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The LIFETEST Procedure |
Let be the number of groups. Let
be the underlying survivor function
th 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
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where is the total number of events in the
th stratum,
,
is the total time on test in the
th stratum, and
. The approximate probability value is computed by treating
as having a chi-square distribution with
–1 degrees of freedom.
Let be the distinct event times in the pooled 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
th sample, respectively. Let
,
, and
.
The choices of the weight function are given in Table 49.3.
Test |
|
---|---|
log-rank |
1.0 |
Wilcoxon |
|
Tarone-Ware |
|
Peto-Peto |
|
modified Peto-Peto |
|
Harrington-Fleming ( |
|
where is the product-limit estimate at
for the pooled sample, and
is a survivor function estimate close to
given by
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The rank statistics (Klein and Moeschberger; 1997, Section 7.3) for testing versus
have the form of a
-vector
with
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and the estimated covariance matrix, , is given by
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where is 1 if
and 0 otherwise. The term
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
, 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.
Suppose the test is to be stratified on levels of a set of STRATA variables. Based only on the data of the
th stratum (
), let
be the test statistic (Klein and Moeschberger; 1997, Section 7.5) for the
th stratum, and let
be its covariance matrix. Let
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A global test statistic is constructed as
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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 chi-squared random variable with
degrees of freedom. Denote
and
as the density function and the distribution function of a standard normal distribution, respectively. Let
be the number of comparisons; that is,
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For a two-sided test comparing the survival of the th group with that of
th group,
, the test statistic is
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and the raw p-value is
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Adjusted p-values for various multiple-comparison adjustments are computed as follows:
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With the first group being the control, let be the
matrix of contrasts; that is,
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Let and
be covariance and correlation matrices of
, respectively; that is,
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and
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The factor-analytic covariance approximation of Hsu (1992) is to find such that
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where is a diagonal matrix with the
th diagonal element being
and
. The adjusted p-value is
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which can be obtained in a DATA step as 1 – PROBMC("DUNNETT2", ,.,.,
).
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which can also be evaluated in a DATA step as 1 – PROBMC("MAXMOD",).
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which can also be evaluated in a DATA step as 1 – PROBMC("RANGE",).
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
samples. The test statistic and its standard error are given by
and
, respectively. Under
, the z-score
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has, asymptotically, a standard normal distribution. PROC LIFETEST provides both one-tail and two-tail p-values for the test.
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Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.