Applied Statistics Using SPSS, STATISTICA, MATLAB and R

372 9 Survival Analysis
the proportional hazards assumption, i.e., the assumption that the hazards can be
expressed as:
h1(t) = ψ h2(t), 9.36
where the positive constant ψ is known as the hazard ratio, mentioned in 9.3.
Let X be an indicator variable such that its value for the ith individual, xi, is 1 or
0, according to the group membership of the individual. In order to impose a
positive value to ψ, we rewrite formula 9.36 as:
βx
h ( t)
= e i h0
( t)
. 9.37
i
Thus h2(t) = h0(t) and ψ = e β . This model can be generalised for p explanatory
variables:
η
h ( t)
e i
i = h0
( t)
, with η i = β1
x1i + β 2x2i
+ K+
β p x pi , 9.38
where ηi is known as the risk score and h0(t) is the baseline hazard function, i.e.,
the hazard that one would obtain if all independent explanatory variables were
zero.
The Cox regression model is the most general of the regression models for
survival data since it does not assume any particular underlying survival
distribution. The model is fitted to the data by first estimating the risk score using a
log-likelihood approach and finally computing the baseline hazard by an iterative
procedure. As a result of the model fitting process, one can obtain parameter
estimates and plots for specific values of the explanatory variables.
Example 9.10
Q: Determine the Cox regression solution for the Heart Valve dataset (eventfree
survival time), using Age as the explanatory variable. Compare the survivor
functions and determine the estimated percentages of an event-free 10-year postoperative
period for the mean age and for 20 and 60 years-old patients as well.
A: STATISTICA determines the parameter βAge = 0.0214 for the Cox regression
model. The chi-square test under the null hypothesis of “no Age influence” yields
an observed p = 0.004. Therefore, variable Age is highly significant in the
estimation of survival times, i.e., is an explanatory variable.
Figure 9.9a shows the baseline survivor function. Figures 9.9b, c and d, show
the survivor function plots for 20, 47.17 (mean age) and 60 years, respectively. As
expected, the probability of a given post-operative event-free period decreases with
age (survivor curves lower with age). From these plots, we see that the estimated
percentages of patients with post-operative event-free 10-year periods are 80%,
65% and 59% for 20, 47.17 (mean age) and 60 year-old patients, respectively.