1. 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.

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

3. 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

4. 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.

5. 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

6. 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?

7. 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.

8. 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.

9. 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

10. 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

11. 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:

12. 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:

13. 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

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

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

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

17. 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

18. 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.

19. 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.