Simultaneous identifications of the minimum effective dose in each of several groups

(1)

This article was downloaded by: [National Chiao Tung University 國立交通大學] On: 25 April 2014, At: 23:21

Publisher: Taylor & Francis

Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Journal of Statistical Computation and

Simulation

Publication details, including instructions for authors and subscription information:

http://www.tandfonline.com/loi/gscs20

Simultaneous identifications of the

minimum effective dose in each of

several groups

Show-Li Jan a , Yuh-Ing Chen b & Gwowen Shieh c

a

Department of Applied Mathematics , Chung Yuan Christian University , Chungli, 32023, Taiwan, R.O.C

b

National Central University

c

National Chiao Tung University Published online: 24 Nov 2006.

To cite this article: Show-Li Jan , Yuh-Ing Chen & Gwowen Shieh (2007) Simultaneous

identifications of the minimum effective dose in each of several groups, Journal of Statistical Computation and Simulation, 77:2, 149-161, DOI: 10.1080/10629360600565038

To link to this article: http://dx.doi.org/10.1080/10629360600565038

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

(2)

and-conditions

(3)

Journal of Statistical Computation and Simulation Vol. 77, No. 2, February 2007, 149–161

Simultaneous identifications of the minimum effective dose in

each of several groups

SHOW-LI JAN*†, YUH-ING CHEN‡ and GWOWEN SHIEH§

†Department of Applied Mathematics, Chung Yuan Christian University, Chungli, 32023, Taiwan, R.O.C.

‡National Central University §National Chiao Tung University (Received 20 October 2004; in final form 8 May 2005)

One primary goal in dose-response studies for drug development is to identify the minimum effective dose (MED) which is the lowest dose level with an effect that exceeds that of the zero dose control. In this paper, we consider step-down closed testing procedures on the basis of pairwise and Helmert contrasts suggested by Tamhane et al. [Tamhane,A.C., Hochberg,Y., and Dunnett, C.W., 1996, Multiple test procedures for dose finding. Biometrics, 52, 21–37.] for the situation where one wishes to make all such MED identifications simultaneously in each of several groups. An example is given to illustrate the proposed tests. The Monte Carlo studies are further implemented to compare the relative error rate and power performances of the proposed tests.

Keywords: Dose-response study; Helmert contrasts; Pairwise contrasts; Simultaneous inference; Step-down closed test.

1. Introduction

In toxicological and drug development studies, a common problem is to investigate the effect of a compound. For this purpose, a dose-response experiment is often conducted in a one-way layout in which several increasing dose levels of the compound, including a zero dose or a placebo to serve as a control, are administered to separate groups of subjects. In drug development studies, the major concern is to identify the minimum effective dose, denoted by MED [1], where the MED is defined to be the lowest dose level with a mean, larger than that of the zero dose control.

Generally, in a one-way layout, the inference about the MED is made by comparing various dose groups with a control. Among the available procedures, the ones proposed by Dunnett [2], Williams [3, 4] and Ruberg [1], are most popular. Note that Williams’ procedure is based on the isotonic regression estimators of ordered means, implemented into a step-down closed testing scheme [5], whereas both Dunnett and Ruberg employed single-step multiple testing

*Corresponding author. Email: sljan@math.cycu.edu.tw

Journal of Statistical Computation and Simulation

DOI: 10.1080/10629360600565038

(4)

procedures based on different contrasts of sample means. Owing to the fact that contrasts of sample means are more convenient to compute than the isotonic regression estimators under ordered restriction, and multi-step multiple tests are more powerful than the related single-step multiple tests, Tamhane et al. [6] further considered some stepwise closed testing procedures based on a variety of contrasts for the dose-finding problem. It was pointed out that the proce-dures based on the so-called step and basin contrasts proposed by Ruberg [1] do not control the familywise error rate (FWE). Moreover, the step-down procedures generally dominate the related step-up procedures. Therefore, Tamhane et al. [6] finally suggested pairwise and Helmert contrasts incorporated into the proposed step-down testing scheme.

In medical and biological researches, however, the experimenters often encounter the multi-group situation due to different gender, age or compounds, where there are r multi-groups, each consisting of c active treatments and one control treatment. In the absence of interaction between treatment and group, one can use the original Dunnett procedure, but with average over all groups, for identifying the treatments which are better than or different from the control. When the interaction is present, however, Cheung and Holland [7, 8] extended Dunnett’s test and proposed a single-step procedure to identify simultaneously the treatments which are more effective than or different from the related control in each of the r groups. Cheung and Holland [9] further suggested an extension of Dunnett and Tamhane’s [10] step-down closed testing procedure to the multi-group problem, which is more powerful than their extension of Dunnett’s test.

