# Biostatistical Assessment of Mutagenicity Studies: A Stepwise Confidence Procedure

Authors: Michael J. Adjabui; Nathaniel K. Howard; Mathias Akamba
Journal: Journal of Probability and Statistics
 (2019)

---

## Abstract

The paper addresses the issue of identifying the maximum safe dose in the context of noninferiority trials where several doses of toxicological compounds exist. Statistical methodology for identifying the maximum safe dose is available for three-arm noninferiority designs with only one experimental drug treatment. Extension of this methodology for several experimental groups exists but with multiplicity adjustment. However, if the experimental or the treatment groups can be ordered a priori according to their treatment effect, then multiplicity adjustment is unneeded. Assuming homogeneity of variances across dose group in normality settings, we employed the generalized Fieller’s confidence interval method in a multiple comparison stepwise procedure by incorporating the partitioning principle in order to control the familywise error rate (FWER). Simulation results revealed that the procedure properly controlled the FWER in strong sense. Also, the power of our procedure increases with increasing sample size and the ratio of mean differences. We illustrate our procedure with mutagenicity dataset from a clinical study.


---

## Body

## 1. Introduction

Assessing an investigational substance for mutagenic activity is one of the vital concerns of genetic toxicologists. This is because it is unacceptable to declare a substance as nonmutagenic when in actual fact it is mutagenic. Hence, the objective of mutagenicity assay in regulatory toxicology is the decision on mutagenicity or nonmutagenicity of an investigational substance (Hothorn et al., [1]). Therefore, it is important to adopt reliable biostatistical procedure to properly control (FWER) in a strong sense. However, a deep-seated problem of a statistical procedure is the possibility of a false decision. A typical experimental design used in this assay for genotoxicity assessment in one-way model in k+2 groups is as follows:(1)Negativecontrol,treatment1,⋯,treatmentk,positivecontrol.In this setup, we have two objectives to achieve. Firstly, we need to assess the sensitivity of the experiment in order to ensure the validity of the study by comparing the the positive control to negative control. Secondly, we simultaneously compare each of the k treatments with the negative control. Statistical decision in this settings involves multiple comparison and stepwise procedures: that is, individual inferences are made in stepwise manner if the sequence of individual inferences is in a specific order, as used in Stefensson et al. [2], Cao et al. [3], Chen [4], and Adjabui et al. [5]. Some simultaneous inferences remit multiplicity adjustments by invoking the partition principle proposed by Finner and Strassburger [6]: where the parameter space is partitioned into many disjoint subsets and only one of these nonempty disjoint subsets contains the true parameter of interest, so that the FWER will be properly controlled. In literature, mutagenicity dataset has been assessed according to the proof of safety by utilizing the concept of the maximum safe dose (Hothorn and Hauschke [7], by numerous authors, among them Hauschke and Hothorn [8], Hauschke et al. [9], Hothorn and Bretz [10]).As a result, this article discusses statistical aspects in terms of design and analysis using stepwise confidence set-based procedure for identification of maximum safe dose: that is, the highest experiment dose with no biological relevant increase in safety effect in comparison with negative control (Hothorn amd Hauschke [9]). We organize the article as follows. In Section 2, we provide both the testing and confidence notations, which are essential for the construction of our proposed stepwise confidence procedure. We proposed stepwise confidence interval procedure for identifying maximum safe dose for a normally distributed data with equal variances across dose group in Section 3. In Section 4, we carried out simulation studies to investigate the performance of our stepwise confidence interval procedure in terms of FWER and power estimation. We apply our proposed procedure to analyze real dataset as an example in Section 5. We end with conclusion of our study in Section 6.


