Practical Issues on Using Weighted Logrank Tests with Interval-Censored Events in Clinical Trials - Interval-Censored-Time-to-Event Data: Methods and Applications - page 360

using Monte Carlo methods within each imputed data set (see Fay and Shih

2012 for details). Fay and Shih (2012) showed by simulation that the Monte-
Carlo method retained the type I error even for very small sample sizes with

ATD.

13.3.7 Other Closely Related Methods
Although we have written the ecient score statistics as U = P z i c i , in fact,
those same ecient scores may be written in a weighted \observed minus ex-

pected" form (Fay, 1999). We call that alternate formulation the weighted
logrank formulation of the ecient score. Because the two forms are equiv-

alent mathematically, any asymptotic results derived under one formalation

hold under the other. Thus, although Sun (2006, p.83) says that Sun (1996)

can be invalid if the percentage of purely right-censored data is large, that may

not be true because the test of Sun (1996) can be derived using the methods

of Section 13.3.5 (see Fay, 1999). In other words, the problems of Sun (1996)

exactly match the problems mentioned in Section 13.3.5 (i.e., number of nui-

sance parameters growing with sample size and those nuisance parameters

approaching the boundary of the parameter space), and those problems may

or may not occur when there is a large percentage of purely right-censored

data. Also, although Sun (1996) was derived for discrete data and Finkelstein

(1986) was derived for continuous data, both methods work under either type

of events (see Fay and Proschan, 2010, Section 6).

Sun et al. (2005) derive a generalized logrank test, where their test statis-
tic denoted U is equivalent to U in this chapter for many cases. They then
derive the variance for U as n n V p , where V p is given in Equation (13.8) (the
notation is dicult in Sun et al. (2005); see the expression of the variance in

simplified notation in So et al. (2010)). Sun et al. (2005) show that U stan-

dardized by its variance is asymptotically multivariate normal under Case II
interval-censoring. Except for the factor of n1
n

in the variance, the test of So

et al. (2010) with (x) = x log(x) is identical to the logrank test proposed by