# The LIFETEST Procedure

#### Comparison of Two or More Groups of Survival Data

Let K be the number of groups. Let be the underlying survivor function of the kth group, . The null and alternative hypotheses to be tested are

for all

versus

at least one of the ’s is different for some

respectively, where is the largest observed time.

##### Likelihood Ratio Test

The likelihood ratio test statistic (Lawless 1982) for test versus 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

where is the total number of events in the kth group, , is the total time on test in the kth stratum, and . The approximate probability value is computed by treating as having a chi-square distribution with K – 1 degrees of freedom.

##### Nonparametric Tests

Let ( denote an independent sample of right-censored survival data, where is the possibly right-censored time, is the censoring indicator (=0 if is censored and =1 if is an event time), and for K different groups. Let be the distinct event times in the sample. At time let be a positive weight function, and let and be the size of the risk set and the number of events in the kth group, respectively. Let , .

The choices of the weight function are given in Table 70.3.

Table 70.3: Weight Functions for Various Tests

Test

Log-rank

1.0

Wilcoxon

Tarone-Ware

Peto-Peto

Modified Peto-Peto

Harrington-Fleming (p,q)

In Table 70.3, is the product-limit estimate at t for the pooled sample, and is a survivor function estimate close to given by

###### Unstratified Tests

The rank statistics (Klein and Moeschberger 1997, Section 7.3) for testing versus have the form of a K-vector with

and the variance of and the covariance of and are, respectively,

The statistic can be interpreted as a weighted sum of observed minus expected numbers of failure for the kth group under the null hypothesis of identical survival curves. Let . The overall test statistic for homogeneity is , where denotes a generalized inverse of . This statistic is treated as having a chi-square distribution with degrees of freedom equal to the rank of for the purposes of computing an approximate probability level.

###### Adjusted Log-Rank Test

PROC LIFETEST computes the weighted log-rank test (Xie and Liu 2005, 2011) if you specify the WEIGHT statement. Let ( denote an independent sample of right-censored survival data, where is the possibly right-censored time, is the censoring indicator (=0 if is censored and =1 if is an event time), for K different groups, and is the weight from the WEIGHT statement. Let be the distinct event times in the sample. At each , and for each , let

Let and denote the number of events and the number at risk, respectively, in the combined sample at time . Similarly, let and denote the weighted number of events and the weighted number at risk, respectively, in the combined sample at time . The test statistic is

and the variance of and the covariance of and are, respectively,

Let . Under , the weighted K-sample test has a statistic given by

with K – 1 degrees of freedom.

###### Stratified Tests

Suppose the test is to be stratified on M levels of a set of STRATA variables. Based only on the data of the sth stratum (), let be the test statistic (Klein and Moeschberger 1997, Section 7.5) for the sth stratum, and let be its covariance matrix. Let

A global test statistic is constructed as

Under the null hypothesis, the test statistic has a distribution with the same degrees of freedom as the individual test for each stratum.

###### Multiple-Comparison Adjustments

Let denote a chi-square random variable with r degrees of freedom. Denote and as the density function and the cumulative distribution function of a standard normal distribution, respectively. Let m be the number of comparisons; that is,

For a two-sided test that compares the survival of the jth group with that of lth group, , the test statistic is

and the raw p-value is

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

• Bonferroni adjustment:

• Dunnett-Hsu adjustment: With the first group being the control, let be the matrix of contrasts; that is,

Let and be covariance and correlation matrices of , respectively; that is,

and

The factor-analytic covariance approximation of Hsu (1992) is to find such that

where is a diagonal matrix with the jth diagonal element being and . The adjusted p-value is

which can be obtained in a DATA step as

• Scheffé adjustment:

• Šidák adjustment:

• SMM adjustment:

which can also be evaluated in a DATA step as

• Tukey adjustment:

which can also be evaluated in a DATA step as

###### Trend Tests

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

with at least one inequality

or

with at least one inequality

Let be a sequence of scores associated with the k samples. The test statistic and its standard error are given by and , respectively. Under , the z-score

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