# Interpretation of age coefficient in relative survival analysis

Suppose we have a Cox proportional hazards model with covariates including the patient's age, where the log hazard ratio for age is positive and significant, indicating increased risk for outcome in older patients.

Then, with the same data and covariates we run a relative survival model. The (raw) coefficient for age now becomes negative and significant. The signs of the coefficients for the other covariates do not change.

In this document, at the end of section 7.2, the negative sign is explained with the following comment:

It is not surprising that age is negatively associated with the outcome in such analysis as older patients will be losing relatively less than the younger ones.

I don't understand this. Could someone expand this explanation to help me make more sense of it? I understand that relative survival models use the risk of death in the general population (using life tables) so that the relative survival ratio is the ratio of observed survival to expected survival in the general population. The example referred to in the above document, along with the Cox model can be obtained in R using

library(relsurv)
data(slopop)
data(rdata)

# cox model
summary(coxmodel <- coxph(Surv(time,cens)~age+sex+year, data=rdata))

# relative survival model
rstrans(Surv(time,cens)~age+sex+year+
ratetable(age=age*365.24,sex=sex,year=year),
data=rdata, ratetable=slopop)


which produces the following for the cox model:

coxph(formula = Surv(time, cens) ~ age + sex + year, data = rdata)

n= 1040, number of events= 547

coef  exp(coef)   se(coef)      z Pr(>|z|)
age   5.938e-02  1.061e+00  4.324e-03 13.734  < 2e-16 ***
sex   9.011e-02  1.094e+00  9.585e-02  0.940  0.34716
year -2.736e-04  9.997e-01  9.177e-05 -2.981  0.00287 **


and this for the the relative survival model:

rstrans(formula = Surv(time, cens) ~ age + sex + year + ratetable(age = age *
365.24, sex = sex, year = year), data = rdata, ratetable = slopop)

coef exp(coef) se(coef)     z       p
age  -0.013904     0.986 4.92e-03 -2.83 4.7e-03
sex   0.528578     1.697 1.01e-01  5.24 1.6e-07
year -0.000228     1.000 9.04e-05 -2.52 1.2e-02