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


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

# relative survival model
    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

In a relative survival model, the survival is compared to expected survival. The expected survival for older people is less than for younger people. So, when an old person gets an illness, he/she loses fewer years of life.

In the simplest case of a completely and quickly fatal disease, people will lose exactly the life expectancy at their age, so a (say) 40 year old would lose almost 10 years more than a 50 year old ("almost" because there is a slight risk of death in your 40s).


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