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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
survfit function in the survival library
in R
(
on-line help).
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.)
Sometimes you just want the interval for one time, say 1000 days.
The summary.survfit function
(on-line help)
does that, as shown in the last line of the example above.
The log-rank or Mantel-Haenszel test of whether there is a difference
between two or more survival curves is performed using the
survdiff function in the survival library
in R
(
on-line help).
P = 0.00115 (Mantel-Haenszel 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
survdiff function only does two-tailed tests.
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.