## 2. Preliminaries

### 2.1. Testing Procedure

Let a random sampleXi1,Xi2,⋯,Xini be the observations from ith group (i=0,1,⋯k+1). Consider a one-way model as follows:(2)Xij=μi+ϵiji=0,1,2,⋯k+1,j=1,2,⋯ni,where Xij represent the genetic response for the jth experimental unit, j=1,2,⋯ni in the (1,2,…,k)th treatment group, where i=0 denote the negative control group and i=k+1 denote a positive control group, respectively. Suppose that the random sample variables Xij are mutually independent and follow a normal distribution with means μi,μk+1, and μ0 with their respective sample sizes ni,nk+1, and n0 which are not necessarily equal. The random error has N(0,σ2), where σ2 is unknown constant variance. Without loss of generality, assume larger values of μi imply better safety of the ith treatment group.The test problem is formulated as(3)H0i:μi-μk+1≥δversusH1i:μi-μk+1<δfori=1,2,⋯k,where δ is a relevant safety threshold. Practitioners, that is, genetic toxicologists, are often reluctant to define δ as an absolute value. However, Hauschke et al. [12] express the δ value as a fraction of difference between negative and positive control groups by δ=(θ-1)(μk+1-μ0), for θ∈(0,1). For some ethical reasons, a negative control group can be included in trial in (3). Therefore, the testing problem can be written as(4)H0:γi≥θversusH1:γi<θ,where γi is the ratio of difference in means denoted as (5)γi=μi-μ0μk+1-μ0fori=1,2,⋯k.Equation (4) is valid if and only if μk+1-μ0>0; this is inescapable condition and must be determined in the first step in our stepwise procedure in order to assess the sensitivity of the trial. We can rearrange and express (3) as(6)H0i:μi-θμk+1-1-θμ0≥0versusH1i:μi-θμk+1-1-θμ0<0.Let the sample mean estimates be(7)Xi¯=1ni∑i=1niXi,i=1,2,⋯kX¯k+1=1nk+1∑j=1nk+1Xk+1,j,X0¯=1n0∑j=1n0X0,j.The unknown and common variance σ2 can be estimated as(8)σ2^=ni-1Si+nk+1-1Sk+1+n0-1S0ni+nk+1+n0-3fori=1,2,⋯k,where σ^2 is the pooled estimator of the variance σ2 and Si2,Sk+12, and S02 denote the sample variances for the experiment and positive and negative groups, respectively. Then, the random variables(9)Ti=Xi¯-θX¯k+1-1-θX0¯σ^1/ni+θ2/nk+1+1-θ2/n0for i=1,2,⋯k are the test statistics for the testing problem in (3), which has t distribution with ν=ni+nk+1+n0-3 degrees of freedom. Pigeot et al. [13] have proved that one can claim safety if (10)Ti>t1-α,νfori=1,2,⋯k,where t1-α,ν is (1-α)-percentile of the central t- distribution with ν d.f. There are two approaches in solving the problem in (2), namely, the p-value approach and the confidence interval approach. It is noted in literature that the confidence interval approach is preferred to p-value approach. Therefore, in this study, we will construct a confidence set-based approach for γi for i=1,2,⋯k that remits multiplicity adjustment. The concept of maximum safe dose (MSD) for the proof of safety was defined by Hothorn and Hauschke [7] as (11)MSD=max⁡i:γi<θ,i=1,2,⋯kwhich means that H0 is rejected if Ti>t1-α,ν (γi<θ) at a given level α. Then, safety can be concluded for treatments i  (i=1,2,⋯k).In solving the testing problem in (3), we construct simultaneous confidence sets using intersection-union principle formulated by Berger [14]: the global null hypothesis can be expressed as the union of the subsets {H0i} of the null hypotheses, H0 against the intersection of the alternatives hypotheses H1, that is,(12)H0=⋃i=1kH0iagainstH1=⋂i=1kH1i.If H0i is rejected, then j=1,2,⋯i-1 are all rejected too in a stepwise fashion. In this case, no multiplicity adjustment is needed. Notice that these hypotheses are a priori ordered according to their importance and one’s interest and beliefs but they assume no order restrictions.


### 2.2. Fieller’s Confidence Interval

We employed the generalized Fieller’s theorem [15] to construct confidence interval for γi for i=1,2,⋯k. We need to solve k quadratic equations and then adapt the following notation from Hasler et al. [11]:(13)Zi=X¯i-X¯0Zk+1=X¯k+1-X¯0Yi=t1-α,ν2ni,Yk+1=t1-α,ν2nk+1,Y0=t1-α,ν2n0,thus yielding the upper confidence bounds as(14)θi,1-α=-∞,ZiZk+1-Y0+ZiZk+1-Y02-Zk+12-Yk+1-Y0Zi2-Yi-Y0Zk+12-Yk+1-Y0.The above confidence interval is only valid as long as Zk+12>Yk+1-Y0 by Fieller’s theorem [15]. The upper confidence limits for one-sided 100(1-α)% confidence interval are(15)θi=ZiZk+1-Y0+ZiZk+1-Y02-Zk+12-Yk+1-Y0Zi2-Yi-Y0Zk+12-Yk+1-Y0fori=1,⋯kfor the parameters γi.


