Bayesian Indexes of Superiority and Equivalence and the p-value of the F -test for the Variances of Normal Distributions

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1 Japanese Journal of Bometrcs Vol. 38, No., 6 (07) Orgnal Artcle Bayesan Indexes of Superorty and Equvalence and the p-value of the F -test for the Varances of Normal Dstrbutons Masaak Do,, Kazuk Ide 3, 4, 5 and Yohe Kawasak 3, 5 Toray Industres, Inc. Clncal Data Scence & Qualty Management Department Graduate School of Scence and Engneerng, Chuo Unversty 3 Department of Pharmacoepdemology, Graduate School of Medcne and Publc Health, Kyoto Unversty 4 Center for the Promoton of Interdscplnary Educaton and Research, Kyoto Unversty 5 Department of Drug Evaluaton & Informatcs, School of Pharmaceutcal Scences, Unversty of Shzuoka e-mal:masaak Do@nts.toray.co.jp We here consder the problem of comparng the varances of two normal populatons. To make a more effcent decson than that made wth the conventonal F -test, we propose usng the Bayesan ndex of the superorty of the varance of one group to the other θ = Pr(σ >σ x, x ). We express ths ndex accordng to the hypergeometrc seres and the cumulatve dstrbuton functons of well-known dstrbutons. Furthermore, we nvestgate the relatonshp between the Bayesan ndex and the p-value of the F -test. In addton, we propose another ndex, the Bayesan ndex of equvalence of two groups, θ e( ) = Pr( <σ /σ < / x, x ) for 0 < <, whch s also expressed accordng to the hypergeometrc seres and the cumulatve dstrbuton functons of well-known dstrbutons. Fnally, we evaluate the propertes of the Bayesan ndex of equvalence usng smulaton, and llustrate the applcaton of the Bayesan ndexes wth data from actual clncal trals. Key words: Bayesan ndex; Condtonal power pror; F -test; Hstorcal data; Hypergeometrc seres; Varance of normal dstrbuton.. Introducton In bomedcal studes, we encounter occasons to compare the varances n varables of nterest across dfferent condtons. Such occasons may be dvded nto two dfferent stuatons. The frst stuaton s when we manly focus on comparng the varances. For example, test-retest varabltes (TRV) of vsual acuty measurements are compared across dfferent degrees of optcal defocus n Rosser et al. (004) and across dfferent methods of scorng n Bosch and Wall (997). The second stuaton s when we manly focus on the locaton parameters (e.g., mean), and we Receved October 06. Revsed January 07. Accepted February 07.

