A TWO-STAGE PROCEDURE ON COMPARING SEVERAL EXPERIMENTAL TREATMENTS AND A CONTROL—THE COMMON AND UNKNOWN VARIANCE CASE

This paper introduces a two-stage selection rule to compare several experimental treatments with a control when the variances are common and unknown. The selection rule integrates the indifference zone approach and the subset selection approach in multipledecision theory. Two mutually exclusive subsets of the parameter space are defined, one is called the preference zone (PZ) and the other, the indifference zone (IZ). The best experimental treatment is defined to be the experimental treatment with the largest population mean. The selection procedure opts to select only the experimental treatment which corresponds to the largest sample mean when the parameters are in the PZ, and selects a subset of the experimental treatments and the control when the parameters fall in the IZ. The concept of a correct decision is defined differently in these two zones. A correct decision in the preference zone (CD1) is defined to be the event that the best experimental treatment is selected. In the indifference zone, a selection is called correct (CD2) if the selected subset contains the best experimental treatment. Theoretical results on the lower bounds for P(CD1) in PZ and P(CD2) in IZ are developed. A table is computed for the implementation of the selection procedure.


Introduction
This study is motivated by the current clinical trials involving protease inhibitors.Since the delta trial pioneered the research in the combination of drugs (e.g., AZT and ddI; AZT and ddC) as an HIV positive treatment, clinicians have experimented with a variety of HIV positive regimens involving different combinations of drugs.Regimens consisting of the combination of protease inhibitors have shown great potential.For instance, studies had shown that triple combination therapy with Saquinavir, Zidovudine, and Lamivudine reduced a mean viral load by 99% in four weeks.Drugs for HIV infection and AIDS are usually classified into two categories: nucleoside analogs and protease inhibitor.Nucleoside analogs constrain HIV replication by incorporation into the elongating strand of DNA and cause chain termination.Protease inhibitors are new drugs that block the action of the viral protease required for protein processing in the viral cycle.Protease inhibitors are usually potent and often used with combination of two nucleoside analogs.Nucleoside analogs include zidovudine (ZVD, AZT), dideoxynosine (ddI, didanosine), dideoxycytidine (ddc, zalcitabine), stavudine (D4T), and so forth.Protease inhibitors include saquinavir, indinavir, ritonavir, and so forth.Many of the combinations show promising results.The best-studied regimens include two nucleoside analog reverse transcriptase inhibitors.The different combinations include zidovudine plus lamivudine, zidovudine plus didanosine, zidovudine plus zalcitabine, stavudine plus didanosine, lamivudine plus stavudine, and didanosine plus lamivudine.
Although many of these treatments are evidently better than the traditional treatments (AZT, AZT, and ddI, or AZT and ddC, etc.), the best treatment is still unknown.This situation is difficult for both the HIV-positive patients and the physicians who are responsible for their well-being.In light of the fact that many protease inhibitors are either approved in the USA or are in advanced clinical testing, it is increasingly important to find the best regimen or a subgroup of equally good regimens.Few studies have been done for this objective.The lack of selection procedures which are designed for this purpose partly contributed to the situation.In addition, among those procedures which can be applied for this purpose, few are communicated effectively to the practitioners in the field.The selection procedure studied in this paper is our attempt to partially address this problem.
The organization of this paper is as follows.In Section 2, we give definitions and state the assumptions and the goal.Section 3 presents the selection procedure.Section 4 reveals the main theoretical results; Section 5 comments on the computation of Table 5.1; Section 6 presents an example and Section 7 gives concluding remarks.

Definitions, assumptions, and the goal
We assume that the treatments are normally distributed with different means and a common but unknown variance (i.e., population π i has distribution N(µ i ,σ 2 ), i = 0,1,...,k).The normal assumption is usually reasonable for HIV clinical trials.The measure of effectiveness of a regimen could be based on the viral load (the amount of virus in the blood stream), CD 4 (the T cell counts), and the clinical symptoms.The effectiveness could be the average of these measurements and thus validate the normal assumption in general.
We order the experimental treatments by their means.The treatment with mean µ i is defined to be better than a treatment with mean µ j if µ i is greater than µ j .We denote the ascending ordered means as µ [1] ,µ [2] ,...,µ [k] and use π (i) to denote the population associated with mean µ [i] .The best experimental treatment is then the treatment π (k) (the treatment associated with the largest population mean µ [k] ).We use µ 0 for the mean of the control treatment.We further assume that the best experimental treatment is better than the control (i.e., µ [k] > µ 0 ).This assumption is reasonable for HIV clinical trials.Indeed, many studies have shown that some regimens involving protease inhibitors are better than the traditional treatment.
The parameter space is defined to be the set of all possible values of the population means together with the possible values of the unknown variance ( Two subsets of the parameter space are of particular interest.One is the preference zone (PZ) which is defined to be the set