## 2.1. Testing Procedure

Let a random sampleXi1,Xi2,⋯,Xini be the observations from ith group (i=0,1,⋯k+1). Consider a one-way model as follows:(2)Xij=μi+ϵiji=0,1,2,⋯k+1,j=1,2,⋯ni,where Xij represent the genetic response for the jth experimental unit, j=1,2,⋯ni in the (1,2,…,k)th treatment group, where i=0 denote the negative control group and i=k+1 denote a positive control group, respectively. Suppose that the random sample variables Xij are mutually independent and follow a normal distribution with means μi,μk+1, and μ0 with their respective sample sizes ni,nk+1, and n0 which are not necessarily equal. The random error has N(0,σ2), where σ2 is unknown constant variance. Without loss of generality, assume larger values of μi imply better safety of the ith treatment group.The test problem is formulated as(3)H0i:μi-μk+1≥δversusH1i:μi-μk+1<δfori=1,2,⋯k,where δ is a relevant safety threshold. Practitioners, that is, genetic toxicologists, are often reluctant to define δ as an absolute value. However, Hauschke et al. [12] express the δ value as a fraction of difference between negative and positive control groups by δ=(θ-1)(μk+1-μ0), for θ∈(0,1). For some ethical reasons, a negative control group can be included in trial in (3). Therefore, the testing problem can be written as(4)H0:γi≥θversusH1:γi<θ,where γi is the ratio of difference in means denoted as (5)γi=μi-μ0μk+1-μ0fori=1,2,⋯k.Equation (4) is valid if and only if μk+1-μ0>0; this is inescapable condition and must be determined in the first step in our stepwise procedure in order to assess the sensitivity of the trial. We can rearrange and express (3) as(6)H0i:μi-θμk+1-1-θμ0≥0versusH1i:μi-θμk+1-1-θμ0<0.Let the sample mean estimates be(7)Xi¯=1ni∑i=1niXi,i=1,2,⋯kX¯k+1=1nk+1∑j=1nk+1Xk+1,j,X0¯=1n0∑j=1n0X0,j.The unknown and common variance σ2 can be estimated as(8)σ2^=ni-1Si+nk+1-1Sk+1+n0-1S0ni+nk+1+n0-3fori=1,2,⋯k,where σ^2 is the pooled estimator of the variance σ2 and Si2,Sk+12, and S02 denote the sample variances for the experiment and positive and negative groups, respectively. Then, the random variables(9)Ti=Xi¯-θX¯k+1-1-θX0¯σ^1/ni+θ2/nk+1+1-θ2/n0for i=1,2,⋯k are the test statistics for the testing problem in (3), which has t distribution with ν=ni+nk+1+n0-3 degrees of freedom. Pigeot et al. [13] have proved that one can claim safety if (10)Ti>t1-α,νfori=1,2,⋯k,where t1-α,ν is (1-α)-percentile of the central t- distribution with ν d.f. There are two approaches in solving the problem in (2), namely, the p-value approach and the confidence interval approach. It is noted in literature that the confidence interval approach is preferred to p-value approach. Therefore, in this study, we will construct a confidence set-based approach for γi for i=1,2,⋯k that remits multiplicity adjustment. The concept of maximum safe dose (MSD) for the proof of safety was defined by Hothorn and Hauschke [7] as (11)MSD=max⁡i:γi<θ,i=1,2,⋯kwhich means that H0 is rejected if Ti>t1-α,ν (γi<θ) at a given level α. Then, safety can be concluded for treatments i  (i=1,2,⋯k).In solving the testing problem in (3), we construct simultaneous confidence sets using intersection-union principle formulated by Berger [14]: the global null hypothesis can be expressed as the union of the subsets {H0i} of the null hypotheses, H0 against the intersection of the alternatives hypotheses H1, that is,(12)H0=⋃i=1kH0iagainstH1=⋂i=1kH1i.If H0i is rejected, then j=1,2,⋯i-1 are all rejected too in a stepwise fashion. In this case, no multiplicity adjustment is needed. Notice that these hypotheses are a priori ordered according to their importance and one’s interest and beliefs but they assume no order restrictions.


## 2.2. Fieller’s Confidence Interval

We employed the generalized Fieller’s theorem [15] to construct confidence interval for γi for i=1,2,⋯k. We need to solve k quadratic equations and then adapt the following notation from Hasler et al. [11]:(13)Zi=X¯i-X¯0Zk+1=X¯k+1-X¯0Yi=t1-α,ν2ni,Yk+1=t1-α,ν2nk+1,Y0=t1-α,ν2n0,thus yielding the upper confidence bounds as(14)θi,1-α=-∞,ZiZk+1-Y0+ZiZk+1-Y02-Zk+12-Yk+1-Y0Zi2-Yi-Y0Zk+12-Yk+1-Y0.The above confidence interval is only valid as long as Zk+12>Yk+1-Y0 by Fieller’s theorem [15]. The upper confidence limits for one-sided 100(1-α)% confidence interval are(15)θi=ZiZk+1-Y0+ZiZk+1-Y02-Zk+12-Yk+1-Y0Zi2-Yi-Y0Zk+12-Yk+1-Y0fori=1,⋯kfor the parameters γi.


## 3. The Proposed Procedure

### 3.1. Stepwise Confidence Interval for Identifying Maximum Safe Dose Based on Ratio of Mean Differences

We identify maximum safe dose via Hsu-Berger [16] stepwise confidence set procedure: In the first step, we establish the assay sensitivity of the procedure by proving that Zk+12>Yk+1-Y0. If not, the procedure stops, indicating that the sensitivity of experiment is inadequate. We estimate the upper confidence limits in the second step as(16)θi^=ZiZk+1-Y0+ZiZk+1-Y02-Zk+12-Yk+1-Y0Zi2-Yi-Y0Zk+12-Yk+1-Y0fori=1,⋯k,where k is the total number of treatment doses to be tested. In step three, we start screening the drug by screening the lowest dose (that is at i=1) for the first safety drug and sequentially screen the subsequent doses for i=2,3,⋯k without adjusting the α levels in each of the steps in ascending manner searching for the first integer M, if it exists 1≤M≤k such that θM<θ and θM+1≥θ (this screens the first unsafe dose that is inferior to the reference dose). In this set up, dose level at step M is estimated as MSD: the highest estimated safe dose that is noninferior to the reference doses, such that it and all lower doses at steps 1,2,⋯M-1 are also noninferior.Once dose at stepM is estimated as MSD^, then the upper confidence bound for doses at M+2,M+3,⋯k steps is unneeded and should not be computed. A discernible property of this procedure is theoretically more powerful than Bonferroni-Holm step-down procedure (Holm [17]). This is because the α value in our procedure is inexhaustible and hence in each step the entire α is used without multiplicity adjustment while in Bonferroni-Holm step-down procedure the α is exhaustible: that is, α/k,α/k-1,⋯,α/2,α is exhausted and hence conservative. This may lead to liberal decision especially when k is large. The conservativeness of Bonferroni-Holm step-down procedure is overcome by the partition principle employed in our procedure.