2 Do et al. want to check the assumpton about the varances n the statstcal method for comparng them. In many clncal trals wth contnuous outcomes, lnear (mxed) models ncludng t-test, ANOVA, and ANCOVA are used as the method of the prmary analyss. Based on whether the varances are equal or not, we may change the statstcal method (e.g., Student s t-test or Welch s t-test) because napproprate choce of the method may lead to ncorrect conclusons. See, for example, Welch (938) and Glass et al. (97). Therefore, to choose the correct method s mportant. However, for many clncal trals n ths stuaton, the tests comparng the varances are known to have lower power than expected. See, for example, Markowsk and Markowsk (990) and Wlcox (995). Ths may occur because the sample szes are calculated for comparng locaton parameters, whch reduces the power of the test for comparng varances. For a general two-group comparson of parameters, Bayesan approaches have ganed ncreasng attenton for ther potental superorty n decson makng compared to conventonal frequentst methods, because a Bayesan approach can borrow strength from the hstorcal data. For example, wth a bnomal dstrbuton B(n,p ), Altham (969), Kawasak and Myaoka (0b), Zaslavsky (03), and Kawasak et al. (04) consdered the posteror probablty Pr(p >p X,X ). For the Posson dstrbuton Po(λ ), Kawasak and Myaoka (0a) and Do (06) consdered Pr(λ <λ X,X ). Kawasak and Myaoka (0b) referred to these types of probabltes as Bayesan ndexes. For both dstrbutons, the Bayesan ndexes were shown to be expressed by the hypergeometrc seres, and the relatonshp between the Bayesan ndexes and the p-values of conventonal frequentst tests were nvestgated. In ths paper, we consder the problem of comparng the varances of two normal populatons. F -test s most frequently used n ths stuaton. To acheve a more effectve decson than possble wth the F -test by borrowng strength from the hstorcal data, we propose a Bayesan ndex of superorty and equvalence for comparng the varances of two groups of normally dstrbuted data. The remander of ths paper s structured as follows. In secton, we propose the Bayesan ndex of superorty for three stuatons, express these ndexes by the hypergeometrc seres and the cumulatve dstrbuton functons of well-known dstrbutons, and nvestgate ther relatonshp wth the p-values of the F -test. In secton 3, we propose the Bayesan ndex of equvalence, whch s also expressed by the hypergeometrc seres and the cumulatve dstrbutons functons. In secton 4, we present the results of a Monte Carlo smulaton to nvestgate the propertes of θ e( ) γ for several and γ values used n the Bayesan ndex of equvalence. In secton 5, we apply the Bayesan ndexes to analyses of real data from actual clncal trals. Fnally, we offer concludng remarks and hghlght the prospects of these ndexes n secton 6. Jpn J Bomet Vol. 38, No., 07

3 Bayesan Index and p-value for Normal Varances 3. Bayesan Index of Superorty. Defnton and the Fundamental Theorem For =,, and n,n N, let X,...,X n be ndependent normal random varables wth mean µ and varance σ. Let the realzed values of X,,X n be denoted by x =(x,...,x n ). For Bayesan analyss, let the pror dstrbuton of σ be the scaled nverse χ dstrbuton Scalednv-χ (ν,τ ) for ν,τ > 0, whose probablty densty functon s `ντ f(σ ν,τ / ν / (σ ) ν / )= exp ντ. Γ(ν /) σ Ths s equvalent to the nverse gamma dstrbuton Inv-Ga`ν /,ν τ /. To compare the varances of two groups, we propose the Bayesan ndex of superorty as follows: θ = Pr(σ >σ x,x ). In the followng descrpton, we frst consder the case where µ and µ are known, and next consder the case where both means are unknown. In each case, the followng theorem s crucal. Theorem If the (margnal) posteror dstrbuton of σ s Inv-Ga(a,b ) for =,, then the Bayesan ndex θ = Pr(σ >σ x,x ) has the followng three expressons: a b b θ = F a, a ;+a ; a B(a,a ) b + b b + b where =I b b +b (a,a ) =F a,a b/a b /a, F (a,b;c;z)= X t=0 (a) t(b) t (c) t zt t! ( z < ) s the hypergeometrc seres, and (k) t = k(k + ) (k + t ) for t N and (k) 0 = s the Pochhammer symbol, Z x F ν,ν (x)= 0 zb ν νz ν ν ν dz, ν ν z + ν ν z + ν s the cumulatve dstrbuton functon of the F dstrbuton F (ν,ν ), and Z x I x(a,b)= t a ( t) b dt B(a, b) 0 s the cumulatve dstrbuton functon of the beta dstrbuton Beta(a, b), also known as the regularzed ncomplete beta functon. proof Let λ =/σ be the precson, then θ = Pr(σ >σ x,x ) Jpn J Bomet Vol. 38, No., 07

