estimate it using a glm poisson with offset log t and when I just use the survreg function from the survival package. I am sure the explanation is perfectly obvious but I can not figure it out.
Example
t <- c(89,74,23,74,53,3,177,44,28,43,25,24,31,111,57,20,19,137,45,48,9,17,4,59,7,26,180,56,36,51,6,71,23,6,13,28,16,180,16,25,6,25,4,5,32,94,106,1,69,63,31)
d <- c(0,1,1,0,1,1,0,1,1,0,1,1,1,1,0,0,1,0,1,1,1,0,1,0,1,1,0,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1)
p <- c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,1,1,1)
df <- data.frame(d,t,p)
# exponential hazards model using poisson with offest log(t)
summary(glm(d ~ offset(log(t)) + p, data = df, family = "poisson"))
Produces:
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -5.3868 0.7070 -7.619 2.56e-14 ***
p 1.3932 0.7264 1.918 0.0551 .
Compared to
# exponential hazards model using survreg exponential
require(survival)
summary(survreg(Surv(t,d) ~ p, data = df, dist = "exponential"))
Produces:
Value Std. Error z p
(Intercept) 5.39 0.707 7.62 2.58e-14
p -1.39 0.726 -1.92 5.51e-02
Why are the coefficients in opposite directions and how would I interpret the results as they stand? Thanks!
In the second model an increased value of p is associated with a decreased expected survival. In the first model the increased p that had a long t value would imply a higher chance of survival and a lower risk. Variations in risk and mean survival time values of necessity go in opposite directions. The fact that the absolute values are the same comes from the mathematical identity log(1/x) = -log(x). The risk is (exactly) inversely proportional to mean lifetime in exponential models.