### 3.2. Validity of the Stepwise Procedure

To construct and validate100(1-α)% simultaneous confidence sets in the above procedure in estimating MSD, the individual confidence intervals should have 100(1-α)% confidence level. For a given parameter space Θ, we set Θic=(-∞,θ) as the rejection region and the alternative Θi=[θ,∞) as the acceptance. We can construct simultaneous confidence set for the parameter vector Γ=γ1,γ2,⋯γk by employing the partitioning principle (Bretz et al. [18]). In identifying the MSD, the parameter space Θ can be decomposed into nonempty disjoint subset as follows:(17)Θ1∗=Θ1Θ2∗=Θ1c∩Θ2⋮Θi∗=Θ1c∩⋯∩Θi-1c∩Θi⋮⋮Θk∗=Θ1c∩Θ2c∩⋯∩Θk-1c∩Θk.Therefore,Θ1∗,Θ2∗,⋯Θk∗ partition the entire parameter space Θ. That is, Θ=Θ1∗∪Θ2∗∪⋯∪Θk∗. Each of these subsets Θi∗ is tested at a local level α with the conviction that the true parameter of interest can be found in one and only one of the nonempty disjoint subsets. This construction leads to multiple comparison procedure which guarantees the control of family-wise error in the strong sense. Hence, (12) can be rewritten as (18)H0=⋃i=1kγi∈ΘiagainstH1=⋂i=1kγi∈Θic.Theorem 1.
Suppose thatθ1,θ2⋯θk are the 100(1-α)% confidence bounds for γ1,γ2⋯γk, respectively, with confidence level 1-α. Then, for all γ1,γ2⋯γk∈Θ, we have(19)Pγ1<θ1,γ2<θ2⋯γM-1<θM-1,γM<θM≥1-α.The proof of Theorem1 is a direct application of Theorem 1 of Hsu and Berger [16].Proof.
Case  1. Let M=1 be the step at which the procedure stops. In such a situation, the assay sensitivity of the experiment cannot be assessed
Case  2. 2≤M≤k: For j = 1,⋯k, let
(i)Cj(X)=γk-j+1<θk-j+1
(ii)Θ1=γk≥θandΘj=⋂l=1j-1γk-l+1<θ∩γk-j+1≥θ
forj=2,⋯,k. Then, the parameter space Θ is partitioned by Θj,j=1,⋯k+1. Moreover,(20)⋃j=1kCjX∩Θjprovides a 100(1-α) confidence set for Γ=γ1⋯γk because if Γ∈Θ then(21)PΓΓ∈⋃j=1kCjX∩Θj=PΓΓ∈CjX≥1-α.In this setup, the unionized confidence set can be decomposed as follows:(22)⋃j=1kCjX∩Θj=⋃j=1M-1CjX∩Θj∪⋃j=MkCjX∩Θj=⋃j=MkCjX∩Θj⊂CMX∩ΘM∪⋂j=1Mγk-j+1<θ=⋂j=1M-1γk-j+1<θ∩γk-M+1≥θ∩CMX∪⋂j=1Mγk-j+1<θ=⋂j=1M-1γk-j+1<θ∩γk-M+1≥θ∩CMX∪⋂j=1Mγk-j+1<θ∩CMX=⋂j=1M-1γk-j+1<θ∩CMX.
Finally, we have(23)PΓΓ∈⋂j=1M-1γk-j+1<θ∩CMX=PΓΓ∈⋃j=1kCjX∩Θj≥1-α.Remark 2.
The resulting proof of Theorem1 warrants the control of FWER at level 1-α in a strong sense.For this reason, we state and prove the following proposition.Proposition 3.
The stepwise simultaneous inferences procedure for ratio of difference in means strongly controls the FWER at levelα.Proof.
LetI be any unknown subset of {1,2,⋯k}. Suppose that I=∅, then no FWER will ever exist. Thus, assume that I≠∅ and I={i1,i2,⋯im}, where 1≤i1<i2<⋯<im≤k. Without loss of generality, let (24)PRejectoneofH0i,i∈I∣H0i,i∈Iistrue=1-PdonotrejectallH0i,i∈I∣H0i,i∈Iistrue≤1-PdonotrejectH0im∣H0i,i∈Iistruetheprocedurethenstopsatstepim=1-Pθim⊄-∞,θ∣H0i,i∈Iistrue≤1-Pγ1<θ1,γ2<θ2⋯γM-1<θM-1,γM<θM≥1-α≤1-1-αByTheorem1=α.Remark 4.
Proposition3 guarantees that FWER is properly controlled at prespecified nominal level α. This is a critical requirement by Food and Drug Administration (FDA) for statistical procedures in dose-findings.To confirm these theoretical results, the following simulation studies were carried out at Section4.


