Joint Statistical Meetings - Biopharmaceutical Section

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1 Iteratve Ch-Square Test for Equvalence of Multple Treatment Groups Te-Hua Ng*, U.S. Food and Drug Admnstraton 1401 Rockvlle Pke, #200S, HFM-217, Rockvlle, MD Key Words: Equvalence Testng; Actve Controls; Nusance Parameter Summary A measure of closeness among three or more means from normal populatons was proposed by Ng (2000). The F-statstc was proposed to test the null hypothess that ths measure of closeness s greater than or equal to some prespecfed δ (>0), aganst the alternatve hypothess that ths measure of closeness s smaller than δ. The null dstrbuton of the F-statstc at the boundary has a non-central F-dstrbuton wth a noncentral parameter that depends on δ and the assumed common varance (σ 2 ) wth equal sample szes. Thus, the crtcal value for the proposed test depends on σ 2, whch must be estmated, resultng n an nflaton of the type I error rate. Ng (2001) descrbed an teratve approach to resolve ths problem. Brefly, multplyng the ch-square statstc by σ 2 results n a test statstc, T 1, whch does not depend on σ 2. However, ts dstrbuton depends on σ 2. To start the teratve process, determne the crtcal value for T 1 as a functon of σ 2, defne a test statstc by subtractng the estmated crtcal value from T 1, and then calculate ts crtcal value as a functon of σ 2. Ths teratve process stops when the crtcal value becomes reasonably flat and the nflaton of the type I error rate becomes neglgble. In ths paper, the crtcal values and the type I error rates at each step wll be presented for a specfc stuaton. 1. Introducton Equvalence testng wth three or more treatment groups can be of the followng three cases. (1) Comparng two or more test groups wth a control, wth or wthout comparsons among the test groups. (2) Comparng three or more test groups wthout a control. (3) Comparng two or more test groups wth two or more controls. Ths paper focuses on the second case. Lot consstency or lot release studes are examples where three or more test groups are compared wthout a control. Comparng three dose regmens s another example. For nstance, we may wsh to compare a dose regmen of once n the mornng versus once n the evenng versus twce daly. A measure of closeness among three means from normal populatons was proposed by Ng (2000). The F-statstc was proposed to test the null hypothess that ths measure of closeness s greater than or equal to some prespecfed δ (> 0), aganst the alternatve hypothess that ths measure of closeness s smaller than δ. However, the crtcal value for the proposed test depends on the unknown varance, whch must be estmated. Usng an estmated crtcal value would nflate the type I error rate. Ng (2001) utlzed the teratve approach proposed by Ng (1993) to resolve ths problem. However, no numercal results were presented. Ths paper presents numercal results n the llustraton of the teratve approach. The measure of closeness s extended to k normal populatons (k 3) n Secton 2. * The vews expressed n ths paper are not necessarly of the U.S. Food and Drug Admnstraton. 2464

2 Furthermore, ths measure of closeness s used to set up the hypotheses. The problem wth the proposed test wll also be dscussed n ths secton. Secton 3 gves a soluton to ths problem and fnally, Secton 4 presents dscussons and further research. 2. A Measure of Closeness and a Problem wth the Proposed Test 2.1. Assumptons and Notatons Suppose that we are comparng k (k 3) treatment groups. We assume normalty wth a common varance σ 2. Let µ be the populaton mean for the th treatment group (=1,, k), and µ = (µ 1,..., µ k ) denotes the vector of the k populaton means. Suppose that there are k ndependent samples wth equal sample sze n. Let X be the sample mean for the th treatment group (=1,, k), S 2 be the pooled sample varance, and X be the grand sample mean Basc ANOVA Results Ths subsecton presents the basc results from one-way analyss of varance (ANOVA). We have and (n/σ 2 ) Σ ( X-X) 2 ~ χ 2 (k-1, λ), (1) [k(n-1)] S 2 /σ 2 ~ χ 2 (k(n-1), 0), (2) where χ 2 (ν, λ) denotes the noncentral chsquare dstrbuton wth noncentralty parameter λ = (n/σ 2 ) Σ (µ - µ) 2. (3) Furthermore, the two ch-square dstrbutons are ndependent. We then form the F-statstc from these two ch-square dstrbutons as follows: F (n/s 2 ) Σ ( X-X) 2 /(k-1). (4) Ths F-statstc follows a noncentral F- dstrbuton wth (k-1) and k(n-1) degrees of freedom and noncentralty paremeter λ gven by (3) A Measure of Closeness A measure of closeness among k means s defned as d(µ) [Σ (µ - µ) 2 ] ½, whch s the dstance between µ and ( µ,, µ), where µ = (µ µ k )/k, and ( µ,, µ) s the projecton of µ on to the vector (1,..., 1)'. Ths s shown graphcally n Fgure 1 for k = 3. Settng d(µ) = δ would result n a crcular cylnder of radus δ (see Fgure 1). Wthn the cylnder, we have d(µ) < δ, and outsde the cylnder, we have d(µ) > δ. Fgure 1. A Measure of Closeness (k = 3) µ µ = (µ 1 +µ 2 +µ 3 )/3 ( µ, µ, µ) µ The Proposed Test and the Problem We test the null hypothess H 0 : d(µ) [Σ (µ - µ) 2 ] ½ δ, (5) aganst the alternatve hypothess H 1 : d(µ) [Σ (µ - µ) 2 ] ½ < δ, [ Σ (µ - µ) 2 ] ½ ( µ 1, µ 2, µ 3 ) [ Σ ( µ - µ ) 2 ] ½ = δ Crcular Cylnder µ 1 where δ (> 0) s prespecfed. Ng (2000) proposed to reject H 0 at sgnfcance level α, and conclude that µ s wthn the cylnder, f the F-statstc gven by (4) s less than the α th 2465