Note that, when the multi-group situation is involved in a drug development study, the MEDs of interest may very well vary in different groups. Therefore, the simultaneous identifications of the MED in each of the groups under study would be of great practical importance. In this paper, we extend the step-down closed testing scheme proposed by Tamhane et al. [6] on the basis of pairwise and Helmert contrasts to the multi-group problem. To provide with a measure of the strength of evidence for the simultaneous MED identifications, we further suggest a modification of the step-down closed testing scheme by using the related adjusted p-values [11].

Section 2 gives the notation and description of the multi-group problem. Section 3 proposes two sets of contrasts for the simultaneous identifications of the MED. The concept of the step-down testing scheme is described both in its classical version and p-value version, where the exact integration and a convenient approximation of the computations of the critical constants and p-values are discussed. Section 4 presents a numerical example to illustrate the proposed step-down procedures. Section 5 shows the results of a Monte Carlo study comparing the proposed tests under the null hypothesis and a variety of alternative hypotheses. Finally, section 6 gives conclusions and recommendations.

2. Preliminaries

Assume the existence of r groups. Denote a set of increasing dose levels by 0, 1, 2, . . . , c, where 0 corresponds to the zero dose level (or placebo control). For i= 1, . . . , r and j = 0, 1, . . . , c, let Xij k denote the kth observation on the j th dose level in the ith group,

where k= 1, . . . , n for all (i, j). We assume that all observations Xij k are mutually

inde-pendent with Xij k∼ N(µij, σ2), i= 1, . . . , r, j = 0, 1, . . . , c, and k = 1, . . . , n. Here µij

represents the effect of the j th dose level in the ith group. Define

Xij = 1 n n k=1 Xij k and S2= r i=1 c j=0 n k=1 (Xij k− Xij)2 f ,

(5)

Simultaneous identifications of the minimum effective dose 151

where f = r(c + 1)(n − 1). Hence the sample means Xij are mutually independent with

Xij ∼ N(µij, σ2/n). The pooled sample variance S2, providing an unbiased estimator of

the common variance σ2 _{based on f degrees of freedom (d.f.), is independent of X}

ij with

S2_{∼ σ}2_χ2

f/f.

Define the MED for the ith group as MEDi = min{j : µij > µi0}, i = 1, . . . , r. In this paper, we wish to identify MEDisimultaneously. The problem is formulated as a sequence of

hypotheses testing problems

H0iji : µi0= µi1= · · · = µiji vs. H1iji : µi0= µi1= · · · = µi,ji−1 < µiji,

ji= 1, . . . , c and i = 1, . . . , r. (1)

If ji∗ is the smallest ji for which H0iji is rejected, then the ji∗th dose is identified to be the

MED for the ith group, that is, MEDi = ji∗.

As noted in ref. [12], the family of hypotheses H= {(H01j1, . . . , H0rjr): 1 ≤ j1, . . . , jr≤ c},

where (H01j1, . . . , H0rjr)represents the hypotheses H01j1, . . . , H0rjrare simultaneously tested,

is closed under intersection in the sense that (H01j1, . . . , H0rjr)∈ H and (H01j1, . . . , H0rjr)∈

Himply that (H01j1∩ H01j1, . . . , H0rjr∩ H0rjr)∈ H . Hence, an α-level closed procedure that

includes separate α-level tests of individual (H01j1, . . . , H0rjr), applied in a step-down manner

can be employed in finding the MEDi simultaneously. Moreover, the closed testing scheme

strongly controls the FWE, which is defined as FWE= P{at least one true H0iji is rejected}.

The step-down procedures proposed in section 3 are of closed type and, hence, control the FWE strongly.

3. The proposed step-down procedures

3.1 Procedures based on contrasts

For testing a hypothesis H0ijof (1), a contrast of the following general form is used: Cij =

a0jXi0+ a1jXi1+ · · · + acjXic, where

c

s=0asj = 0. The corresponding t-statistic is given

by Tij = Cij − E(Cij) V ar(Cij) = c s=0 asj(Xis− µis) S c s₌₀ a_sj2/n , 1≤ i ≤ r, 1 ≤ j ≤ c. (2)

The critical points of the procedure depend on the joint distribution of Tij. Letting

Ti= (Ti1, . . . , Tic), we note that, under H0ic, Ti has a general c-variate t-distribution with f d.f. and associated correlation matrix A= {ρjj}, where ρjj is the correlation

coef-ficient between the j th and the jth contrasts, 1≤ j = j≤ c. Similarly, under H0ic, i= 1, . . . , r, (T11, . . . , T1c, . . . , Tr1, . . . , Trc)= (T1, . . . , Tr) has a r× c-variate t-distri-bution with f d.f. and correlation matrix Rr×c= diag(A, . . . ,A) [13, 14].