## 3.1. Stepwise Confidence Interval for Identifying Maximum Safe Dose Based on Ratio of Mean Differences

We identify maximum safe dose via Hsu-Berger [16] stepwise confidence set procedure: In the first step, we establish the assay sensitivity of the procedure by proving that Zk+12>Yk+1-Y0. If not, the procedure stops, indicating that the sensitivity of experiment is inadequate. We estimate the upper confidence limits in the second step as(16)θi^=ZiZk+1-Y0+ZiZk+1-Y02-Zk+12-Yk+1-Y0Zi2-Yi-Y0Zk+12-Yk+1-Y0fori=1,⋯k,where k is the total number of treatment doses to be tested. In step three, we start screening the drug by screening the lowest dose (that is at i=1) for the first safety drug and sequentially screen the subsequent doses for i=2,3,⋯k without adjusting the α levels in each of the steps in ascending manner searching for the first integer M, if it exists 1≤M≤k such that θM<θ and θM+1≥θ (this screens the first unsafe dose that is inferior to the reference dose). In this set up, dose level at step M is estimated as MSD: the highest estimated safe dose that is noninferior to the reference doses, such that it and all lower doses at steps 1,2,⋯M-1 are also noninferior.Once dose at stepM is estimated as MSD^, then the upper confidence bound for doses at M+2,M+3,⋯k steps is unneeded and should not be computed. A discernible property of this procedure is theoretically more powerful than Bonferroni-Holm step-down procedure (Holm [17]). This is because the α value in our procedure is inexhaustible and hence in each step the entire α is used without multiplicity adjustment while in Bonferroni-Holm step-down procedure the α is exhaustible: that is, α/k,α/k-1,⋯,α/2,α is exhausted and hence conservative. This may lead to liberal decision especially when k is large. The conservativeness of Bonferroni-Holm step-down procedure is overcome by the partition principle employed in our procedure.


## 3.2. Validity of the Stepwise Procedure

To construct and validate100(1-α)% simultaneous confidence sets in the above procedure in estimating MSD, the individual confidence intervals should have 100(1-α)% confidence level. For a given parameter space Θ, we set Θic=(-∞,θ) as the rejection region and the alternative Θi=[θ,∞) as the acceptance. We can construct simultaneous confidence set for the parameter vector Γ=γ1,γ2,⋯γk by employing the partitioning principle (Bretz et al. [18]). In identifying the MSD, the parameter space Θ can be decomposed into nonempty disjoint subset as follows:(17)Θ1∗=Θ1Θ2∗=Θ1c∩Θ2⋮Θi∗=Θ1c∩⋯∩Θi-1c∩Θi⋮⋮Θk∗=Θ1c∩Θ2c∩⋯∩Θk-1c∩Θk.Therefore,Θ1∗,Θ2∗,⋯Θk∗ partition the entire parameter space Θ. That is, Θ=Θ1∗∪Θ2∗∪⋯∪Θk∗. Each of these subsets Θi∗ is tested at a local level α with the conviction that the true parameter of interest can be found in one and only one of the nonempty disjoint subsets. This construction leads to multiple comparison procedure which guarantees the control of family-wise error in the strong sense. Hence, (12) can be rewritten as (18)H0=⋃i=1kγi∈ΘiagainstH1=⋂i=1kγi∈Θic.Theorem 1.
Suppose thatθ1,θ2⋯θk are the 100(1-α)% confidence bounds for γ1,γ2⋯γk, respectively, with confidence level 1-α. Then, for all γ1,γ2⋯γk∈Θ, we have(19)Pγ1<θ1,γ2<θ2⋯γM-1<θM-1,γM<θM≥1-α.The proof of Theorem1 is a direct application of Theorem 1 of Hsu and Berger [16].Proof.
Case  1. Let M=1 be the step at which the procedure stops. In such a situation, the assay sensitivity of the experiment cannot be assessed
Case  2. 2≤M≤k: For j = 1,⋯k, let
(i)Cj(X)=γk-j+1<θk-j+1
(ii)Θ1=γk≥θandΘj=⋂l=1j-1γk-l+1<θ∩γk-j+1≥θ
forj=2,⋯,k. Then, the parameter space Θ is partitioned by Θj,j=1,⋯k+1. Moreover,(20)⋃j=1kCjX∩Θjprovides a 100(1-α) confidence set for Γ=γ1⋯γk because if Γ∈Θ then(21)PΓΓ∈⋃j=1kCjX∩Θj=PΓΓ∈CjX≥1-α.In this setup, the unionized confidence set can be decomposed as follows:(22)⋃j=1kCjX∩Θj=⋃j=1M-1CjX∩Θj∪⋃j=MkCjX∩Θj=⋃j=MkCjX∩Θj⊂CMX∩ΘM∪⋂j=1Mγk-j+1<θ=⋂j=1M-1γk-j+1<θ∩γk-M+1≥θ∩CMX∪⋂j=1Mγk-j+1<θ=⋂j=1M-1γk-j+1<θ∩γk-M+1≥θ∩CMX∪⋂j=1Mγk-j+1<θ∩CMX=⋂j=1M-1γk-j+1<θ∩CMX.
Finally, we have(23)PΓΓ∈⋂j=1M-1γk-j+1<θ∩CMX=PΓΓ∈⋃j=1kCjX∩Θj≥1-α.Remark 2.
The resulting proof of Theorem1 warrants the control of FWER at level 1-α in a strong sense.For this reason, we state and prove the following proposition.Proposition 3.
The stepwise simultaneous inferences procedure for ratio of difference in means strongly controls the FWER at levelα.Proof.
LetI be any unknown subset of {1,2,⋯k}. Suppose that I=∅, then no FWER will ever exist. Thus, assume that I≠∅ and I={i1,i2,⋯im}, where 1≤i1<i2<⋯<im≤k. Without loss of generality, let (24)PRejectoneofH0i,i∈I∣H0i,i∈Iistrue=1-PdonotrejectallH0i,i∈I∣H0i,i∈Iistrue≤1-PdonotrejectH0im∣H0i,i∈Iistruetheprocedurethenstopsatstepim=1-Pθim⊄-∞,θ∣H0i,i∈Iistrue≤1-Pγ1<θ1,γ2<θ2⋯γM-1<θM-1,γM<θM≥1-α≤1-1-αByTheorem1=α.Remark 4.
Proposition3 guarantees that FWER is properly controlled at prespecified nominal level α. This is a critical requirement by Food and Drug Administration (FDA) for statistical procedures in dose-findings.To confirm these theoretical results, the following simulation studies were carried out at Section4.


