Download AN ADAPTIVE ALGORITHM TO COMPUTE POWER FOR THE HOCHBERG SEQUENTIAL REJECTIVE PROCEDURE: A CORRELATED DATA EXAMPLE

Paper 2581-2007
AN ADAPTIVE ALGORITHM TO COMPUTE POWER FOR THE HOCHBERG
SEQUENTIAL REJECTIVE PROCEDURE: A CORRELATED DATA EXAMPLE
Jimmy Thomas Efird, Cancer Research Center of Hawaii, and John A. Burns School of
Medicine, University of Hawāi’i at Manoa, Honolulu, HI, Susan Searles Nielsen, Public
Health Sciences Division, Fred Hutchinson Cancer Research Center, Cancer
Epidemiology Research Cooperative, Seattle, WA
ABSTRACT
This paper presents an IML-based SAS MACRO for computing power for the Hochberg
sequential rejective procedure. A simple adaptation of the algorithm also allows for the
computation of restricted conditional power. A correlated data example involving 2 active
formulations of a new investigational drug versus placebo is presented. Significantly increased
study power is observed for the Hochberg procedure when treatment arms versus control are
positively correlated.
INTRODUCTION
Simultaneously testing the statistical significance of multiple null hypotheses is a routine
practice in biomedical research. For example, medical researchers may be interested in
determining if different formulations of a new investigational drug are more effective than a
standard FDA-approved regimen or placebo. The classical Bonferroni inequality provides a
simple distribution-free method for p-value adjustment in the multiple testing situation that does
not require any specific dependence structure among the groups being compared (Sarkar and
Chang 1997). Letting ai denote the probability that hypothesis Si is incorrect, and applying
DeMorgan’s law followed by Boole's inequality, the Bonferroni probability for the joint null
hypothesis may be written as
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C
⎡ v-1
⎤
v-1
⎛
⎛
⎞
C⎞ ⎥
⎢
P ⎜ ∩ S ⎟ = P ⎜ ∪ S i ⎟ ≥ 1 − ∑ α i.
⎢
⎥
⎝ i =1 i ⎠
i =1
⎠ ⎥⎦
⎢⎣⎝ i =1
v-1
(1)
i
The Bonferonni method rejects the (v-1) set of null hypothesis if Pi <∑α/(v-1) for at least one i,
where Pi denotes the p-value corresponding to the ith null hypothesis (Sarkar and Chang 1997).
Although the familywise error rate is preserved, the power for the Bonferroni method is
particularly low when the test statistics are highly correlated. Examples of a correlated design
include twin, family and friend-based studies. Placebo-treatment pairs are selected from these
groups as a means to increase study efficiency and reduce residual confounding (Jewell 2004).
Recently, Hochberg (1988) developed a sequential multiplicity test that controls the familywise
error rate yet has greater power than the Bonferroni procedure when the individual test statistics
are positively dependent (Sarkar and Chang 1997; Benjamini and Yekutieli 2001). Under the
Hochberg method for any j=k, k-1,...,1, all hypotheses Si, i<j, are rejected if zj<α/(k-j+1), where
zj denotes the jth ordered p-value (Denne and Koch 2002). Given the sequential rejective nature
of this method, no closed form exists for computing the power for the Hochberg procedure. The
current paper provides an efficient computer algorithm to simulate the power for the Hochberg
procedure and a restrictive conditional adaptation of this method.
SIMULATION METHODOLOGY
Consider the case where we wish to compare the mean responses for t treatments against placebo
given n observations per arm. Assuming that the underlying paired data are normally distributed
with mean mv and standard deviation sv, 1<v<t+1<n, the test statistic
x −x
σˆ x + σˆ x − 2σˆ x x
v
2
0
(2)
2
v
0
v
0
follows a standard t distribution with n-1 degrees of freedom (df) under the null hypothesis
(Davis 2002). Equation (2) may be used to compute an unadjusted p-value for each of the i
comparisons, i.e., P1, P2, … , Pi. To obtain a corresponding set of p-values adjusted for
multiplicity, rank the unadjusted p-values from largest to smallest, i.e.,
Pi>Pi-1> … >P2>P1. Then compare the largest p-value against the critical value α and declare the
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entire set of comparison significant if Pi<α. Otherwise, compare the next largest p-value to α/2
and declare this difference and all smaller p-values as statistically significant if Pi-1<α/2, and so
on (e.g., the next denominator would be 3). The adjusted p-values are computed by multiplying
the divisor of a by the ranked unadjusted p-values (Rochon 1996).
Power for the Hochberg procedure may be computed by sampling from a non-central T2
−1
distribution with parameters 1, n-1 and n (cx ) ' (csc ') (cx ), where s denotes the sample covariance
matric and c the contrast vector of response differences (Davis 2002), and tabulating the
percentage of instances in which the null hypothesis is rejected. Similarly, a restrictive conditional
power is computed at the trial level by sampling from a standard t distribution as given in equation
(2) under the alternative hypothesis and considering only instances in which the null hypothesis is
rejected and the value for t is negative (assuming that the treatment response means are smaller
than for placebo), and visa versa. In the latter case, random variates from a multivariate normal
distribution with covariance matrix S are obtained using the formula T ' z + µ , where Σ = T ' T.
The method of Odell and Feiveson (1966) is used to simulate values from a Wishart
distribution in order to compute the sample covariance matrix necessary for the denominator
'
term in equation (2). Accordingly, a Wishart identity matrix is formed by V ( t +1) V ( t +1) where the
diagonal elements of the upper-triangular matrix V are independent variates from a χ2
distribution with (n-v+1) df, and the off-diagonal elements are independent variates from a
standard normal distribution. It follows that a Wishart matrix with parameters n and covariance
matrix Σ = V ' V is given by L ' V ' VL (Hocking and Smith 1967; Everson and Morris 2000).
