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.
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 chisquare distribution with K – 1 degrees of freedom.
Let ( denote an independent sample of rightcensored survival data, where is the possibly rightcensored 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 


Logrank 
1.0 
Wilcoxon 

TaroneWare 

PetoPeto 

Modified PetoPeto 

HarringtonFleming (p,q) 

In Table 70.3, is the productlimit estimate at t for the pooled sample, and is a survivor function estimate close to given by
The rank statistics (Klein and Moeschberger 1997, Section 7.3) for testing versus have the form of a Kvector 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 chisquare distribution with degrees of freedom equal to the rank of for the purposes of computing an approximate probability level.
PROC LIFETEST computes the weighted logrank test (Xie and Liu 2005, 2011) if you specify the WEIGHT statement. Let ( denote an independent sample of rightcensored survival data, where is the possibly rightcensored 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 Ksample test has a statistic given by
with K – 1 degrees of freedom.
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.
Let denote a chisquare 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 twosided test that compares the survival of the jth group with that of lth group, , the test statistic is
and the raw pvalue is
Adjusted pvalues for various multiplecomparison adjustments are computed as follows:
Bonferroni adjustment:
DunnettHsu 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 factoranalytic covariance approximation of Hsu (1992) is to find such that
where is a diagonal matrix with the jth diagonal element being and . The adjusted pvalue 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 (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 zscore
has, asymptotically, a standard normal distribution. PROC LIFETEST provides both onetail and twotail pvalues for the test.