## 4. Simulation Studies

### 4.1. FWER

We conducted simulation studies to investigate the performance of the (FWER). Without loss of generality, we setθ=0.8, α=0.025. In this study, observations were generated with 1million replications from a normal distribution based on the assumption of equal variance across dose groups. This is indicated in Table 1 as HOMO. We also explored the effect of violation of this assumption as a way of comparing the two situations and this is indicated in Table 1 as HETRO. We used Hasler et al. [11] means configuration μP=16.5,μR=16.5,μE1=32.66,μE2=32.66. For HOMO=(σp=σR=σEi=5 for i=1,2) and the HETRO= (σp=5,σR=12,σEi=9 for i=1,2). In the simulation study, we considered only k=2 experimental treatment. Results from Table 1 indicated that the FWER is properly controlled at a nominal value α=0.025 in the case of equal variances but that of unequal variances is seriously conservative because simulated values are far below or above 0.025, the nominal level, and hence, poorly controlled the FWER.Table 1
Simulated FWER, givenα=0.05,nR=20,nP=20, and, θ=0.8.
nE1(nE2) HOMO HETRO 5(6) 0.0252 (0.0250) 0.0299 (0.0305) 7(8) 0.0249 (0.02480 0.0184 (0.0177) 9 (10) 0.0251 (0.0251) 0.0109 (0.0160) 11 (12) 0.0249 (0.0247) 0.0157 (0.0153) 13 (14) 0.0244 (0.0249) 0.0149 (0.0114) 15 (16) 0.0247 (0.0248) 0.0141 (0.0136) 17 (18) 0.0249 (0.0250) 0.0129 (0.0128) 19 (20) 0.0249 (0.0248) 0.0124 (0.0119) 21 (22) 0.0249 (0.0250) 0.0117 (0.0115) 23 (24) 0.0205 (0.0247) 0.0110 (0.0109) 25 (26) 0.0250 (0.0249) 0.0106 (0.0106) 27 (28) 0.0251 (0.0245) 0.0100 (0.0009) 29 (30) 0.0250 (0.0248) 0.0096 (0.0093)