In this paper, we consider the following two sets of contrasts: (1) Pairwise (P) contrasts. The j th pairwise contrast is defined by

asj =      −1, s = 0, 1, s = j, 1 ≤ j ≤ c. 0, o.w,

(6)

Thus, under H0ic, we can rewrite (2) as

TPij =

Xij − Xi0

S√2/n , 1≤ i ≤ r and 1 ≤ j ≤ c. (3)

The correlation coefficients are given by ρjj = 1/2, 1 ≤ j = j≤ c. And hence,

under H0ic, i= 1, . . . , r, (TP11, . . . , TP1c, . . . , TPr1, . . . , TPrc) has a r× c-variate

t-distribution with f d.f. and correlation matrix Rr×c= diag(A, . . . ,A), where

A=    1 1/2 . ._. 1/2 1    . (4)

(2) Helmert (H) contrasts. The j th Helmert contrast is defined by

asj =    −1, s = 0,1, . . . , j − 1, j, s= j, 1 ≤ j ≤ c. 0, o.w,

This contrast compares the j th dose level mean with the average of all the lower dose level means, including the zero dose control. In this case, under H0ic, (2) specifies to

THij =

j Xij− (Xi0+ · · · + Xi,j−1)

S√j (j + 1)/n , 1≤ i ≤ r, and 1 ≤ j ≤ c. (5)

These contrasts are mutually orthogonal and hence ρjj= 0, 1 ≤ j = j≤ c. Therefore,

under H0ic, i= 1, . . . , r, (TH11, . . . , TH1c, . . . , THr1, . . . , THrc) has a r× c-variate

t-distribution with f d.f. and correlation matrix Rr×c= Ir×c, the identity matrix.

3.2 Step-down testing scheme

We now show how to extend the step-down procedures proposed by Tamhane et al. [6] for identifying MED in a one-way layout to the situation where one wishes to make all such MED identifications simultaneously in each of r groups [12].

We shall describe the step-down procedure first in its classical version, based on criti-cal constants for a specified α-level. Let Trα×c,f,Rr×c be the upper αth percentile of T(r×c)=

max{Tij : 1 ≤ i ≤ r, 1 ≤ j ≤ c}, and let t(r×c)be the observed maximum of T(r×c). At the first

step, let c1i = c be the number of hypotheses to be tested in the ith group, i = 1, . . . , r, and k1=

r

i=1c1i be the total number of hypotheses still to be tested. Compute t(k1)and define

(g(k1), d(g(k1))), which represents the group and dose level, to be the antirank vector of t(k1), i.e., t(k1)= tg(k1),d(g(k1)). If t(k1)≥ T

α

k1,f,Rk1, then reject H0g(k1)j, j = d(g(k1)), . . . , c1g(k1),

and go to the second step with k2= r

i=1c2i, where c2i= d(g(k1))− 1 if i = g(k1)and c1i otherwise. Otherwise, stop testing and accept all null hypotheses. In general, at the th step, set k=

r

i=1ci, where ci = d(g(k−1))− 1 if i = g(k−1)and c−1,iotherwise. Let

(g(k), d(g(k)))be the antirank vector of t(k), where t(k)is the observed maximum of the

corresponding t-statistics associated with the knull hypotheses. If t(k)≥ T

α

k,f,Rk, then reject

H0g(k)j, j = d(g(k)), . . . , cg(k); otherwise, stop testing. When testing stops, say at the mth

step, estimate the MEDi as cmi+ 1, that is, MEDi = cmi+ 1, i = 1, . . . , r.

(7)

Next we describe how to apply the concept of adjusted p-values, proposed by Dunnett and Tamhane [10], to this step-down procedure. In general, at the th step for testing H0(k), i.e.,

H0g(k)dg(k), first compute the probability

p = P{at least one Ta≥ t(k), a= 1, . . . , k}

= P{ max

1≤j≤ci 1≤i≤r

Tij ≥ t(k)}, = 1, 2, . . . .

(6)

Then define the p-value for H0(k)to be

p= max(p, p₋₁, . . . , p1), = 1, 2, . . . . (7) On the basis of these p-values, hypotheses tests can be conducted at any fixed specified α-level by comparing any p with α and rejecting H0g(k)j, j= d(g(k)), . . . , cg(k) if

p≤ α, = 1, 2, . . .. Note that the p-values are monotonically ordered, that is, p1≤ p2 ≤ · · · . Thus if p> αand hence H0(k)is accepted, then monotonicity ensures acceptance

also of H0(1), . . . ,H0(k−1).