4 4 Do et al. When the (margnal) posteror dstrbuton of σ = Pr(λ <λ x,x ). () s Inv-Ga(a,b ), the (margnal) posteror dstrbuton of λ s Ga(a,b ), whose probablty densty functon s f(λ a,b )=b a /Γ(a) λ a exp( b λ ). Hence, () s the Bayesan ndex for the Posson parameters defned n Kawasak and Myaoka (0a). Therefore, theorem follows from Kawasak and Myaoka (0a) and theorem of Do (06). From the cumulatve dstrbuton functon expressons n theorem, θ can be qute easly calculated usng standard statstcal software. Remark For theorem, snce only the (margnal) posteror dstrbuton of σ s supposed as the nverse gamma dstrbuton, the pror dstrbuton of σ may be mproper as long as the posteror s the nverse gamma.. Case : µ and µ are Known.. Calculaton of the Bayesan Index of Superorty In ths case, we denote the lkelhood of σ by L(σ x,µ ). Snce we suppose that the pror dstrbuton of σ s Scaled-nv-χ (ν,τ ), the posteror dstrbuton of σ can be derved as follows: where f(σ x,µ,ν,τ ) L(σ x,µ ) f(σ ν,τ )! n = (πσ exp X (x j µ ) `ντ / ν / (σ ) ν / )n / σ j= Γ(ν /), `σ (ν +n )/ exp ντ + n T σ T = n X (x j µ ). n j= exp ντ σ Hence, the posteror dstrbuton of σ s Inv-Ga((ν + n )/, (ν τ + n T )/). Therefore, from theorem, we have (ντ + n T )/(ν + n ) θ = F ν +n,ν +n (ν τ + n T. )/(ν + n) ().. The Relatonshp between the Bayesan Index of Superorty and the p-value of the one-sded F -test Here, we consder the F -test wth H 0 : σ = σ versus H : σ >σ when µ and µ are known. Under H 0, the test statstcs T /T follow F (n,n ). Hence, the p-value s calculated as Z p = T /T zb n nz n n n dz, n n z + n n z + n Jpn J Bomet Vol. 38, No., 07

5 Bayesan Index and p-value for Normal Varances 5 = F n,n (T /T ). (3) Then, the followng theorem holds. Theorem If µ and µ are known and the pror dstrbuton of σ s Scaled-nv-χ (ν,τ ) for =,, then the followng relaton holds between the Bayesan ndex θ = Pr(σ >σ x,x ) and the one-sded p-value of the F -test wth H 0 : σ = σ versus H : σ >σ : proof From () and (3), lm θ = p. (ν,ν ) (0,0) holds. lm (ν,ν ) (0,0) θ = Fn,n (T /T )= p Remark For the pror dstrbuton, `ντ f(σ ν,τ / ν / (σ ) ν / )= exp ντ Γ(ν /) σ (σ ) ν/ exp ντ σ ν 0 (σ ), when the pror dstrbuton of σ s f(σ ) (σ ), whch s mproper, the posteror dstrbuton of σ s Inv-Ga(n /,n T /). Therefore, as stated n remark, the Bayesan ndex can be expressed by theorem as θ = F n,n `T /T. Then, the followng theorem holds. Theorem 3 If µ and µ are known and the pror dstrbuton of σ s f(σ ) (σ ) for =,, then θ = p holds. Snce ν s the pror effectve sample sze of Scaled-nv-χ (ν,τ ) as defned n Morta et al. (008), theorem 3 can be nterpreted as follows: the Bayesan ndex wth pror effectve sample sze 0 for both groups s equal to ( p) of the one-sded F -test. Furthermore, wth the pror Scaled-nvχ (ν,τ ), the Bayesan ndex can be nterpreted as equal to the F -test wth the pror nformaton of α addtonal samples. Jpn J Bomet Vol. 38, No., 07

