The likelihood ratio test for non-standard hypotheses near the boundary of the null – with application to the assessment of non-inferiority

Abstract

We consider a class of testing problems where the null space is the union of k − 1 subgraphs of the form h j (θ j ) ≤ θ k , with j = 1, . . . , k − 1, (θ 1 , . . . , θ k ) the unknown parameter, and h j given increasing functions. The data consist of k independent samples, assumed to be drawn from a distribution with parameter θ j , j = 1, . . . , k, respectively. An important class of examples covered by this setting is that of non-inferiority hypotheses, which have recently become important in the evaluation of drugs or therapies. When the true parameter approaches the boundary at a 1/ √ n rate, we give the explicit form of the asymptotic distribution of the log-likelihood ratio statistic. This extends previous work on the distribution of likelihood ratio statistics to local alternatives. We consider the prominent example of binomial data and illustrate the theory for k = 2 and 3 samples. We explain how this can be used for planning a non-inferiority trial. To this end we calculate the optimal sample ratios yielding the maximal power in a binomial non-inferiority trial.

Blinded sample size re‐estimation in three‐arm trials with ‘gold standard’ design

“…For the 'gold standard' design multiple hypotheses and testing strategies have been proposed [7][8][9][10]. and planning and analzing studies in this design received great attention in past years [11][12][13][14][15][16][17][18].…”

“…Three-arm trials with the 'gold standard' design are the recommended design in indications such as asthma, schizophrenia, and migraine [8][9][10]. For the 'gold standard' design, multiple hypotheses and testing strategies have been proposed [7,[11][12][13], and planning and analyzing studies in this design received great attention in the past years [14][15][16][17][18][19][20][21]. In general, in three-arm trials with the 'gold standard' design, one distinguishes two approaches to defining the effect size and the statistical hypothesis: the absolute margin approach and the retention-of-effect approach.…”

Design and analysis of three‐arm trials with negative binomially distributed endpoints

“…Pigeot et al , , Hauschke and Pigeot , and Hasler et al , studied the retention of effect hypothesis for normally distributed endpoints, Kieser and Friede for binary distributed endpoints, and Mielke et al , for censored exponentially distributed endpoints. Moreover, Mielke and Munk and Balabdaoui et al , established the maximum‐likelihood theory for a generalization of the retention of effect hypothesis. Munzel derived a statistical test for the non‐parametric equivalent.…”

Maximum Likelihood Theory for Retention of Effect Non-Inferiority Trials

“…the hypothesis of demonstrating that a new treatment is non-inferior to two standard treatments, whereas the hypothesis H I is suitable for showing that a new treatment is as effective as one of two given standards, or that one of two treatments is as effective as a standard, the latter being for example the aim in Hesketh et al (1996). Furthermore, the extension of H U to k-samples (Balabdaoui, Mielke and Munk, 2009), i.e. the comparison of treatment to an arbitrary number of other treatments, can be found in dose finding studies.…”

“…Note that the likelihood ratio converges to zero under a parameter in the alternative. To this end, Balabdaoui et al (2009) derive the asymptotic distribution of the likelihood ratio statistic for H U with k-samples under a local alternative. Another approach fixes the alternative and considers a higher-order expansion of the likelihood ratio statistic to obtain a non-degenerated limit distribution (see Mielke, 2007).…”