time-dependent covariates: those that can be defined by a mathematical equa-
tion (external) and those measured directly from patients (internal); the former
categories are implemented much more easily.
Simple Test of Goodness of Fit Treatment of time-dependent covariates
leads to a simple test of goodness of fit. Consider the case of a fixed covariate,
denoted byX 1. Instead of the basic proportional hazards model,
lðtjX 1 ¼x 1 Þ¼lim
d# 0PrðtaTatþdjtaT;X 1 ¼x 1 Þ
d
¼l 0 ðtÞebx^1we can define an additional time-dependent covariateX 2 ,
X 2 ¼X 1 tConsider the expanded model,
lðt;X 1 ¼x 1 Þ¼l 0 ðtÞeb^1 x^1 þb^2 x^2¼l 0 ðtÞeb^1 x^1 þb^2 x^1 tand examine the significance of
H 0 :b 2 ¼ 0The reason for interest in testing whether or notb 2 ¼0 is thatb 2 ¼0 implies a
goodness of fit of the proportional hazards model for the factor under investi-
gation,X 1. Of course, in defining the new covariateX 2 ,tcould be replaced by
any function oft; a commonly used function is
X 2 ¼X 1 logðtÞThis simple approach results in a test of a specific alternative to the propor-
tionality. The computational implementation here is very similar to the case of
cumulative exposures; however,X 1 may be binary or continuous. We may even
investigate the goodness of fit for several variables simultaneously.
Example 11.6 Refer to the data set in Example 11.1 (Table 11.1), where the
remission times of 42 patients with acute leukemia were reported from a clini-
cal trial undertaken to assess the ability of the drug 6-mercaptopurine (6-MP)
to maintain remission. Each patient was randomized to receive either 6-MP or
placebo. The study was terminated after one year; patients have di¤erent fol-
MULTIPLE REGRESSION AND CORRELATION 403