University of Minnesota, Twin Cities School of Statistics Stat 5601 Rweb Computing Examples
Source: Marshall and Proschan (1965), Annals of Mathematical Statistics 65:69-77, pp. 70-71, esp. equations (3.6) and (3.2).
rate | lower limit | upper limit |
---|---|---|
0.0000 | -∞ | 42 |
0.0210 | 42 | 61 |
0.0333 | 61 | 66 |
0.0566 | 66 | 81 |
0.5000 | 81 | 82 |
∞ | 82 | ∞ |
The IFR point estimate is a function (the rate function),
as in many cases, the nonparametric function estimate is a step function
(like the empirical c. d. f.). Failure rate infinity
past x = 82 means all individuals surviving to that time
fail immediately. Similarly, failure rate zero before x = 42,
means no failures occur before then.
Thus the failure time distribution is concentrated on the observed range
of the data 42 < x < 82.
For comparison, the estimator assuming constant failure rate on (0, &infin),
the exponential failure time distribution, has failure rate
0.0154.
There is a similar DFR point estimate, also given by Marshall and Proschan
(1965) cited above. Since we have decided that this example is IFR rather
than DFR, we will skip it.
The Kaplan-Meier survival curve is estimated using the
The log-rank or Mantel-Haenszel test of whether there is a difference
between two or more survival curves is performed using the
P = 0.00115 (Mantel-Haenzel test).
The reason this disagrees with the book (Hollander and Wolfe, Section 11.7,
page 553) is that Hollander and Wolfe do a one-tailed test, and the
Of course, one can always convert between the two using two tails is
twice one tail. Indeed Hollander and Wolfe's P-value is
half of R's.
DFR Point Estimate
Kaplan-Meier
Point Estimate (Survival Curve)
survfit
function in the survival
library
in R
(
on-line help).
Example 11.7 in Hollander and Wolfe.
Confidence Interval
Example 11.7 in Hollander and Wolfe.
Comment
This is a pointwise not (!) simultaneous confidence interval for
the curve. Hollander and Wolfe describe simultaneous confidence bands
for the curve, but apparently the survival
package in R
does not implement them. (I have no idea why.)
Hypothesis Test
survdiff
function in the survival
library
in R
(
on-line help).
Example 11.7 in Hollander and Wolfe.
Summary
Comment
survdiff
function only does two-tailed tests.