### 4.2. Power Estimation

Power estimation is imperative for a well-design clinical study. There are many definitions of power in multiple comparisons procedures, but in this study, we will define power in the case of maximum safe dose. The maximum safe dosei is established when θj<θ and θj+1≥θ for j=1,2,⋯i. That is,(25)PMSD^=i=P⋂j=1iTj>t1-α,vi∩Ti+1≤t1-α,vi.Hence, in this setting, power is defined as the probability of rejecting the incorrect null hypotheses. This power concept is directly related to all-pairs power definition introduced by Ramsay [19]. Therefore, (25) expression can be rewritten as(26)PRejectHjoforj=1,2,⋯i=P⋂j=1iTj>t1-α,vi.Therefore, (26) can be calculated from a k variate noncentral t-distribution with νi degree of freedom and noncentrality parameters for i=1,2,⋯,k:(27)Θi=μi-θμk+1-1-θμ0σ1/ni+θ2/nk+1+1-θ2/n0.It is possible to express common varianceσ as a fraction of difference μk+1-μ0, that is, σ=ϵ(μk+1-μ0),  ϵ>0. Hence, the following representation of noncentrality parameter based on the ratio of mean differences is stated as(28)Θi=γi-θϵ1/ni+θ2/nk+1+1-θ2/n0.From (28), it is clear that the expected values of power are a function of γi, the ratio of mean differences, and the sample sizes. From Table 2, it can be seen that power increases with increasing γi and sample size but decreases with increasing values of ϵ. This is consistent with the results of Pigeot et al. [13].Table 2
Power Estimation of the confidence intervals forσR=10,σP=10,σEi=10i=1,2.
Ratio(γi) n E i = 1,2 ϵ = 0.25 ϵ = 0.5 ϵ = 1 0.85 5 0.0623 0.0402 0.0319 0.85 20 0.1161 0.0574 0.00385 0.85 30 0.1409 0.0644 0.0410 0.85 40 0.1606 0.700 0.00429 0.90 5 0.1332 0.0623 0.0402 0.90 20 0.3336 0.1410 0.0573 0.90 30 0.4234 0.1409 0.0645 0.90 40 0.4903 0.1606 0.0700 0.95 5 0.2460 0.0928 0.0503 0.95 20 0.6312 0.2082 0.0828 0.95 30 0.7550 0.2627 0.1085 0.95 40 0.8273 0.3056 0.1086 1.00 5 0.3964 0.1332 0.0623 1.00 20 0.8643 0.3336 0.1161 1.00 30 0.9422 0.4230 0.1409 1.00 40 0.9720 0.4903 0.1606 1.05 5 0.5641 0.11842 0.0764 1.05 20 0.9689 0.4830 0.1578 1.05 30 0.9930 0.5982 0.1961 1.05 40 0.9980 0.6771 0.2266 1.10 5 0.7300 0.2460 0.0928 1.10 20 0.9957 0.6312 0.2082 1.10 30 0.9996 0,7550 0.2630 1.10 40 0.9999 0.8273 0.3057 1.15 5 0.8437 0.3124 0.1113 1.15 20 0.9965 0.7635 0.2672 1.15 30 0.9996 0.9232 0.3335 1.15 40 0.9999 0.9232 0.3951 1.20 5 0.9242 0.3963 0.1332 1.20 20 0.9999 0.8643 0.3335 1.20 30 0.9999 0.9422 0.4235 1.20 40 1.0000 0.9730 0.4903


## 4.1. FWER

We conducted simulation studies to investigate the performance of the (FWER). Without loss of generality, we setθ=0.8, α=0.025. In this study, observations were generated with 1million replications from a normal distribution based on the assumption of equal variance across dose groups. This is indicated in Table 1 as HOMO. We also explored the effect of violation of this assumption as a way of comparing the two situations and this is indicated in Table 1 as HETRO. We used Hasler et al. [11] means configuration μP=16.5,μR=16.5,μE1=32.66,μE2=32.66. For HOMO=(σp=σR=σEi=5 for i=1,2) and the HETRO= (σp=5,σR=12,σEi=9 for i=1,2). In the simulation study, we considered only k=2 experimental treatment. Results from Table 1 indicated that the FWER is properly controlled at a nominal value α=0.025 in the case of equal variances but that of unequal variances is seriously conservative because simulated values are far below or above 0.025, the nominal level, and hence, poorly controlled the FWER.Table 1
Simulated FWER, givenα=0.05,nR=20,nP=20, and, θ=0.8.
nE1(nE2) HOMO HETRO 5(6) 0.0252 (0.0250) 0.0299 (0.0305) 7(8) 0.0249 (0.02480 0.0184 (0.0177) 9 (10) 0.0251 (0.0251) 0.0109 (0.0160) 11 (12) 0.0249 (0.0247) 0.0157 (0.0153) 13 (14) 0.0244 (0.0249) 0.0149 (0.0114) 15 (16) 0.0247 (0.0248) 0.0141 (0.0136) 17 (18) 0.0249 (0.0250) 0.0129 (0.0128) 19 (20) 0.0249 (0.0248) 0.0124 (0.0119) 21 (22) 0.0249 (0.0250) 0.0117 (0.0115) 23 (24) 0.0205 (0.0247) 0.0110 (0.0109) 25 (26) 0.0250 (0.0249) 0.0106 (0.0106) 27 (28) 0.0251 (0.0245) 0.0100 (0.0009) 29 (30) 0.0250 (0.0248) 0.0096 (0.0093)