To conclude this section, we discuss the methods for calculation of critical constants and p-values. Among others, Somerville [15–17] and Genz and Bretz [18–20] have developed exact integration methods for the calculation of these values. (The SAS and FORTRAN programs by Genz and Bretz are available from the web-sites with URL http://www.bioinf.uni-hannover.de/mcp_home/software.html and URL http://www.sci.wsu.edu/math/faculty/genz/homepage, respectively.) While exact methods exist for these computations, it is sometimes necessary for practical purpose to consider some simple and convenient approximations [21–24]. To this end, we introduce the use of the SAS function PROBMC along with some approximations for computing the critical constants and p-values of the proposed procedures. When (T1, . . . , Tk) has a k-variate t-distribution with

product correlation structure ρij = λiλj,−1 < λi, λj <1, the critical constants and p-values

for T(k)= max(T1, . . . , Tk)can be easily obtained, through computing

PROBMC(‘DUNNETT1’,·, 1 − α, f, k, λ1, . . . , λk), (8)

and

1− PROBMC(‘DUNNETT1’, t(k),·, f, k, λ1, . . . , λk), (9)

respectively. For Helmert procedure, the correlation matrix Rr×c is an identity matrix, thus

formulas (8) and (9) can be applied readily. (A small positive number substituted for 0 in λi

may be used to improve the convergence of the PROBMC method.) For pairwise procedure, the Rr×cmatrix is not of product correlation structure, moreover, it is changing at each step

of testing; in this case, we consider a convenient approximation by replacing ρij with their

arithmetic average ρ (computed from Rr×c) when applying formulas (8) and (9) for the

cal-culations. Due to the nature of the correlation matrix Rr×c, where most entries are zero, this

approximation is simple and quite acceptable. The approximations for pairwise and Helmert procedures are justified through an example in the next section.

4. An example

In this section, we illustrate the step-down procedures using a drug-dose-response data set which was collected for a study in pharmacochemistry [25]. To investigate the effect of drug and dose on the analgesic potency, a two-way layout was implemented which involved five

(8)

drug groups, including three mixed-type opioids (groups 1, 3 and 4), one placebo containing 0.9% saline solution (group 2) and one opioid antagonist (group 5), and five doses, ED5, ED10, ED20, ED40and ED80determined from an earlier in vitro trial. For each drug group and dose, n= 10 mice were used. The analgesic response of each mouse was measured at certain time points.

Note that the analysis results of the time-dependent analgesia under the assumption of normal distribution in ref. [25] reveal that the interaction effect between the drug group and dose is present. Therefore, the nature of analgesia is a function of the dose levels at various drug groups. In this data set, for the purpose of dose-finding, the MED among c= 4 doses (ED10− ED80) relative to the control dose ED5 would be of interest. However, the MEDs should be identified for each drug group at each time point. Therefore, we show how to apply the step-down procedures under study to simultaneously identify the MEDs for r= 5 drug groups based on the analgesic response measured at 10 min after the mouse receiving the drug. The mean values of the analgesic potency in each cell are given in table 1, where the dose levels are recorded as 0, 1, 2, 3, 4. The variance estimate and the associated d.f. are s2_{= 8.825} and f = 225, respectively. The t-statistics computed by using formulas (3) and (5) are then shown in table 2.

First, considering pairwise contrasts and using α= 0.05. Here ρ = 0.5 × 6 × 5/(20 × 19/2)= 0.079 is used at each step, the critical values and probabilities p are computed,

respectively, using formulas (8) and (9). The pand pprobabilities in parentheses are exact

values computed by using the SAS program from Benz and Bretz. The testing results are summarized in table 3. Note that the testing stops at the 11th step, and hence estimate the

Table 1. Mean values of the analgesic potency at time= 10 min from ref. [25].

Group Dose Mean

1 0 7.07 1 1 9.56 1 2 14.78 1 3 21.62 1 4 23.16 2 0 1.25 2 1 1.26 2 2 1.08 2 3 1.04 2 4 1.39 3 0 6.91 3 1 9.12 3 2 15.13 3 3 24.63 3 4 22.63 4 0 2.79 4 1 1.85 4 2 3.48 4 3 5.75 4 4 11.66 5 0 18.26 5 1 27.50 5 2 40.19 5 3 46.04 5 4 57.21

Note: Each cell mean is based on 10 observations.

