Download - 1 Superior Safety in Noninferiority Trials David R. Bristol To appear in Biometrical Journal, 2005
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Superior Safety in Noninferiority Trials
David R. Bristol
To appear in Biometrical Journal, 2005
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Abstract
Noninferiority of a new treatment to a reference treatment with respect to
efficacy is usually associated with the superiority of the new treatment to the
reference treatment with respect to other aspects not associated with efficacy.
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Abstract
When the superiority of the new treatment to the reference treatment is with respect
to a specified safety variable, the between-treatment comparisons with
respect to safety may also be performed. Here techniques are discussed for the
simultaneous consideration of both aspects.
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Background
ICH (1998) guidelines E-9 and E-10 discuss noninferiority trials, but only with
respect to the efficacy comparison.
The efficacy problem has been discussed by several authors.
Bristol (1999) provides a review.
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Notation
Treatment 0 = Reference treatment, (efficacious with an associated adverse
effect on a specified safety variable)
Treatment 1 = New treatment.
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GOAL
Show that Treatment 1 is superior to Treatment 0 with respect to the specified
safety variable and noninferior with respect to a specified efficacy variable.
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Study Design
A randomized parallel-group study is to be conducted to compare Treatment 0
and Treatment 1, with n subjects / group.
A placebo group could be included in this design for completeness and sensitivity testing, but its inclusion will not have a direct impact on the primary analysis,
which is discussed here.
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Notation
Let Xij and Yij denote the efficacy and
safety responses, respectively, for Subject j on Treatment i, i=0,1, j=1, …,n.
It is assumed that
(Xij,Yij)' ~BVN(μXi, μYi, σ2
X, σ2
Y, ρ),
where all parameters are unknown. Assume small values of efficacy and
safety are preferable.
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Testing
It is desired to show that μX1 < μX0 +Δ and
μY1 < μY0, where the noninferiority margin
Δ is a specified positive number and is defined by clinical importance, often as a proportion of the average efficacy seen
previously for Treatment 0.
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Testing
This goal can be achieved by simultaneously testing
H0X: μX1 ≥μX0+Δ against H1X: μX1 < μX0 +Δ,
and
H0Y: μY1 ≥μY0 against H1Y: μY1 < μY0.
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Testing
Let H0=H0X U H0Y and let H1=H1X ∩H1Y.
It is desired to test H0 against H1.
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Testing
The noninferiority (NI) aspect differs from that seen in most NI problems, as the
response is bivariate.
The reverse multiplicity (RM) aspect pertains to the “all-pairs” multiple
comparisons problem,
where both H0X and H0Y must be rejected.
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Test Procedures
Univariate approach
composite score or a global statistic: O’Brien (1984)
Pocock, Geller, Tsiatis (1987)
And many others
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Test Procedures
The multiplicity problem is solved by reducing the dimensionality of the
response variable used for the comparison. This approach suffers from
the possible impact of one variable on the new response variable. Thus, this
approach should not be considered for this problem. However, it is briefly
discussed for completeness.
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Notation
Let
and
where and are (pooled) unbiased estimates of σ2
X and σ2Y, respectively.
1/ 21 0 ˆ(.5 ) ( ) /X XZ n X X
1/ 21 0 ˆ(.5 ) ( ) /Y YZ n Y Y
2ˆ X 2ˆY
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Rejection Rule(s)
The rejection rule for efficacy is to
Reject H0X: μX1 ≥μX0 +Δ in favor of
H1X: μX1 < μX0 +Δ if ZX≤ -zα
and the rejection rule for safety is to
Reject H0Y: μY1 ≥μY0 in favor of
H1Y: μY1 < μY0 if ZY≤ -zα,
where zα is the 100 (1-α)-th percentile of
the standard normal distribution.
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Notation
Let ΔX= μX1 -μX0 and ΔY = μY1 - μY0. Then
the problem is to simultaneously test
H0X: ΔX≥ Δ against H1X : ΔX< Δ
and
H0Y: ΔY ≥ 0 against H1Y: ΔY < 0.
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Notation
(ZX,ZY)' ~
BVN((.5n)1/2(ΔX-Δ)/ σX,(.5n)1/2ΔY/σY,1,1,ρ).
(approx.)
Tests could be based on linear combinations of ZX and ZY.
Such tests will be inappropriate for the RM formulation.
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Max Test (“Bivariate” Approach)
The simultaneous comparison is performed using a test based on
W=max{ZX,ZY}.
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Max Test
The rejection rule is
Reject H0 in favor of H1 if W≤ C,
where C is chosen such that
P(Reject H0| ΔX =Δ and ΔY = 0)=α.
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Max Test
Let G(.,.| ρ) is the joint cdf of a bivariate normal distribution with zero means, unit
variances, and correlation ρ.
Then
P(Reject H0| ΔX =Δ and ΔY = 0) =G(C, C | ρ).
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Max TestGiven ρ, C can be chosen such that
G(C,C| ρ)= α.
However, ρ is unknown. The critical value can be estimated by satisfying
where r is an estimate of ρ
(pooled or average).
ˆ ˆ( , | ) ,G C C r
C
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Stepwise Approach
Stepwise approaches to the multiple endpoints problem were considered by
Lehmacher, Wassmer, and Reitmer (1991) and several others.
However, because of the RM formulation, these results are not directly applicable.
A stepwise procedure could be used here.
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Stepwise Approach
(I) Test H0X.
If H0X is not rejected in favor of H1X, stop.
If H0X is rejected in favor of H1X,
(II) Test H0Y.
If H0Y is not rejected in favor of H1Y, stop.
If H0Y is rejected in favor of H1Y,
Reject H0 in favor of H1.
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Stepwise Approach
The choice of level for each test has an important impact on the overall level, and
using an α-level test for each of the univariate tests results in the overall level
being much less than α.
The properties of this testing procedure are examined below using simulations.
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Simulation Results
The following results are based on 10,000 for each set of parameters, unit
variances and n=50 subjects per treatment. Each test is conducted at the
α=0.05 level. The simulations were conducted with the same seed for
comparison.
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Simulation Results
Let PX & PY be the estimated power for
the univariate tests based on X and Y respectively.
Let P denote the estimated power of the stepwise procedure of testing H0Y only if
H0X is rejected, where both tests are
performed at the 0.05 level.
“Maximum” is test using W, with “pooled” or “average” estimate of correlation.
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Power Estimates (%)
Maximum ΔX -Δ ΔY ρ PX PY P Pooled Average
0 0 0.1 5.19 5.22 0.27 4.80 4.86 0.5 5.19 5.00 1.09 4.83 4.92 0.9 5.19 4.98 3.13 4.85 4.88
- 0.5 0 0.1 79.95 5.22 4.45 19.67 19.62 0.5 79.95 5.00 4.92 13.32 13.33 0.9 79.95 4.98 4.98 7.26 7.29
-0.25 -0.5 0.1 34.64 80.64 28.73 63.34 63.06 0.5 34.64 80.36 32.45 54.34 53.93 0.9 34.64 79.93 34.64 43.80 42.67
-0.5 -0.5 0.1 79.95 80.64 65.21 91.09 90.75 0.5 79.95 80.36 68.91 86.40 86.09 0.9 79.95 79.93 75.02 81.85 80.89
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Discussion and SummaryNoninferiority trials are often conducted
when the new treatment has an advantage, other than efficacy, over the reference treatment. To simultaneously
test superiority with respect to safety and noninferiority with respect to efficacy, the single-stage testing approach based on
maximum is easy to use and easy to interpret.
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