Practical Issues on Using Weighted Logrank Tests with Interval-Censored Events in Clinical Trials - Interval-Censored-Time-to-Event Data: Methods and Applications - page 358

and c and z are averages over the n individuals. We can either use exact

methods using Q 0 as the test statistic, or use the permutational central limit
theorem, which shows that Q 0 is approximately distributed chi-square with g
1 degrees of freedom under the null. Alternatively, zi i may be a g-dimensional
vector and the generalized inverse of V p is used to define the quadratic form
(see, for example, Fay and Shih, 1998, Equation 4).

13.3.5 Score Tests with a High-Dimensional Nuisance

Parameter

Another way to derive the test is to treat S 0 as a high-dimensional nuisance
parameter and perform the score test in the usual way (see, for example, Cox

and Hinkley, 1974). Recall that the NPMLE is uniquely described at m points,
t 1 ;:::;t m , so we need m parameters to describe S 0 . The usual assumptions
for the score test may be violated in two ways. First, if S 0 (t j ) = S 0 (t j+1 )
for some j, then one of the associated parameters is on the boundary of the

parameter space, the first derivative of the log likelihood is not 0 (i.e., U 6= 0)
at the NPMLE, and the derivation of the score test cannot proceed in the

usual way. The second violation could occur if the number of parameters m

increases with sample size. Fay (1996) proposed an ad hoc solution: whenever
S 0 (t j ) = S 0 (t j+1 ), then eliminate one of the nuisance parameters, so that after
the ad hoc adjustment, none of the remaining nuisance parameters approach

the boundary of the parameter space.

The advantage of the score test, at least theoretically, is that if the as-

sessments are regular and the sample size is large so that neither of the two

assumptions mentioned above are violated, then we only need CIA and can

allow assessment treatment dependence.