(9)

Simultaneous identifications of the minimum effective dose 155 Table 2. t-Statistics for pairwise and Helmert contrasts.

t-Statistcs

Group Dose Pairwise Helmert

1 1 1.87 1.87 1 2 5.80 5.62 1 3 10.95 10.28 1 4 12.11 9.43 2 1 0.01 0.01 2 2 −0.13 −0.15 2 3 −0.16 −0.14 2 4 0.11 0.22 3 1 1.66 1.66 3 2 6.19 6.18 3 3 13.34 13.13 3 4 11.83 8.27 4 1 −0.71 −0.71 4 2 0.52 1.01 4 3 2.23 2.81 4 4 6.68 7.80 5 1 6.96 6.96 5 2 16.51 15.05 5 3 20.91 16.03 5 4 29.32 23.05

Table 3. Testing results for using pairwise contrasts (f = 225, ρ = ρ = .079). Step() k (c1, c2, c3, c4, c5) t(k) Group Dose T

.05 k,f,ρ p p 1 20 (4, 4, 4, 4, 4) 29.32 5 4 2.82 .0000 .0000 (.0000 .0000) 2 19 (4, 4, 4, 4, 3) 20.91 5 3 2.80 .0000 .0000 (.0000 .0000) 3 18 (4, 4, 4, 4, 2) 16.51 5 2 2.78 .0000 .0000 (.0000 .0000) 4 17 (4, 4, 4, 4, 1) 13.34 3 3 2.76 .0000 .0000 (.0000 .0000) 5 15 (4, 4, 2, 4, 1) 12.11 1 4 2.72 .0000 .0000 (.0000 .0000) 6 14 (3, 4, 2, 4, 1) 10.95 1 3 2.70 .0000 .0000 (.0000 .0000) 7 13 (2, 4, 2, 4, 1) 6.96 5 1 2.67 .0000 .0000 (.0000 .0000) 8 12 (2, 4, 2, 4, 0) 6.68 4 4 2.65 .0000 .0000 (.0000 .0000) 9 11 (2, 4, 2, 3, 0) 6.19 3 2 2.62 .0000 .0000 (.0000 .0000) 10 10 (2, 4, 1, 3, 0) 5.80 1 2 2.58 .0000 .0000 (.0000 .0000) 11 9 (1, 4, 1, 3, 0) 2.23 4 3 2.54 .1106 .1106 (.1013 .1013)

(10)

Table 4. Testing results for using Helmert contrasts (f = ∞, ρ = .00011).

Step() k (c1, c2, c3, c4, c5) t(k) Group Dose Tk.05,f,ρ p p

1 20 (4, 4, 4, 4, 4) 23.05 5 4 2.80 .0000 .0000 (.0000 .0000) 2 19 (4, 4, 4, 4, 3) 16.03 5 3 2.78 .0000 .0000 (.0000 .0000) 3 18 (4, 4, 4, 4, 2) 15.05 5 2 2.77 .0000 .0000 (.0000 .0000) 4 17 (4, 4, 4, 4, 1) 13.13 3 3 2.75 .0000 .0000 (.0000 .0000) 5 15 (4, 4, 2, 4, 1) 10.28 1 3 2.71 .0000 .0000 (.0000 .0000) 6 13 (2, 4, 2, 4, 1) 7.80 4 4 2.66 .0000 .0000 (.0000 .0000) 7 12 (2, 4, 2, 3, 1) 6.96 5 1 2.63 .0000 .0000 (.0000 .0000) 8 11 (2, 4, 2, 3, 0) 6.18 3 2 2.60 .0000 .0000 (.0000 .0000) 9 10 (2, 4, 1, 3, 0) 5.62 1 2 2.57 .0000 .0000 (.0000 .0000) 10 9 (1, 4, 1, 3, 0) 2.81 4 3 2.53 .0224 .0224 (.0240 .0240) 11 8 (1, 4, 1, 2, 0) 1.87 1 1 2.49 .2193 .2193 (.2243 .2243)

MEDias c11,i+ 1, i = 1, 2, 3, 4, 5. That is, MED1= 2, MED2= 5, MED3= 2, MED4= 4 and MED5= 1.The adjusted p-value associated with this conclusion is p10 = 0.0000. Second, the test based on Helmert contrasts also stops at the 11th step, but the results are MED1= 2,

MED2= 5, MED3= 2, MED4 = 3, and MED5= 1, and the corresponding approximate p-value is p10= 0.0224, where the exact p-value is p10 = 0.0240. The sequential testing results are further presented in table 4.

The conclusion by using the step-down procedures for this set of data is finally obtained. For the two mixed-type opioids in groups 1 and 3, both the pairwise and Helmert tests identify the MED as the second dose level, ED20. For the other mixed-type opioid in group 4, the pairwise test identifies the MED as ED80, while the Helmert test identifies ED40 to be the MED. Moreover, both the tests find that, for the placebo in group 2, the MED is not under study, and claim that the first dose level, ED10, is the MED for the opioid antagonist in group 5.

