## 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.