Biomedical Engineering Reference
In-Depth Information
ecient rank test for proportional hazards, the logrank test, and the logrank
scores are (Peto and Peto, 1972; Finkelstein, 1986; Fay, 1999)
n
S
0
(`
i
)
o
S
0
(r
i
) log
n
S
0
(r
i
)
o
S
0
(`
i
) log
c
i
=
;
(13.5)
S
0
(`
i
) S
0
(r
i
)
where we let 0log(0) = 0. Sun (1996) proposed a slightly dierent version of
these logrank scores (see also Fay, 1999; Fay and Shaw, 2010),
S
0
(`
i
) log
n
S
0
(`
i
)
o
S
0
(r
i
) log
n
S
0
(r
i
)
o
S
0
(`
i
) S
0
(r
i
)
c
i
=
;
(13.6)
where S
0
(t) is a function of S
0
(t) that is like a Nelson{Aalen-type survival
estimator and reduces to that estimator for right-censored data. For right-
censored data with no ties in the deaths, these scores are equivalent to those
derived from the linear rank tests on the accelerated failure time model with
the extreme value distribution (see Kalbfleisch and Prentice, 2002, p. 221).
If F is logistic, this leads to the proportional odds model and the scores
reduce to a linear function of the ranks when there is no censoring (Peto and
Peto, 1972; Fay, 1996):
S
0
(`
i
) +
S
0
(r
i
) 1:
c
i
=
(13.7)
13.3.4
Permutation-Based Rank Tests
If there is no assessment treatment dependence (e.g., the data are TIA), then
under the null hypothesis the scores ci
i
are exchangeable and we can simply
perform a permutation test on those scores. As with all linear permutation
tests, we can estimate the p-value in at least three different ways: (i) exactly
calculate the p-value from all possible permutations, (ii) estimate the exact
p-value by Monte Carlo simulation, or (iii) estimate the p-value by the per-
mutational central limit theorem (PCLT).
It is conceptually easy to determine the exact p-value by complete enu-
meration. Consider, for example, the one-sided test for the two-sample case.
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