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The other is the indifference zone (IZ) which is defined to be the set In PZ, we have an outstanding treatment and therefore, our selection rule will insist on selecting only the best treatment.On the contrary, there is no one treatment which stands out in comparison with the other treatments or the control in the IZ.Thus, we select a subset of treatments that are comparable with the control.
The goal of this study is to derive a selection procedure P c which selects population π (k) if µ ∈ PZ or selects a random sized subset containing π (k) if µ ∈ IZ.Since we assume the common variance of the populations is unknown, we use a two-stage selection procedure in which the first stage is used to estimate the unknown variance.
This goal reflects a conservative "don't rock the boat" philosophy.Conservative approaches are often adopted in clinical trials.Regimens involving protease inhibitors, for instance, often develop resistance and even cross-resistance after a patient stops using the drug for a period of time.Switching to a new regimen involving protease inhibitors should take place only when the new regimen is clearly significantly better than the control treatment.Because a failed regimen involving protease inhibitors will reduce the effectiveness of the other regimens involving different protease inhibitors, it will be in the patients best interest to start with the best possible regimen.
We define a correct selection differently in the PZ and the IZ.We call a decision to be correct (CD 1 ) if our selection rule selects the population associated with π (k) when µ ∈ PZ.When µ ∈ IZ, a correct decision (CD 2 ) is defined to be the event that the selected subset contains the best population π (k) .
The probability requirement is defined to be an ordered pair (P * 1 ,P * 2 ).We say that a selection rule R satisfies a given probability requirement (P * 1 ,P * 2 ) if P(CD 1 |R) ≥ P * 1 and P(CD 2 |R) ≥ P * 2 .The two-stage selection rule proposed in this paper satisfies any given probability requirement (P * 1 ,P * 2 ) by allocating a sufficiently large sample size.Since this procedure combines selection and screening, the required sample size will be larger than either the indifference zone approach or the subset selection approach.However, our procedures sample size is smaller than the combined sample sizes of both selection procedures.From this point of view, our procedure is more efficient.

The selection procedure P c
We use X0 to denote the sample mean of the control regimen and X(i) for the experimental sample mean associated with µ [i] , i = 1,2,...,k.The selection rule is formulated as follows.
(1) Take an initial sample of size n 0 from the populations (n 0 ≥ 2).Denote the observations by X i j and compute (3.1) 50 Comparing several experimental treatments and a control (2) Define where h = max{h 2 ,h 3 } and h 1 , h 2 , and h 3 are chosen to satisfy the probability requirement.
(3) Take (n − n 0 ) additional observations from each population and compute we select the population which corresponds to X[k] .Otherwise, we select all populations π i with sample means satisfying Xi ≥ X0 − d.
In this selection procedure, for a specified δ * , the experimenter has the freedom to select a different size for c by selecting an appropriate value of a.The probability of selecting one best population increases as a increases (i.e., as c decreases).However, the value of d increases as a increases, thus the subset size increases.The tradeoff between selecting one regimen or the size of a subset of regimens subset size is controlled by a.

The main theorem
We need to specify n, c, and d such that the procedure P c will satisfy a given probability requirement (P * 1 ,P * 2 ).To derive lower bounds for the probability of correct decisions, we first investigate the infimum of P(CD 1 |P c ) in the PZ (denoted as the least favorable configuration, LFC) and P(CD 2 |P c ) in the IZ (denoted as the worst configuration, WC).
The monotonicity of P(CD 1 |P c ) in the PZ is easily seen.Under the assumption that the best experimental treatment is better than the control, the monotonicity of P(CD 2 |P c ) in the IZ can be shown in a similar way to that of Chen [1].We state the result in the following lemma.
Lemma 4.1.Given k + 1 normal populations N(µ i ,σ 2 ), i = 0,1,...,k, then for any fixed σ, Using Lemma 4.1, we evaluate the lower bounds of the P(CD 1 |P c ) in the PZ and the P(CD 2 |P c ) in the IZ.These bounds are used to compute h 1 , h 2 , and h 3 which are used to compute c, d, and n.The following is the main theorem of this paper (the proof of the theorem is given in Appendix 7).
Theorem 4.2.The values of h 1 , h 2 , and h 3 which simultaneously satisfy y dΦ(z) dF(y)dΦ(x) are the values for the selection rule P c to satisfy a given probability requirement (P * 1 ,P * 2 ).Here Φ and F are the distribution functions for the standard normal and X = √ Y variables, respectively; Y has Chi-square distribution with N = (k + 1)(n 0 − 1) degree of freedom.

The computation of the tables
The computation of Table 5.1 is carried out using FORTRAN 77.The density functions were programmed using FORTRAN except the normal distribution function which was an IMSL standard function.Integrations were computed using Gaussian quadrature and IMSL subroutines.
There are three parameters in our selection rule that need to be determined, namely, h 1 , h 2 , and h 3 .The parameter h 3 is selected to satisfy the first probability requirement and h 1 , h 2 are selected to satisfy the second probability requirement.To simplify the computation, we first find the smallest h 3 that satisfies the first probability requirement.We then set h 1 = h 3 and search for an h 2 = h 2 (a − 1) −1 value to satisfy the second probability requirement.where Conditioning on the sample standard deviation S, applying Lemma 4.1 simultaneously to T i j , i = 1,2,3; j = 1,2, estimating n using (3.2), and choosing d such that n = (Sh 1 d −1 ) 2 , we have the following inequalities simultaneously: dΦ(z) dF(y)dφ(x)≥ (k − 1) (z) dF(y)dφ(x), Table 5.1 was computed for a = 2. Thus h 2 = h 2 and we use the notation h 2 in the table.For other values of a, different tables are necessary.Interested parties can contact the authors to obtain the h 1 , h 2 , and h 3 values for other values of a.