Applied Statistics Using SPSS, STATISTICA, MATLAB and R

368 9 Survival Analysis
The exponential model can be fitted to the data using a maximum likelihood
procedure (see Appendix C). Concretely, let the data consist of n survival times, t1,
t2, …, tn, of which r are death times and n – r are censored times. Then, the
likelihood function is:
n
−λti
δi
−λt
L e e i 1−δ
⎧0
ith
individual is censored
( λ)
= λ
i
∏ ( ) ( ) with δ i = ⎨
. 9.25
i=
1
⎩1
otherwise
Equivalently:
n
∏
i=
1
δi
−λti
L(
λ)
= λ e , 9.26
from where the following log-likelihood formula is derived:
n
∑δilog λ − λ∑
n
log L(
λ ) =
t = r log λ − λ t . 9.27
i
i=
1 i=
1
n
∑
i
i=
1
The maximum log-likelihood is obtained by setting to zero the derivative of
9.27, yielding the following estimate of the parameter λ:
n
= ∑ i=
i t ˆλ
1
. 9.28
1 r
The standard error of this estimate is ˆ λ / r.
The following statistics are easily
derived from 9.24:
ˆ
0 . 5 =
λˆ t ln 2 / . 9.29a
λˆ ˆ = ln( 1/(
1−
p))
/ . 9.29b
t p
The standard error of these estimates is tˆ p / r .
Example 9.8
Q: Consider the survival data of Example 9.5 (Heart Valve dataset). Determine
the exponential estimate of the survivor function and assess the validity of the
model. What are the 95% confidence intervals of the parameter λ and of the
median time until an event occurs?
A: Using STATISTICA, we obtain the survival and hazard functions estimates
shown in Figure 9.7. STATISTICA uses a weighted least square estimate of the
model function instead of the log-likelihood procedure. The exponential model fit
shown in Figure 9.7 is obtained using weights nihi, where ni is the number of
observations at risk in interval i of width hi. Note that the life-table estimate of the
hazard function is suggestive of a constant behaviour. The chi-square goodness of
fit test yields an observed significance of 0.59; thus, there is no evidence leading to
the rejection of the null, goodness of fit, hypothesis.