3 percentle of the noncentral F-dstrbuton wth (k-1) and k(n-1) degrees of freedom and noncentralty parameter gven by λ 0 = (n/σ 2 )Σ (µ - µ) 2 = nδ 2 /σ 2. (6) The second equalty n (6) above follows under H 0 at the boundary. Note that the null dstrbuton for testng H 0 gven by (5) has a noncentral F-dstrbuton whle the null dstrbuton for testng equalty of k means n one-way ANOVA has a central F-dstrbuton. Therefore, the crtcal value depends on σ 2 that must be estmated. Usng an estmated crtcal value would nflate the type I error rate. An teratve approach proposed by Ng (1993) s used to resolve ths problem, whch wll be dscussed next. n=20, k=3, δ=1.5 and α= However, when we use the estmated crtcal value for T 1, the type I error rate wll be nflated as shown n Fgure 2b. Thus, we proceed to step A Soluton 3.1 An Iteratve Approach The crtcal values and the type I error rates are determned for n=20, k=3, δ=1.5 and α=0.025 to llustrate the teratve approach. Note that the crtcal values are defned as a functon of σ n Ng (2001). In ths paper, the crtcal values are defned as a functon of σ 2 nstead of σ. In Step 1, we start wth T 1 Σ ( X-X) 2. From (1), the crtcal value of T 1, say, c 1 (σ 2 ), such that P[T 1 c 1 (σ 2 )] = α, can be determned as (7) c 1 (σ 2 ) = (σ 2 /n) χ -1 (α; k-1, λ 0 ), where χ -1 (α; k-1, λ 0 ) denotes α th percentle of the noncentral ch-square dstrbuton wth (k- 1) degrees of freedom and noncentralty parameter λ 0 gven by (6). The crtcal value, c 1 (σ 2 ), s shown graphcally n Fgure 2a, for In step 2, we defne T 2 T 1 - c 1 (S 2 ), and then fnd the crtcal value of T 2, say, c 2 (σ 2 ), such that P[T 2 c 2 (σ 2 )] = α. The computaton of c 2 (σ 2 ) nvolves the dstrbuton functon of T 2 whch wll be derved n Appendx A. The crtcal value, c 2 (σ 2 ), s shown graphcally n Fgure 3a. When we use the estmated crtcal value for T 2, the type I error rate wll be nflated or deflated as shown n Fgure 3b. Thus, we proceed to step

4 In step 3, we defne T 3 T 2 c 2 (S 2 ), and then fnd the crtcal value of T 3, say, c 3 (σ 2 ), such that P[T 3 c 3 (σ 2 )] = α. The crtcal value, c 3 (σ 2 ), s shown graphcally n Fgure 4a. When we use the estmated crtcal value for T 3, the type I error rate wll be nflated or deflated as shown n Fgure 4b. Thus, we proceed to step 4. Note that T 3 can be expressed n terms of T 1 as follows: T 3 T 2 c 2 (S 2 ) = T 1 - c 1 (S 2 ) - c 2 (S 2 ). Contnue ths process. In step j, we defne T j as T j T j-1 - c j-1 (σ 2 ), and then fnd the crtcal value of T j, say, c j (σ 2 ), such that P[T j c j (σ 2 )] = α. When we use the estmated crtcal value for T j, the type I error rate wll be nflated or deflated. Thus, we proceed to step (j+1). Note that T j can be expressed n terms of T 1 as follows. (8) T j T j-1 - c j-1 (S 2 ) = T 1 - c 1 (S 2 ) - - c j-1 (S 2 ). The computaton of c j (σ 2 ) nvolves the dstrbuton functon of T j whch wll be derved n Appendx B. Hopefully, at some step, say, J, c J (σ 2 ) 0, for all σ 2. We then stop and reject H 0, f T J c J (S 2 ). The nflaton of the type I error rate wll be neglgble because c J (S 2 ) c J (σ 2 ), for all σ 2. If we express T J n term of T 1 as gven by (8), we would reject H 0, f where T 1 < c*(s 2 ), c*(σ 2 ) c 1 (σ 2 ) + + c J (σ 2 ). 2467

