General Instructions
To do each example, just click the Submit
button.
You do not have to type in any R instructions or specify a dataset.
That's already done for you.
Exponential versus IFR or DFR
Hypothesis Tests
Example 11.1 in Hollander and Wolfe.
IFR Point Estimate
Source: Marshall and Proschan (1965), Annals of Mathematical Statistics 65:69–77, pp. 70–71, esp. equations (3.6) and (3.2).
Summary
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, ∞), the exponential failure time distribution, has failure rate 0.0154.
DFR Point Estimate
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.
Kaplan-Meier
Point Estimate (Survival Curve)
The Kaplan-Meier survival curve is estimated using the
survfit
function in the survival
library
in R
(on-line help).
A Simple Toy Example.
Times are the numbers 1 to 10. The 3rd, 5th, and 9th are censored.
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.)
Single Confidence Interval
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.
Hypothesis Test
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).
Example 11.7 in Hollander and Wolfe.
Summary
P = 0.00115 (Mantel-Haenszel test).
Comment
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.