Transcript Sequentially rejective test procedures for partially

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.
Partially closed test procedures
Outline
•
•
•
•
Motivating example
Some theory
Examples
Summary
17 July 2015
Slide no 2
Partially closed test procedures
17 July 2015
Slide no 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
17 July 2015
Motivating Example
• We could consider
Test inferiority
of high dose
Test inferiority
of low dose
if rejected
if rejected
Test non-superiority
of high dose
Test non-superiority
of low dose
This gives strong FWE control for each dose, but not
overall.
Slide no 4
Partially closed test procedures
17 July 2015
Motivating Example…
• Or we could consider
Test inferiority
for high dose
Test non-superiority
for high dose
if rejected
if rejected
Test inferiority for
low dose
Test non-superiority
for low dose
Again, this does not give overall FWE control
Slide no 5
Partially closed test procedures
17 July 2015
Motivating Example…
• But what about a ’two-dimensional’ sequentially
rejective procedure?
Test inferiority for
high dose
if rejected
if rejected
Test non-superiority
for high dose
if both
rejected
Test inferiority for
low dose
• Does this control the FWE?
Test non-superiority
for low dose
Slide no 6
Partially closed test procedures
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Slide no 7
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
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Slide no 8
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.
Partially closed test procedures
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Slide no 9
Applied to the ’motivating example’
original
There is 1
antichain
with  2
elements
p-closed
so 1 intersection
hypothesis is inserted
Hi in
Hi in
Hi ns & Lo in
Hi ns
Lo in
Hi ns
Lo ns
Lo in
Lo ns
Partially closed test procedures
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Slide no 10
Example: gold standard design
Comparing experimental treatment with placebo
and an active control
inferiority to control
IN C
NS C
NS P
non-superiority
to placebo
non-superiority
to control
Partially closed test procedures
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Slide no 11
Example: gold standard design with 2 doses
• Now suppose there are two doses of the experimental
drug. We would like an inference strategy like:
Lo ns P
Lo in C
Hi ns P
Lo ns C
Hi in C
Hi ns C
Partially closed test procedures
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Slide no 12
Example: gold standard design with two doses
For FWE control we insert 3
intersection hypotheses:
Hi ns P
Hi in C & Lo ns P
Hi in C
Hi ns C & Lo ns P
Hi ns C & Lo in C
Lo ns P
Hi ns C
Lo in C
Lo ns C
Partially closed test procedures
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Slide no 13
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:
H2:
H3:
H4:
inferior wrt X
non-superior wrt X
inferior wrt Y
non-superior wrt Y
Since (H1  H3)c = H1c  H3c
we must first test H1  H3
Partially closed test procedures
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Slide no 14
Non-inferiority/superiority for two endpoints
original
p-closed
H1 or H3
H1 or H3
H2 & H4
H2
H4
H2
H4
Partially closed test procedures
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Closed test procedures are a special case
original
p-closed
2&3&4
234
2&3
2&4
3&4
2
3
4
Slide no 15
Partially closed test procedures
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Slide no 16
Serial gatekeeper procedures are a special case
original
p-closed
1&2&3
1
2
3
1&2
1&3
2&3
1
2
3
4&5&6
4
5
6
4&6
4&5
5&6
4
6
5
Partially closed test procedures
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Slide no 17
A ’modified’ serial gatekeeping procedure
original
p-closed
1&2&3
1
3
2
1&3
1&2
2&3
1&6
3
2
Omit arrow
from 1 to 6
Entanglement:
5
4
6
1 & 6 precedes 1
1
4&5&6
4&5
5&6
4&6
5
4
6
Partially closed test procedures
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Slide no 18
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 nonstatisticians, and
• include as special cases: closed test procedures,
hierarchical (fixed sequence) test procedures, and
serial gatekeeping procedures.
Partially closed test procedures
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Slide no 19
Reference
• Edwards, D and Madsen, J. Constructing multiple test
procedures for partially ordered hypothesis sets,
Statistics in Medicine, to appear.