Statistical Methods in Medical Research 4ed

17.9 Diagnostic methods 589
is often used for this purpose. The integrated hazard may be obtained from the
Kaplan±Meier estimate of S…t† using (17.29), or from the cumulative hazard,
evaluated as the sum of the estimated discrete hazards at all the event times up to
t. A plot of ln H…t† against ln t is linear with a slope of g for the Weibull (17.21),
or a slope of 1 for the exponential (17.20).
For a more general model (17.25), the plot of ln H…t† against ln t has no
specified form but plots made for different subgroups of individualsÐfor
example, defined by categories of a qualitative covariate or stratified ranges of
a continuous covariateÐmay give guidance on whether a proportional-hazards
or accelerated failure time model is the more appropriate choice for the effect
of the covariates. For a proportional-hazards model the curves are separated
by constant vertical distances, and for an accelerated failure time model by
constant horizontal distances. Both of these conditions are met if the plots
are linear, reflecting the fact that the Weibull and exponential are both proportional-hazards
and accelerated failure time models. Otherwise it may be
difficult to distinguish between the two possibilities against the background of
chance variability, but then the two models may give similar inferences
(Solomon, 1984).
The graphical approach to checking the proportional-hazards assumption
does not provide a formal diagnostic test. Such a test may be constructed by
including an interaction term between a covariate and time in the model. In an
analysis with one explanatory variable x, suppose that a time-dependent variable
z is defined as x ln …t†, and that in a regression of the log hazard on x and z the
regression coefficients are, respectively, b and g. Then the relative hazard for an
increase of 1 unit in x is t g exp…b†. The proportional-hazards assumption holds if
g ˆ 0, whilst the relative hazard increases or decreases with time if g > 0or
g < 0, respectively. A test of proportional hazards is, therefore, provided by the
test of the regression coefficient g against the null hypothesis that g ˆ 0.
As discussed in §11.9, residual plots are often useful as a check on the
assumptions of the model and for determining if extra covariates should be
included. With survival data it is not as clear as for a continuous outcome
variable what is meant by a residual. A generalized residual (Cox & Snell,
1968) for a Cox proportional-hazards model is defined for the ith individual as
ri ˆ ^H0…t† exp…b T xi†, …17:30†
where b is the estimate of b, and ^H0…t† is the fitted cumulative hazard corresponding
to the time-dependent part of the hazard, l0…t† in (17.25), which may be
estimated as a step function with increment 1/exp…b T xj† for each death. These
residuals should be equivalent to a censored sample from an exponential distribution
with mean 1, and, if the ri are ordered and plotted against the estimated
cumulative hazard rate of the ri, then the plot should be a straight line through
the origin with a slope of 1.