Dividing the latter equation by n gives the sample covariance matrix. Removing the restriction
for the t-value in the above formula closely matches the power derived via the non-central t
distribution using an F approximation to the distribution of Hotelling's T2.
EXAMPLE
Medical researchers are interested in determining if 2 active formulations of a new
investigational drug, differing in manufacturing cost, provide improved mean responses for a
specified biomarker indicative of disease progression compared with placebo. Participants in the
active arms are selected as pairs from relatives in the placebo group. Consider the case where
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µplacebo=0.36, µtreatment1=µtreatment2=0.21 and common standard deviation=0.36. Given values for
the correlation between treatment arms and placebo equal to {0, 0.25, 0.50), a sample size of
n=268 per arm yields unconditional power ={81%, 91%, 98%) based upon a simulation of
250,000 trials. The SAS code for this example is shown in Appendix 1.
DISCUSSION
In this paper we have provided a computer algorithm to compute power for the Hochberg
sequential rejective method. Further, we provided an example showing that a study design in
which the treatment arms are positively correlated with the placebo group is considerably more
powerful than when compared with an uncorrelated design.
CONFLICT OF INTEREST
None declared.
REFERENCES
Benjamini Y, Yekutieli D. The control of the false discovery rate in multiple testing under
dependency. Ann Stat 2001;29:1165-1168.
Davis C. Statistical Methods for the Analysis of Repeated Measurements. Springer-Verlag:New
York, 2002.
Denne J, Koch G. A sequential procedure for studies comparing multiple doses. Pharmaceut
Statistics 2002;1:107-118.
Everson P, Morris C. Simulation from Wishart distributions with eigenvalue constraints. J
Comp Graph Stat 2000;9:380-389.
Hochberg Y. A sharper Bonferroni procedure for multiple tests of significance. Biometrika
1988;75:800-803.
Hocking R, Smith W. Generation of random samples from a Wishart distribution. Technical
4
Report #6, Institute of Statistics, Texas A & M University, February, 1967.
Jewell N. Statistics for Epidemiology. Chapman & Hall/CRC:Boca Raton, 2004.
Odell P, Feiveson A. A numerical procedure to generate a sample covariance matrix. JASA
1966;61:199-203.
Rochon J. An Update on Repeated Measures Analysis. Course notes, 1996.
Sarkar S, Chang C. The Simes method for multiple hypothesis testing with positively dependent
test statistics. JASA 1997;42:1601-1608.
ACKNOWLEDGMENTS
This manuscript was made possible by grants P20 MD000173 from NCMHD and G12RR003061 from NCRR. Its
contents are solely the responsibility of the authors.
CONTACT INFORMATION
Dr. Jimmy Thomas Efird
Director, Biostatistics and Data Management Facility
650 Ilalo Street, Biosciences Building 320-B
Honolulu, HI 96822
Tel: 650.248.8282
Fax: 808.692.1979
E-mail: [email protected]
* Programme: Appendix1.sas
* Programmer: Dr. J. T. Efird
* Telephone: 650.248.8282
* Email:
[email protected]
***********************************************************************;
options ls=80 ps=40 cleanup compress=yes center nodate nonotes nosource;
%macro lp2(cnd,pvaluea,pvalueb,tvalue,cnt);
&cnd if
&pvaluea>=&pvalueb & &pvaluea<=0.05 then do;
cnt1=cnt1+1;
cnt2=cnt2+1;
end;
else if &pvaluea>=&pvalueb & &pvaluea> 0.05 then do;
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if &pvalueb<=0.025 then &cnt;
end;
%mend lp2;
%macro lp1(mu,s,ss,repeat);
proc iml worksize=512000 symsize=2097024;
start main;
tvalue=j(2,1,0);
pvalue=j(2,1,0);
cnt1=j(1,1,0);
cnt2=j(1,1,0);
&s;
&mu;
c1={1,-1,0}`;
c2={1,0,-1}`;
%do i=1 %to 2;
delta&i=&ss*(c&i*mu)`*inv(c&i*s*c&i`)*(c&i*mu);
%end;
%do j=1 %to &repeat;
%do k=1 %to 2;
tvalue[&k]=finv(ranuni(%eval(&j&k&r+512)),1,&ss-1,delta&k);
pvalue[&k]=1-probf(tvalue[&k],1,&ss-1);
%end;
%lp2(%str(),%str(pvalue[2]),%str(pvalue[1]),%str(tvalue[1]),%str(cnt1=cnt1+1));
%lp2(%str(else),%str(pvalue[1]),%str(pvalue[2]),%str(tvalue[2]),%str(cnt2=cnt2+1));
%end;
free _all_;
file 'cnt';
put cnt1 cnt2;
finish;
%mend lp1;
%macro lp3(repeat1,repeat2);
%do r=1 %to &repeat2;
%lp1(%str(mu={0.36,0.21,0.21}),
%str(s={0.36 0.18 0.18, 0.18 0.36 0.0, 0.18 0.0 0.36}),268,100);
run;
data a&r;
infile 'cnt';
input cnt1 cnt2;
%end;
data combo1;
set %do m=1 %to %eval(&r-1);
a&m
%end;;
proc univariate data=combo1 noprint;
var cnt1 cnt2;
output out=out1 sum=sum1 sum2;
data power(drop=sum1 sum2);
set out1;
power1=(sum1/%eval(&repeat1*&repeat2));
power2=(sum2/%eval(&repeat1*&repeat2));
%mend lp3;
%lp3(100,2500);
proc print data=power;
run;
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