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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

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sequentially rejective test procedures for partially ordered sets of hypotheses

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
Partially closed test proceduresOutline
  • Motivating example
  • Some theory
  • Examples
  • Summary
motivating example
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

motivating example1
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.

slide5
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

slide6
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?
some theory
Partially closed test proceduresSome 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.

some theory1
Partially closed test proceduresSome 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 proceduresApplied to the ’motivating example’

so 1 intersection hypothesis is inserted

There is 1 antichain with  2 elements

example gold standard design
Partially closed test proceduresExample: 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 proceduresExample: 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 proceduresExample: gold standard design with two doses

For FWE control we insert 3 intersection hypotheses:

non inferiority superiority for two endpoints
Partially closed test proceduresNon-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

slide15
Partially closed test procedures

Closed test procedures are a special case

slide16
Partially closed test procedures

Serial gatekeeper procedures are a special case

slide17
Partially closed test procedures

A ’modified’ serial gatekeeping procedure

Omit arrow from 1 to 6

Entanglement:

1 & 6 precedes 1

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