5. Monte Carlo study

We conducted a Monte Carlo study to compare the two procedures proposed in section 3. In the study, r, the number of groups was fixed at five, and c, the number of dose levels excluding control was fixed at four. A common sample size, n, was assumed in each group by dose combination. The standard error of the means, σ/√n, was fixed at one. The d.f., f , was assumed to be∞, using α = 0.05. For convenience, the critical constants for α = 0.05 were computed by formula (8) with ρ= ρ = 0.079 for TP and ρ = 0.00011 for TH.

(11)

Without loss of generality, µio was fixed at 0 for all i= 1, . . . , r. For the positive dose

responses, we considered 16 configurations of monotone case which include step and lin-ear responses, and another 12 configurations of nonmonotone case. For each configuration (µij, i= 1, . . . , r, j = 0, 1, . . . , c), the sample means Xij, which are independent N (µij,1)

r.v.s, were generated using the RANNOR function in SAS. This was replicated 10,000 times for each of the 28 configurations. Table 5 contains the estimates of the FWE, which is the proportion of replications that identified a noneffective dose in at least one group, and the

Table 5. Estimated FWE and power for α= 0.05, r = 5, and c = 4. µ10 µ11 µ12 µ13 µ14

µ20 µ21 µ22 µ23 µ24

µ30 µ31 µ32 µ33 µ34

FWE Power

µ40 µ41 µ42 µ43 µ44

Case µ50 µ51 µ52 µ53 µ54 True MED Pairwise Helmert Pairwise Helmert

1 0 0 0 0 5 4 .0445 .0491 .2937 .7721 0 0 0 0 5 4 0 0 0 0 5 4 0 0 0 0 5 4 0 0 0 0 5 4 2 0 0 0 5 5 3 .0448 .0466 .3658 .7628 0 0 0 5 5 3 0 0 0 5 5 3 0 0 0 5 5 3 0 0 0 5 5 3 3 0 0 5 5 5 2 .0456 .0477 .4800 .7437 0 0 5 5 5 2 0 0 5 5 5 2 0 0 5 5 5 2 0 0 5 5 5 2 4 0 5 5 5 5 1 – – .8175 .7133 0 5 5 5 5 1 0 5 5 5 5 1 0 5 5 5 5 1 0 5 5 5 5 1 5 0 0 0 0 5 4 .0423 .0449 .4505 .6096 0 0 0 5 5 3 0 0 5 5 5 2 0 5 5 5 5 1 0 5 5 5 5 1 6 0 0 0 5 5 3 .0435 .0446 .4795 .6259 0 0 0 5 5 3 0 0 5 5 5 2 0 5 5 5 5 1 0 5 5 5 5 1 7 0 0 0 5 5 3 .0439 .0474 .4509 .6771 0 0 0 5 5 3 0 0 5 5 5 2 0 0 5 5 5 2 0 5 5 5 5 1 8 0 0 0 5 5 3 .0423 .0436 .5138 .5906 0 0 0 5 5 3 0 5 5 5 5 1 0 5 5 5 5 1 0 5 5 5 5 1 (continued)

(12)

Table 5. Continued. µ10 µ11 µ12 µ13 µ14

µ20 µ21 µ22 µ23 µ24

µ30 µ31 µ32 µ33 µ34 _FWE _Power

µ40 µ41 µ42 µ43 µ44

9 0 0 0 8 12 3 .0481 .0479 .9464 .9521 0 0 0 8 12 3 0 0 0 8 12 3 0 0 0 8 12 3 0 0 0 8 12 3 10 0 0 4 8 12 2 .0405 .0424 .1271 .3325 0 0 4 8 12 2 0 0 4 8 12 2 0 0 4 8 12 2 0 0 4 8 12 2 11 0 4 6 8 10 1 – – .4994 .4792 0 4 6 8 10 1 0 4 6 8 10 1 0 4 6 8 10 1 0 4 6 8 10 1 12 0 0 0 8 12 3 .0415 .0423 .2224 .3071 0 0 4 8 12 2 0 0 4 8 12 2 0 4 6 8 10 1 0 4 6 8 10 1 13 0 0 0 0 5 4 .0397 .0443 .2516 .3504 0 0 0 5 5 3 0 0 5 5 5 2 0 4 6 8 10 1 0 4 6 8 10 1 14 0 0 0 5 5 3 .0398 .0425 .2183 .3217 0 0 0 5 5 3 0 0 4 8 12 2 0 4 6 8 10 1 0 4 6 8 10 1 15 0 0 0 5 5 3 .0435 .0448 .3143 .4014 0 0 5 5 5 2 0 0 5 5 5 2 0 4 6 8 10 1 0 4 6 8 10 1 16 0 0 5 5 5 2 .0423 .0421 .3568 .4511 0 5 5 5 5 1 0 0 0 8 12 3 0 0 4 8 12 2 0 4 6 8 10 1 17 0 0 5 0 0 2 .0412 .0454 .4293 .7129 0 0 5 0 0 2 0 0 5 0 0 2 0 0 5 0 0 2 0 0 5 0 0 2 18 0 5 5 0 0 1 – – .7789 .6866 0 5 5 0 0 1 0 5 5 0 0 1 0 5 5 0 0 1 0 5 5 0 0 1 (continued)

