PROC LIFETEST: Comparison of Two or More Groups of Survival Data :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The LIFETEST Procedure

Comparison of Two or More Groups of Survival Data

Let $\text{[math]}$ be the number of groups. Let $\text{[math]}$ be the underlying survivor function $\text{[math]}$ th group, $\text{[math]}$ . The null and alternative hypotheses to be tested are

$\text{[math]}$ for all $\text{[math]}$

versus

$\text{[math]}$ at least one of the $\text{[math]}$ ’s is different for some $\text{[math]}$

respectively, where $\text{[math]}$ is the largest observed time.

Likelihood Ratio Test

The likelihood ratio test statistic (Lawless; 1982) for test $\text{[math]}$ versus $\text{[math]}$ assumes that the data in the various samples are exponentially distributed and tests that the scale parameters are equal. The test statistic is computed as

$\text{[math]}$

where $\text{[math]}$ is the total number of events in the $\text{[math]}$ th stratum, $\text{[math]}$ , $\text{[math]}$ is the total time on test in the $\text{[math]}$ th stratum, and $\text{[math]}$ . The approximate probability value is computed by treating $\text{[math]}$ as having a chi-square distribution with $\text{[math]}$ –1 degrees of freedom.

Nonparametric Tests

Let $\text{[math]}$ be the distinct event times in the pooled sample. At time $\text{[math]}$ , let $\text{[math]}$ be a positive weight function, and let $\text{[math]}$ and $\text{[math]}$ be the size of the risk set and the number of events in the $\text{[math]}$ th sample, respectively. Let $\text{[math]}$ , $\text{[math]}$ , and $\text{[math]}$ .

The choices of the weight function $\text{[math]}$ are given in Table 49.3.

Table 49.3 Weight Functions for Various Tests
Test	$\text{[math]}$
log-rank	1.0
Wilcoxon	$\text{[math]}$
Tarone-Ware	$\text{[math]}$
Peto-Peto	$\text{[math]}$
modified Peto-Peto	$\text{[math]}$
Harrington-Fleming ( $\text{[math]}$ , $\text{[math]}$ )	$\text{[math]}$

where $\text{[math]}$ is the product-limit estimate at $\text{[math]}$ for the pooled sample, and $\text{[math]}$ is a survivor function estimate close to $\text{[math]}$ given by

$\text{[math]}$

Unstratified Tests

The rank statistics (Klein and Moeschberger; 1997, Section 7.3) for testing $\text{[math]}$ versus $\text{[math]}$ have the form of a $\text{[math]}$ -vector $\text{[math]}$ with

$\text{[math]}$

and the estimated covariance matrix, $\text{[math]}$ , is given by

$\text{[math]}$

where $\text{[math]}$ is 1 if $\text{[math]}$ and 0 otherwise. The term $\text{[math]}$ can be interpreted as a weighted sum of observed minus expected numbers of failure under the null hypothesis of identical survival curves. The overall test statistic for homogeneity is $\text{[math]}$ , where $\text{[math]}$ denotes a generalized inverse of $\text{[math]}$ . This statistic is treated as having a chi-square distribution with degrees of freedom equal to the rank of $\text{[math]}$ for the purposes of computing an approximate probability level.

Stratified Tests

Suppose the test is to be stratified on $\text{[math]}$ levels of a set of STRATA variables. Based only on the data of the $\text{[math]}$ th stratum ( $\text{[math]}$ ), let $\text{[math]}$ be the test statistic (Klein and Moeschberger; 1997, Section 7.5) for the $\text{[math]}$ th stratum, and let $\text{[math]}$ be its covariance matrix. Let

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

A global test statistic is constructed as

$\text{[math]}$

Under the null hypothesis, the test statistic has a $\text{[math]}$ distribution with the same degrees of freedom as the individual test for each stratum.

Multiple-Comparison Adjustments

Let $\text{[math]}$ denote a chi-squared random variable with $\text{[math]}$ degrees of freedom. Denote $\text{[math]}$ and $\text{[math]}$ as the density function and the distribution function of a standard normal distribution, respectively. Let $\text{[math]}$ be the number of comparisons; that is,

$\text{[math]}$

For a two-sided test comparing the survival of the $\text{[math]}$ th group with that of $\text{[math]}$ th group, $\text{[math]}$ , the test statistic is

$\text{[math]}$

and the raw p-value is

$\text{[math]}$

Adjusted p-values for various multiple-comparison adjustments are computed as follows:

Bonferroni Adjustment

$\text{[math]}$

Dunnett-Hsu Adjustment

With the first group being the control, let $\text{[math]}$ be the $\text{[math]}$ matrix of contrasts; that is,

$\text{[math]}$

Let $\text{[math]}$ and $\text{[math]}$ be covariance and correlation matrices of $\text{[math]}$ , respectively; that is,

$\text{[math]}$

and

$\text{[math]}$

The factor-analytic covariance approximation of Hsu (1992) is to find $\text{[math]}$ such that

$\text{[math]}$

where $\text{[math]}$ is a diagonal matrix with the $\text{[math]}$ th diagonal element being $\text{[math]}$ and $\text{[math]}$ . The adjusted p-value is

$\text{[math]}$

which can be obtained in a DATA step as 1 – PROBMC("DUNNETT2", $\text{[math]}$ ,.,., $\text{[math]}$ ).

Scheffé Adjustment

$\text{[math]}$

idák Adjustment

$\text{[math]}$

SMM Adjustment

$\text{[math]}$

which can also be evaluated in a DATA step as 1 – PROBMC("MAXMOD", $\text{[math]}$ ).

Tukey Adjustment

$\text{[math]}$

which can also be evaluated in a DATA step as 1 – PROBMC("RANGE", $\text{[math]}$ ).

Trend Tests

Trend tests (Klein and Moeschberger; 1997, Section 7.4) have more power to detect ordered alternatives as

$\text{[math]}$ with at least one inequality

Let $\text{[math]}$ be a sequence of scores associated with the $\text{[math]}$ samples. The test statistic and its standard error are given by $\text{[math]}$ and $\text{[math]}$ , respectively. Under $\text{[math]}$ , the z-score

$\text{[math]}$

has, asymptotically, a standard normal distribution. PROC LIFETEST provides both one-tail and two-tail p-values for the test.

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