The Essentials of Biostatistics for Physicians, Nurses, and Clinicians

10.4 Parametric Survival Curves 169

Table 10.5

Negative Exponential Survival Estimates for Patients From

Table 10.3

Time

T

Number

of

deaths

Dj

Number of

withdrawals

Wj

Number

at risk

n j

Est.

prop,

of

deaths

q j

Est. prop.

surviving

p j

K M

survival

estimate

Negative

exp.

survival

estimate

1.5 1 0 10 0.1 0.9 0.9 0.940

4.3 1 1 8 0.125 0.875 0.788 0.838

5.4 1 0 7 0.143 0.857 0.675 0.801

11.8 1 0 6 0.167 0.833 0.562 0.616

18 0 5 5 0 1 0.562 0.478

h ( t ) = λ exp( − λ t )/exp( − λ t ) = λ. In this case, we will fi t an exponential
model to the data used to fi t the Kaplan – Meier curve in Table 10.3.
Table 10.5 compares the estimated negative exponential survival curve
with the Kaplan – Meier estimate.
The exponential survival curve differs markedly from the Kaplan –
Meier curve, indicating that the negative exponential does not ade-
quately fi t the data.

10.4.2 Weibull Family of Survival Distributions

The Weibull model is more general and involves two parameters λ and
β. The negative exponential is the special case of a Weibull model,
when β = 1. The Weibull is common in reliability primarily because it
is the limiting distribution for the minimum of a sequence of indepen-
dent identically distributed random variables. In some situations, a
failure time can be the fi rst of many possible event times, and hence is
a minimum. So under common conditions, the Weibull occurs as an
extreme value limiting distribution similar to the way the normal dis-
tribution is the limiting distribution for sums or averages.
For the Weibull model S ( t ) = exp( − ( λ t ) β^ ), F ( t ) = 1 − exp( − ( λ t ) β^ ),
f ( t ) = λ β ( λ t ) β^ − 1 exp[ − ( λ t ) β^ ], and h ( t ) = λ β ( λ t ) β^ −^1. For the Weibull model,
λ > 0 and β > 0.