6 6 Do et al..3 Case : µ and µ are Unknown In ths case, we consder two types of the pror dstrbutons of (µ,σ ). In each type, we denote the lkelhood of (µ,σ ) by L(µ,σ x ). Furthermore, n the followng, we denote the probablty densty functon of the normal nverse gamma dstrbuton NIG(µ 0,k,α,β) for µ 0 R,k,α,β > 0 by f(µ,σ µ 0,k,α,β) r k = πσ exp k(µ µ0) σ βα Γ(α) (σ ) α exp βσ. When (µ,σ ) follows NIG(µ 0,k,α,β), the margnal dstrbuton of σ s Inv-Ga(α,β)..3. Calculaton of the Bayesan ndex of superorty for the scaled nverse χ varance pror We frst suppose that the pror dstrbuton of µ s non-nformatve,.e., f(µ ), and the pror dstrbuton of σ s Scaled-nv-χ (ν,τ ). Then, the pror dstrbuton of (µ,σ ) s Scaled-nv-χ (ν,τ ). Here, the posteror dstrbuton of (µ,σ ) can be derved as where f(µ,σ x,ν,τ ) L(µ,σ x ) f(µ,σ ν,τ )! n = (πσ exp X (x j µ ) `ντ / ν / (σ ) ν / )n / Γ(ν /) `σ / exp j= σ n(µ x) `σ σ x = n n X j= x j,s = n Therefore, the posteror dstrbuton of (µ,σ ) s exp ντ σ ν + n exp ντ +(n ) S, σ n X j= (x j x ). NIG` x,n, (ν + n )/, (ν τ +(n ) S )/, and the margnal posteror dstrbuton of σ s Inv-Ga((ν + n )/, (ν τ +(n ) S )/). Then, the Bayesan ndex can be expressed by theorem as (ντ +(n ) S θ = F )/(ν + n ) ν +n,ν +n (ν τ +(n ). (4) S )/(ν + n ).3. The relatonshp between the Bayesan ndex of superorty for the scaled nverse χ varance pror and the p-value of the one-sded F -test Here, we consder the F -test wth H 0 : σ = σ versus H : σ >σ when µ and µ are unknown. Under H 0, the test statstcs S /S follow F (n,n ). Therefore, the p-value s Jpn J Bomet Vol. 38, No., 07

7 calculated as Z p = S /S Bayesan Index and p-value for Normal Varances 7 n zb, n (n )z (n )z +(n ) n n n (n )z +(n ) = F n,n (S /S ). (5) Then, the followng theorem holds. Theorem 4 If the pror dstrbuton of µ s non-nformatve,.e., f(µ ), and that of σ s Scaled-nv-χ (ν,τ ) for =,, respectvely, then the followng relaton holds between the Bayesan ndex θ = Pr(σ >σ x,x ) and the one-sded p-value of the F -test wth H 0 : σ = σ versus H : σ >σ : proof From (4) and (5), lm θ = p. (ν,ν ) (0,0) dz holds. lm θ = F n,n (S/S )= p (ν,ν ) (0,0) Remark 3 For the pror dstrbuton, `ντ f(µ,σ ν,τ / ν / (σ ) ν / )= exp ντ Γ(ν /) σ (σ ) ν/ exp ντ σ ν 0 (σ ), when the pror dstrbuton of (µ,σ ) s f(µ,σ ) (σ ), the posteror dstrbuton of (µ,σ ) s NIG( x,n, (n )/, (n ) S /) and the margnal posteror dstrbuton of σ s Inv- Ga((n )/, (n ) S /). Therefore, the Bayesan ndex can be expressed by theorem as θ = F n,n `S /S. Then, the followng theorem holds. Theorem 5 If the pror dstrbuton of (µ,σ ) s f(µ,σ ) (σ ) for =,, then θ = p holds. Jpn J Bomet Vol. 38, No., 07

