David Edwards and Jesper Madsen Novo Nordisk Sequentially rejective test procedures for partially ordered sets of hypotheses David Edwards and Jesper Madsen Novo Nordisk Or: a way to construct inference strategies for clinical trials that closely reflect the trial objectives and strongly control the FWE.

Outline Motivating example Some theory Examples Summary Partially closed test procedures 14 May 2019 Outline Motivating example Some theory Examples Summary

Partially closed test procedures 14 May 2019 Motivating Example Consider a three-arm trial, comparing a high and a low dose of an experimental drug with an active control. The goal is to demonstrate non-inferiority and, if possible, superiority of each dose to the active control. There are four null hypotheses: inferiority of high dose inferiority of low dose non-superiority of high dose non-superiority of low dose

Motivating Example We could consider Partially closed test procedures 14 May 2019 Motivating Example We could consider Test inferiority of high dose if rejected Test non-superiority of high dose Test inferiority of low dose if rejected Test non-superiority of low dose This gives strong FWE control for each dose, but not overall.

Motivating Example… Or we could consider Partially closed test procedures 14 May 2019 Motivating Example… Or we could consider Test inferiority for high dose if rejected Test inferiority for low dose Test non-superiority for high dose if rejected Test non-superiority for low dose Again, this does not give overall FWE control

Partially closed test procedures 14 May 2019 Motivating Example… But what about a ’two-dimensional’ sequentially rejective procedure? Test inferiority for high dose Test non-superiority for high dose if rejected if both rejected if rejected Test inferiority for low dose Test non-superiority for low dose Does this control the FWE?

Partially closed test procedures 14 May 2019 Some theory… Let F = {H1, .. HK} be a partially ordered set of null hypotheses. A partial ordering  (precedes) is a binary relation that is irreflexive and transitive, that is, no element precedes itself, and v  w and w  x  v  x. We draw F as a directed acyclic graph (DAG): draw an arrow from v to w whenever v  w, but there is no element x with v  x  w. We consider sequentially rejective procedures on F: ie each hypothesis is tested using an -level test if and only if all preceding hypotheses have been tested and rejected at the  level.

Partially closed test procedures 14 May 2019 Some theory… A subset of a partially ordered set is called an antichain if no element of the subset precedes any other element of the subset. Consider the antichains of F with  2 elements. Let I={I1, … It} be the corresponding intersection hypotheses. The p-closed version of F is defined as F*=F  I endowed with the natural partial ordering (see paper). Theorem: A sequentially rejective procedure on F* strongly controls the FWE with respect to F.

Applied to the ’motivating example’ Partially closed test procedures 14 May 2019 Applied to the ’motivating example’ so 1 intersection hypothesis is inserted There is 1 antichain with  2 elements

Example: gold standard design Partially closed test procedures 14 May 2019 Example: gold standard design Comparing experimental treatment with placebo and an active control inferiority to control non-superiority to control non-superiority to placebo

Example: gold standard design with 2 doses Partially closed test procedures 14 May 2019 Example: gold standard design with 2 doses Now suppose there are two doses of the experimental drug. We would like an inference strategy like:

Example: gold standard design with two doses Partially closed test procedures 14 May 2019 Example: gold standard design with two doses For FWE control we insert 3 intersection hypotheses:

Non-inferiority/superiority for two endpoints Partially closed test procedures 14 May 2019 Non-inferiority/superiority for two endpoints Two co-primary endpoints X and Y. The goal is to show that the experimental treatment is non-inferior (and if possible superior) to the control for both X and Y. Null hypotheses: H1: inferior wrt X H2: non-superior wrt X H3: inferior wrt Y H4: non-superior wrt Y Since (H1  H3)c = H1c  H3c we must first test H1  H3

Non-inferiority/superiority for two endpoints Partially closed test procedures 14 May 2019 Non-inferiority/superiority for two endpoints

Closed test procedures are a special case Partially closed test procedures 14 May 2019 Closed test procedures are a special case

Serial gatekeeper procedures are a special case Partially closed test procedures 14 May 2019 Serial gatekeeper procedures are a special case

A ’modified’ serial gatekeeping procedure Partially closed test procedures 14 May 2019 A ’modified’ serial gatekeeping procedure Omit arrow from 1 to 6 Entanglement: 1 & 6 precedes 1

Partially closed test procedures 14 May 2019 Summary We have shown how to construct multiple test procedures that strongly control the FWE, which are closely tailored to the study objectives, are transparent and easily understood by non-statisticians, and include as special cases: closed test procedures, hierarchical (fixed sequence) test procedures, and serial gatekeeping procedures.

Partially closed test procedures 14 May 2019 Reference Edwards, D and Madsen, J. Constructing multiple test procedures for partially ordered hypothesis sets, Statistics in Medicine, to appear.