## 4.2. Power Estimation

Power estimation is imperative for a well-design clinical study. There are many definitions of power in multiple comparisons procedures, but in this study, we will define power in the case of maximum safe dose. The maximum safe dosei is established when θj<θ and θj+1≥θ for j=1,2,⋯i. That is,(25)PMSD^=i=P⋂j=1iTj>t1-α,vi∩Ti+1≤t1-α,vi.Hence, in this setting, power is defined as the probability of rejecting the incorrect null hypotheses. This power concept is directly related to all-pairs power definition introduced by Ramsay [19]. Therefore, (25) expression can be rewritten as(26)PRejectHjoforj=1,2,⋯i=P⋂j=1iTj>t1-α,vi.Therefore, (26) can be calculated from a k variate noncentral t-distribution with νi degree of freedom and noncentrality parameters for i=1,2,⋯,k:(27)Θi=μi-θμk+1-1-θμ0σ1/ni+θ2/nk+1+1-θ2/n0.It is possible to express common varianceσ as a fraction of difference μk+1-μ0, that is, σ=ϵ(μk+1-μ0),  ϵ>0. Hence, the following representation of noncentrality parameter based on the ratio of mean differences is stated as(28)Θi=γi-θϵ1/ni+θ2/nk+1+1-θ2/n0.From (28), it is clear that the expected values of power are a function of γi, the ratio of mean differences, and the sample sizes. From Table 2, it can be seen that power increases with increasing γi and sample size but decreases with increasing values of ϵ. This is consistent with the results of Pigeot et al. [13].Table 2
Power Estimation of the confidence intervals forσR=10,σP=10,σEi=10i=1,2.
Ratio(γi) n E i = 1,2 ϵ = 0.25 ϵ = 0.5 ϵ = 1 0.85 5 0.0623 0.0402 0.0319 0.85 20 0.1161 0.0574 0.00385 0.85 30 0.1409 0.0644 0.0410 0.85 40 0.1606 0.700 0.00429 0.90 5 0.1332 0.0623 0.0402 0.90 20 0.3336 0.1410 0.0573 0.90 30 0.4234 0.1409 0.0645 0.90 40 0.4903 0.1606 0.0700 0.95 5 0.2460 0.0928 0.0503 0.95 20 0.6312 0.2082 0.0828 0.95 30 0.7550 0.2627 0.1085 0.95 40 0.8273 0.3056 0.1086 1.00 5 0.3964 0.1332 0.0623 1.00 20 0.8643 0.3336 0.1161 1.00 30 0.9422 0.4230 0.1409 1.00 40 0.9720 0.4903 0.1606 1.05 5 0.5641 0.11842 0.0764 1.05 20 0.9689 0.4830 0.1578 1.05 30 0.9930 0.5982 0.1961 1.05 40 0.9980 0.6771 0.2266 1.10 5 0.7300 0.2460 0.0928 1.10 20 0.9957 0.6312 0.2082 1.10 30 0.9996 0,7550 0.2630 1.10 40 0.9999 0.8273 0.3057 1.15 5 0.8437 0.3124 0.1113 1.15 20 0.9965 0.7635 0.2672 1.15 30 0.9996 0.9232 0.3335 1.15 40 0.9999 0.9232 0.3951 1.20 5 0.9242 0.3963 0.1332 1.20 20 0.9999 0.8643 0.3335 1.20 30 0.9999 0.9422 0.4235 1.20 40 1.0000 0.9730 0.4903


## 5. Example

To illustrate our procedure, we used raw data published by Adler and Kliesch [20] for a micronucleus assay by applying 30mg/kg, 50mg/kg, 75mg/kg, and 100mg/kg doses of hydroquinone (Hydro) with positive control 25mg/kg cyclophosphamide. Their primary interest is to demonstrate whether the underlying substance is able to induce chromosome damage or interact with spindle apparatus. The male mice studies results of 24h sampling time are given in Table 3. and summary of the test for micronucleus assay data from Hasler et al. [11] is given in Table 4.Table 3
Number of micronuclei per animal and 2000 scored cells for the negative control, four doses of hydroquinone and positive control cyclophosphamide.
Experimental group Mean Standard deviation Sample size Vehicle control 2.57 1.27 7 Hydro30 3.80 1.10 5 Hydro50 6.30 1.48 5 Hydro75 14.0 3.97 5 Hyro100 20.0 4.06 5 Positive control 25 8.91 4Table 4
Summary of the test for micronucleus assay data from Hasler et al. [11].
Treatment groups Unadjusted p-values Upper bound 30 mg/kg 0.0088 0.24 50 mg/kg 0.0182 0.35 75 mg/kg 0.0288 0.74 100 mg/kg 0.09639 1.04In evaluation of the mutagenicity data from Table3 and setting α=0.05 and θ=0.5, where θ is the safety threshold, the following results were obtained:(29)θ1^=0.24<θ=0.5werejectH01θ2^=0.35<θ=0.5werejectH02θ3^=0.74≮θ=0.5wedonotrejectH03;the procedure then stop at step 3, which implies that it is needless to step it further down.From this analysis, the doses 30mg/kg and 50mg/kg are declared safe while doses 75mg/kg and 100mg/kg are unsafe at levelα. Since θ3^=0.74≮θ=0.5, 50mg/kg is recommended as the maximum safe dose, which the highest dose that is noninferior to the reference drug at level α. Note that 30mg/kg is also noninferior to the reference drug but lower.


## 6. Conclusion

In this paper, we have proposed a stepdown confidence set approach for identification of maximum safe dose within the framework of noninferiority clinical trials. The classical three-arm trial for noninferiority investigations involves only one experimental treatment but in clinical trials some therapeutic situations necessitate comparisons with several experimental compounds. Therefore, the proposed(k+2)-arm trial is an extended three-arm noninferiority trial with only one treatment to multiple treatments without multiplicity adjustment. Our simulations results revealed strong control of the familywise type I error rate when we assumed equal variances across dose groups for a normally distributed dataset. This was validated by the partitioning principle.



---

*Source: 3249097-2019-04-14.xml*