(13)

Simultaneous identifications of the minimum effective dose 159 Table 5. Continued. µ10 µ11 µ12 µ13 µ14 µ20 µ21 µ22 µ23 µ24 µ30 µ31 µ32 µ33 µ34 _FWE _Power µ40 µ41 µ42 µ43 µ44

19 0 0 5 5 0 2 .0448 .0475 .4744 .7411 0 0 5 5 0 2 0 0 5 5 0 2 0 0 5 5 0 2 0 0 5 5 0 2 20 0 5 5 5 0 1 – – .8130 .7080 0 5 5 5 0 1 0 5 5 5 0 1 0 5 5 5 0 1 0 5 5 5 0 1 21 0 0 0 5 0 3 .0413 .0438 .4621 .6179 0 0 0 5 0 3 0 0 5 5 0 2 0 5 5 0 0 1 0 5 5 0 0 1 22 0 0 0 5 0 3 .0437 .0439 .5429 .6058 0 0 5 5 0 2 0 5 5 0 0 1 0 5 5 5 0 1 0 5 5 5 0 1 23 0 5 6 8 10 1 .0389 .0397 .0877 .1664 0 0 0 0 2 4 0 5 6 10 8 1 0 0 0 5 8 3 0 5 6 8 10 1 24 0 0 6 8 12 2 .0431 .0465 .3505 .4534 0 0 0 0 0 5 0 0 6 12 10 2 0 0 4 6 8 2 0 4 5 8 12 1 25 0 0 5 5 8 2 .0415 .0448 .1859 .3417 0 0 0 0 0 5 0 0 4 6 5 2 0 0 0 4 8 3 0 4 5 8 10 1 26 0 0 6 8 12 2 .0430 .0464 .4480 .5500 0 0 0 0 0 5 0 0 6 12 10 2 0 0 0 5 5 3 0 4 5 8 12 1 27 0 0 4 6 8 2 .0422 .0458 .1761 .3134 0 0 0 0 0 5 0 0 4 8 5 2 0 0 0 0 5 4 0 4 6 6 8 1 28 0 0 6 8 12 2 .0444 .0475 .4308 .5378 0 0 0 0 0 5 0 0 6 12 10 2 0 0 0 0 5 4 0 4 5 8 12 1 Average power .4274 .5545

(14)

estimates of the power, which is the proportion that identified the true MED vector. This table also includes the average powers which can be used to compare the power performances of the two tests over all the configurations considered in the study. Note that the configurations with true MED vector= 1 involve no type I error, and hence the entry of estimated FWE = .000 is omitted for all procedures.

First, from the simulation results, we note that all two procedures control the FWE quite accurately at the nominal α= .05 under all partial null configurations. (Recall that the esti-mated FWE must exceed .05+ 1.96√.05× .95/10, 000 = .0543 in order to conclude that it is significantly different from [higher than] α= .05.)

Next, looking at the estimates of the power, we see that TP has higher power than TH only when the true MED vector= 1 (i.e. cases 4, 11, 18 and 20), otherwise the TH test is better than the TP test. This is as expected since the pairwise contrasts are good for detecting low MEDs, whereas Helmert contrasts are good for detecting high MEDs. Also, it is worth noting that their relative performances are not affected by the true type of response function (monotone vs. nonmonotone or linear vs. step). Finally, on an average, the TH test gains about 30% more power than the TP test.

6. Conclusions

In this paper, we develop two step-down closed procedures based on pairwise and Helmert contrasts for simultaneous identifications of MEDs in each of several groups while main-taining control of familywise error rate FWE strongly. We described how to obtain close approximations for the critical constants and p-values of the proposed procedures using the PROBMC function in SAS. A simulation study including both monotone and nonmonotone configurations was conducted to compare the two procedures. We note that both procedures control the FWE quite accurately under all partial null configurations. Also, we observed that TP has higher power than TH when MED= 1 in each group, otherwise the TH test is better than the TP test. And these general trends are not affected by the type of the response function. On an average, the TH test is better than the TP test.

In the present study, we considered the equal sample size case, and its conclusions need to be generalized to the unequal sample size case. Although unequal sample sizes make the correlations ρjj unequal for both pairwise and Helmert contrasts (in particular, the Helmert

contrasts are no longer uncorrelated), whereas this causes no difficulty because exact inte-gration methods exist for the calculations of the required critical constants and p-values, or, alternatively, they can still be approximately calculated by formulas (8) and (9), respectively, using the average correlation ρ.