8 8 Do et al..3.3 Calculaton of the Bayesan ndex of superorty for the normal nverse gamma pror We next suppose that the pror dstrbuton s µ σ N(µ 0,,σ /k ) and σ Scaled-nvχ (ν,τ )=Inv-Ga(ν /,ν τ /). Then, the pror dstrbuton of (µ,σ ) s the normal nverse gamma dstrbuton NIG`µ 0,,k,ν /,ν τ /. Hence, the posteror dstrbuton of (µ,σ ) can be derved as f(µ,σ x,µ 0,,k,ν,τ ) L(µ,σ x ) f(µ,σ µ 0,,k,ν,τ )! n = (πσ exp X (x j µ ) / k k(µ µ0,) exp )n / σ j= πσ σ `ντ / ν / (σ ) ν / exp ντ Γ(ν /) σ `σ / (k + n)(µ µn) exp σ 0 `σ ν + n ν τ +(n ) S kn(µ0, x) + exp k + n C A, σ where µ n = a n = kµ0, + n x k + n,k n = k + n, ν + n,b n = ντ +(n ) S + kn(µ0, x). (k + n ) Then, the posteror dstrbuton of (µ,σ ) s NIG(µ n,k n,a n,b n). Hence, the margnal posteror dstrbuton of σ theorem as s Inv-Ga(a n,b n). Therefore, the Bayesan ndex can be expressed from 0 θ = F an,a n ν τ +(n ) S ν + n ν τ +(n ) S ν + n kn(µ0, x) + (ν + n )(k + n ) C kn(µ0, A. (6) x) + (ν + n )(k + n ).3.4 The relatonshp between the Bayesan ndex of superorty for the normal nverse gamma pror and the p-value of the one-sded F -test Then, the followng theorem holds. Theorem 6 If the pror dstrbuton of (µ,σ ) s NIG(µ 0,,k,ν /,ν τ /) for =,, then the followng relaton holds between the Bayesan ndex θ = Pr(σ >σ x,x ) and the onesded p-value of the F -test wth H 0 : σ = σ versus H : σ >σ : lm (ν,k,τ,ν,k,τ ) (,0,0,,0,0) θ = p. Jpn J Bomet Vol. 38, No., 07

9 proof From (6) and (5), Bayesan Index and p-value for Normal Varances 9 lm (ν,k,τ,ν,k,τ ) (,0,0,,0,0) θ = F n,n (S /S ) = p holds. Remark 4 For the pror dstrbuton, f(µ,σ µ 0,,k,ν,τ ) s k k(µ µ0,) `ντ = exp / / ν πσ σ (σ ) ν/ exp ντ Γ(ν /) σ (σ ) (ν+)/ exp ντ + k (µ µ 0,) (σ (ν,τ,k ). ) (,0,0) σ As already shown n theorem 5, f the pror dstrbuton of (µ,σ ) s f(µ,σ ) (σ ), then holds. θ = p.4 Remark on the Pror Dstrbuton To utlze the hstorcal data effectvely, we here consder how to construct the pror dstrbuton of (µ,σ ). For =, and j =,,n 0,, let the hstorcal data x 0,j ndependently follow N(µ,σ ), and x 0, =(x 0,,...,x 0,n0, ), and let f 0(µ,σ ) `σ. Here, for 0 α, an example of the condtonal power pror dstrbuton, defned n Ibrahm and Chen (000), s where f(µ,σ ) L(µ,σ x 0,) α f 0(µ,σ ) ( (πσ exp n 0,!) α X (x )n 0,/ σ 0,j µ ) j= =`σ / exp `σ x 0, = n X 0, n 0, αn0,(µ x0,) σ (α n 0, )/ exp α(n0, ) S 0, σ j= x 0,j,S 0, = n X 0, (x 0,j x 0,). n 0, j= `σ, (7) Jpn J Bomet Vol. 38, No., 07

