Sequentially rejective test procedures for partially ordered sets of hypotheses

1. 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.

2. Partially closed test procedures Outline • Motivating example • Some theory • Examples • Summary

3. 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 Partially closed test procedures

4. Motivating Example Partially closed test procedures • 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.

5. Partially closed test procedures 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

6. Partially closed test procedures 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?

7. Partially closed test procedures 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.

8. Partially closed test procedures 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.

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

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

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

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

13. Partially closed test procedures 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

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

15. Partially closed test procedures Closed test procedures are a special case

16. Partially closed test procedures Serial gatekeeper procedures are a special case

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

18. Partially closed test procedures 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.

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