Finally, we note that this study has assumed that the observations are normally distributed. It is possible that the normal assumption is not reasonable or the sample sizes are too small to rely on the central limit theorem for normality, in which case nonparametric procedures are necessary [12].

Acknowledgements

The authors wish to thank the associate editor and a referee for their suggestions for improving the clarity of the exposition. The research of the first author was partially supported by National Science Council Grant NSC-89-2118-M-033-002.

(15)

Simultaneous identifications of the minimum effective dose 161 References

[1] Ruberg, S.J., 1989, Contrasts for identifying the minimum effective dose. Journal of the American Statistical Association, 84, 816–822.

[2] Dunnett, C.W., 1955, A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association, 50, 1096–1121.

[3] Williams, D.A., 1971, A test for differences between treatment means when several dose levels are compared with a zero dose level. Biometrics, 27, 103–117.

[4] Williams, D.A., 1972, The comparison of several dose levels with a zero dose control. Biometrics, 28, 519–531. [5] Marcus, R., Peritz, E., and Gabriel, K.R., 1976, On closed testing procedures with special reference to ordered

analysis of variance. Biometrika, 63, 655–660.

[6] Tamhane, A.C., Hochberg, Y., and Dunnett, C.W., 1996, Multiple test procedures for dose finding. Biometrics,

52, 21–37.

[7] Cheung, S.H. and Holland, B., 1991, Extension of Dunnett’s multiple comparison procedure to the case of several groups. Biometrics, 47, 21–32.

[8] Cheung, S.H. and Holland, B., 1992, Extension of Dunnett’s multiple comparison procedure with differing sample sizes to the case of several groups. Computational Statistics and Data Analysis, 14, 165–182. [9] Cheung, S.H. and Holland, B., 1994, A step-down procedure for multiple tests of treatment versus control in

each of several groups. Statistics in Medicine, 13, 2261–2267.

[10] Dunnett, C.W. and Tamhane, A.C., 1991, Step-down multiple tests for comparing treatments with a control in unbalanced one-way layouts. Statistics in Medicine, 10, 939–947.

[11] Wright, S.P., 1992, Adjusted p-values for simultaneous inference. Biometrics, 48, 1005–1013.

[12] Jan, S.L. and Chen, Y.I., 2004, Nonparametric procedures for simultaneous identification of the minimum effective dose in each of several groups. Journal of Biopharmaceutical Statisticas, 14, 781–789.

[13] Hochberg, Y. and Tamhane, A.C., 1987. Multiple Comparison Procedures (New York: 21 John Wiley & Sons); pp. 374–375.

[14] Johnson, N.L. and Kotz, S., 1972, Distributious in Statistics: Continuous Multivariate Distributions. (New York: John Wiley & Sons); p. 134.

[15] Somerville, P.N., 1997, Multiple testing and simultaneous confidence intervals: calculation of constants. Computational Statistics and Data Analysis, 25, 217–223.

[16] Somerville, P.N., 1998, Numerical computation of multivariate normal and multivariate t probabilities over convex regions. Journal of Computational and Graphical Statistics, 7, 529–545.

[17] Somerville, P.N., 1999, Critical values for multiple testing and comparisons: one step and step down procedures. Journal of Statistical Planning and Inference, 82, 129–138.

[18] Genz, A. and Bretz, F., 1999, Numerical computation of the multivariate t probabilities with application to power calculation of multiple contrasts. Journal of Statistical Computation and Simulation, 63, 361–378. [19] Genz, A. and Bretz, F., 2000, Numerical computation of critical values for multiple comparison problem. ASA

Proceedings of the Statistical Computing Section, 84–87.

[20] Genz, A. and Bretz, F., 2002, Comparison of methods for the computation of multivariate t-Probabilities. Journal of Computational and Graphical Statistics, 11, 950–971.

[21] Royen, T., 1987, An approximation for multivariate normal probabilities of rectangular regions. Statistics, 18, 389–400.

[22] Iyengar, S., 1988, Evaluation of normal probabilities of symmetric regions. SIAM Journal on Scientific and Statistical Computing, 9, 418–423.

[23] Iyengar, S. and Tong, Y.L., 1989, Convexity properties of elliptically contoured distributions with applications. Sankhy¯a, 51, 13–29.

[24] Hsu, J.C., 1992, The factor analytic approach to simultaneous inference in the general linear model. Journal of Computational and Graphical Statistics, 1, 151–168.

[25] Mager, P.P., 1991, Design Statistics in Pharmacochemistry (New York: John Wiley & Sons).