is almost significant at the 5% levelðp¼ 0 : 0732 Þ; that is, the presence of the
morphologic characteristic modifies substantially the e¤ect of WBC.
11.4.3 Time-Dependent Covariates and Applications
In prospective studies, since subjects are followed over time, values of many
independent variables or covariates may be changing; covariates such as
patient age, blood pressure, and treatment. In general, covariates are divided
into two categories: fixed and time dependent. A covariate istime dependent
if thedi¤erencebetween covariate values from two di¤erent subjects may be
changing with time. For example, gender and age are fixed covariates;a
patient’s age is increasing by one a year, but the di¤erence in age between two
patients remains unchanged. On the other hand, blood pressure is an obvious
time-dependent covariate.
Examples The following are three important groups of time-dependent co-
variates.
1.Personal characteristics whose measurements are made periodically during
the course of a study. Blood pressure fluctuates; so do cholesterol level
and weight. Smoking and alcohol consumption habits may change.
2.Cumulative exposure. In many studies, exposures such as smoking are
often dichotomized; subjects are classified as exposed or unexposed. But
this may be oversimplified, leading to loss of information; the length of
exposure may be important. As time goes on, a nonsmoker remains a
nonsmoker, but ‘‘years of smoking’’ for a smoker increases.
3.Switching treatments. In a clinical trial, a patient may be transferred from
one treatment to another due to side e¤ects or even by a patient’s request.
Organ transplants form another category with switching treatments;
when a suitable donor is found, a subject is switched from the non-
transplanted group to the transplanted group. The case of intensive care
units is even more complicated, as a patient may be moved in and out
more than once.Implementation Recall that in analysis using proportional hazards model, we
order the death times and form the partial likelihood function:
TABLE 11.4
Factor Coe‰cient
Standard
Error zStatistic pValueWBC 0.14654 0.17869 0.821 0.4122
AG group 5.85637 2.75029 2.129 0.0332
Product 0.49527 0.27648 1.791 0.0732
MULTIPLE REGRESSION AND CORRELATION 401