ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

CHAPTER 9 ST 520, A. TSIATIS and D. Zhang
• γ > 0 implies that individuals on treatment 1 have worse survival (i.e. die faster)
• γ = 0 implies the null hypothesis
• γ < 0 implies that individuals on treatment 1 have better survival (i.e. live longer)
If the proportional hazards assumption were true; that is,
then this would imply that
or
Consequently,
and
d log S1(t)
−
dt
λ1(t) = λ0(t) exp(γ),
d log S0(t)
= − exp(γ),
dt
− log S1(t) = − log S0(t) exp(γ).
S1(t) = {S0(t)} exp(γ) ,
log{− log S1(t)} = log{− log S0(t)} + γ.
This last relationship can be useful if we want to assess whether a proportional hazards as-
sumption is a reasonable representation of the data. By plotting the two treatment-specific
Kaplan-Meier curves on a log{− log} scale we can visually inspect whether these two curves
differ from each other by a constant over time.
Also, in the special case where we feel comfortable in assuming that the survival distributions
follow an exponential distribution; i.e. constant hazards, the proportional hazards assumption is
guaranteed to hold. That is,
λ1(t)
λ0(t)
λ1
= .
λ0
In section 9.1 we showed that the median survival time for an exponential distribution with
hazard λ is equal to m = log(2)/λ. Therefore, the ratio of the median survival times for two
treatments whose survival distributions are exponentially distributed with hazard rates λ1 and
λ0 is
m1
m0
= {log(2)/λ1}
{log(2)/λ0}
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λ0
= .
λ1