Comparative poisson clinical trials of multiple experimental treatments vs a single control using the negative multinomial distribution

Abstract

This paper introduces a method which conditions on the number of events that occur in the control group to determine rejection regions and power for comparative Poisson trials with multiple experimental treatment arms that are each compared to one control arm. This leads to the negative multinomial as the statistical distribution used for testing. For one experimental treatment and one control with curtailed sampling, this is equivalent to Gail's (1974) approach. We provide formulas to calculate exact one‐sided overall Type I error and pointwise power for tests of treatment superiority and inferiority (vs the control). Tables of trial design parameters for combinations of one‐sided overall Type I error = 0.05, 0.01 and pointwise power = 0.90, 0.80 are provided. Curtailment approaches are presented to stop follow‐up of experimental treatment arms or to stop the study entirely once the final outcomes for each arm are known.

Economizing comparative Poisson clinical trials of multiple experimental treatments by testing against the same control

“…So, the expected number of subjects needed in an uncurtailed Design C trial is false( K + 1 false) × d C i C and the standard deviation is false( K + 1 false) 2 × d C i C 2. 12…”

“…The study is stopped once the control arm reaches a prespecified number of events d C. 6,7 It is shown in 6 that under equal allocation (i.e., N 1 = N 2 = … = N K = N C), the conditional distribution of D 1 , D 2 , … , D K falsefalse| d C is negative multinomial with parameters d C , i C i C + false∑ k = 1 K 0.2em i k , i 1 i C + false∑ k = 1 K 0.2em i k , i 2 i C + false∑ k = 1 K 0.2em i k , … , i K i C + false∑ k = 1 K 0.2em …”

“…The Design C procedure controls the family-wise error rate to be ≤ α across the K experimental treatments. The Type 1 error and pointwise power (the probability to declare any experimental treatment with rate ratio r to be superior to the control) are given, respectively, by 7 and As for Design A, the power under Design C (and hence the calculation of the parameters d C and m) 7 depends on the incidence rates of events in the experimental and control arms only through r .…”

“…For a curtailed Design C study, the expected number of subjects to enroll and corresponding standard deviation under the composite null hypothesis can be estimated using the following simulation algorithm 12 : Generate K random variables from a Gamma ( m + 1 , 1 i C ) distribution to represent the number of subjects needed to enroll in the K experimental treatment arms to reach m + 1 events. Let bold-italicT x be a vector containing these K values. Generate a random variable, denoted by C , from a Gamma ( d C , 1 i C ) distribution to represent the number of subjects needed to enroll for the control arm to reach d C events. Create the vector T bold-italicx bold∗ as follows: For each entry in bold-italicT x, if the entry is greater than or equal to C , the corresponding entry in T bold-italicx bold∗ is set to C .…”

“…Subjects are enrolled and observation continues until the control arm reaches d C events, at which time the numbers of events D k , k = 1 , 2 , … , K , in the K experimental treatment arms are evaluated. 6,7 Experimental treatment arms with a sufficiently small number of events at the time of study stoppage are declared superior to the control 7 (see the right-hand side of Figure 1 for a sample schematic of this approach).…”