5 3.2. Computaton of c j (σ 2 ) Note that c 1 (σ 2 ) s gven by (7). Appendx A gves a dervaton of the dstrbuton functon of T 2 from whch the crtcal value c 2 (σ 2 ) may be determned numercally. The dstrbuton functon of T 2 nvolves a double ntegral of one central and one noncentral chsquare dstrbuton densty functons. Appendx B gves a dervaton of the dstrbuton functon of T j (j 3) from whch the crtcal value c j (σ 2 ) may be determned numercally. The dstrbuton functon of T j also nvolves a double ntegral of one central and one noncentral ch-square dstrbuton densty functons. SAS/IML s used n the computatons of the crtcal values and the type I error rates. 4. Dscussons and Further Research Based on the numercal experence for the one sample t-test and the Behrens-Fsher problem n Ng (1993), t s a conjecture that c j (σ 2 ) would converge to zero unformly as j approaches to nfnty because T 1 and S 2 are ndependent. Ths conjecture s also supported by the numercal results as shown n Fgures 5a and 5b. Wth unequal sample szes, t s not clear how to test the null hypothess gven by (5) for the followng reason. The F-statstc for the one-way ANOVA wth unequal sample szes s gven by (1/S 2 ) Σ n ( X-X) 2 /(k-1), where n denotes the sample sze of th treatment group. Ths statstc follows a noncentral F-dstrbuton wth noncentralty parameter gven by (1/σ 2 )Σ n (µ - µ) 2. Therefore, the null dstrbuton (at the boundary) of the test statstc depends on the µ s, whch cannot be expressed as a functon of δ. On the other hand, usng the geometrc mean sample sze may provde a good approxmate soluton when the sample szes are not too unbalanced. For comparng three or more dose regmens, δ may be determned by a small fracton of the overall expected effect sze of the dose regmens. Furthermore, t s clear that a fxed effects model as opposed to a random effects model s applcable n comparng dose regmens. On the other hand, t s not clear how to choose δ for lot consstency studes. Furthermore, t may be more approprate to assume a random effects model n lot consstency studes. In that case, the man nterest s to show that the future lots wll not vary too much n terms of ther true means. In other words, we would be testng the 2468

6 varablty of the true means. Ths s an nterestng research topc. Appendx A. A Dervaton of the Dstrbuton Functon of T 2 Let h 2 (t 2 ) be the dstrbuton functon of T 2 under H 0 at the boundary. Wrte λ 0 gven by (6) as λ 0 λ 0 (σ 2,δ) = nδ 2 /σ 2. From (1) and (7), we have h 2 (t 2 ) P[T 2 t 2 ] = P[T 1 - c 1 (S 2 ) t 2 ] = P[T 1 - (S 2 /n) χ -1 (α,k-1,λ 0 (S 2,δ)) t 2 ] = P[(n/σ 2 )T 1 - (S 2 /σ 2 ) χ -1 (α;k-1,λ 0 (S 2,δ)) (n/σ 2 )t 2 ] = P[χ 2 (k-1,λ 0 (σ 2,δ)) (n/σ 2 )t 2 + (S 2 /σ 2 )χ -1 (α;k-1,λ 0 (S 2,δ))]. Expressng S 2 as a ch-square dstrbuton n (2) and due to the fact that the two ch-square dstrbutons are ndependent, we have h 2 (t 2 ) = 0 u(y; t 0 2 ) f x (x) f y (y) dx dy, where f x (x) and f y (y) denote the probablty densty functons of χ 2 (k-1, λ 0 (σ 2,δ)) and χ 2 (k(n-1), 0), respectvely, and u(y;t 2 ) = (n/σ 2 ) t 2 + χ -1 (α; k-1, λ*)y/[k(n-1)], λ* = k(n-1)nδ 2 /(yσ 2 ). Appendx B. A Dervaton of the Dstrbuton Functon of T j (j 3) h j (t j ) P[T j t j ] = P[T 1 - c 1 (S 2 ) - - c j-1 (S 2 ) t j ] = P[(n/σ 2 )T 1 - (n/σ 2 ){c 1 (S 2 ) + + c j-1 (S 2 )} (n/σ 2 )t j ] = P[χ 2 (k-1,λ 0 (σ 2,δ)) (n/σ 2 )t j + (n/σ 2 ){c 1 (S 2 )+ +c j-1 (S 2 )}]. Expressng S 2 as a ch-square dstrbuton n (2) and due to the fact that the two ch-square dstrbutons are ndependent, we have h j (t j ) = 0 u (y; t ) 0 j j f x (x) f y (y) dx dy, where f x (x) and f y (y) are defned as n Appendx A, and u j (y; t j ) = (n/σ 2 )t j + (n/σ 2 ){c 1 (y*)+ +c j-1 (y*)}], References y* = σ 2 y/[k(n-1)] Ng, T-H. (1993). A soluton to hypothess testng nvolvng nusance parameters. Proceedngs of the Bopharmaceutcal Secton, Amercan Statstcal Assocaton, Ng, T-H. (2000), Equvalence Testng wth Three or More Treatment Groups. Proceedngs of the Bopharmaceutcal Secton, Amercan Statstcal Assocaton, Ng, T-H. (2001), F-Test for Equvalence of Three or More Treatment Groups Proceedngs of the Amercan Statstcal Assocaton. Let h j (t j ) be the dstrbuton functon of T j under H 0 at the boundary. From (1) and (7), we have 2469