10 0 Do et al. Then, the pror dstrbuton of (µ,σ ) s the normal nverse gamma dstrbuton NIG( x 0,, α n 0,, (α n 0, )/, α (n 0, ) S0,/) when α n 0, >. Hence, the margnal pror dstrbuton of σ s Inv-Ga((α n 0, )/, α (n 0, ) S0,/) when α n 0, >. In ths stuaton, the next corollary drectly follows from theorem 6. Corollary If the pror dstrbuton of (µ,σ ) s the condtonal power pror descrbed above for =,, then holds. lm (α,α ) (0,0) θ = p 3. Bayesan Index of Equvalence Next, we propose the Bayesan ndex of equvalence for satsfyng < as follows: θ e( ) = Pr(/ <σ /σ < x,x ). Here, we compare and / not to the rato of the varances but rather to the rato of the standard devatons. Then, the followng theorem holds. Theorem 7 If the (margnal) posteror dstrbuton of σ s Scaled-nv-χ (a,b ) for =,, then b/a b/a θ e( ) =F a,a F a,a b /a b /a =I b (a,a ) I b / (a,a ) b +b b / +b a b b = F a B(a,a ) b / a, a ;+a ; + b b / + b a b b F a B(a,a ) b a, a ;+a ;. + b b + b proof Snce Pr(σ /σ = ) = 0, θ e( ) = Pr(/ <σ /σ < x,x ) = Pr(σ /σ < x,x ) Pr(σ /σ < / x,x ) = Pr(σ/σ < x,x ) Pr(σ/σ < / x,x ). Then, consder the posteror dstrbuton of σ/σ = λ /λ, where λ =/σ s the precson for =,. From theorem 3 n Do (06) or (.0) n Prce and Bonett (000), b/a Pr(λ /λ <c x,x )=F a,a c, b /a therefore θ e( ) =Pr(λ /λ < x,x ) Pr(λ /λ < / x,x ) Jpn J Bomet Vol. 38, No., 07

11 Bayesan Index and p-value for Normal Varances b /a j b/a = F a,a b /a =F a,a b/a ( F m,n(/x)= F n,m(x)) b/a =F a,a b /a b/a F a,a b /a ff F a,a b/a b /a The rest of the proof follows from theorem 3 n Do (06). j F a,a b/a b /a. ff 4. Smulaton We conducted a Monte Carlo smulaton to nvestgate the property of θ e( ) γ for several values of and γ. We used the condtonal power pror dstrbuton. The hstorcal data x 0,j ndependently follow N(0,σ ) for =,; j =,...,n 0,, and we consder α = α =. The present data x j ndependently follow N(0,σ ) for =,; j =,...,n. Here, let n = n = n = 5,50,00,00, and n 0 = n 0, = n 0, =0,5,50,00,00, wth n 0, n for =,. Further, we take γ = 0.90, 0.95 and =.0,.5,.50,.00. We conducted 00,000 teratons for each scenaro. For the frst scenaro, we set σ = σ = 0; that s, the varances are equal. As shown n Table, the percentage satsfyng ths condton heavly depended on the sample sze. For the second scenaro, we set σ = 5 and σ = 0; that s, σ /σ =.5, so that the varances of group are greater than those of group. As shown n Table, the percentages satsfyng θ e(.50) 0.90 and θ e(.50) 0.95 show mnmal dependence on the sample sze when n 50. These results suggest that the decson of sutable values of and γ must be consdered dependng on the Table. The percentage satsfyng θ e( ) γ when σ = σ = 0. n 0 n γ =0.90 γ =0.95 θ e(.0) θ e(.5) θ e(.50) θ e(.00) θ e(.0) θ e(.5) θ e(.50) θ e(.00) Jpn J Bomet Vol. 38, No., 07

12 Do et al. Table. The percentage satsfyng θ e( ) γ when σ = 5, and σ = 0. n 0 n γ =0.90 γ =0.95 θ e(.0) θ e(.5) θ e(.50) θ e(.00) θ e(.0) θ e(.5) θ e(.50) θ e(.00) stuaton. For example, f n = 50 and n 0 5, then θ e(.50) 0.90 or θ e(.50) 0.95 may be approprate. 5. Applcaton The applcaton of the Bayesan ndexes of superorty and equvalence was evaluated usng data from actual clncal trals, as shown n Table 3. Tral (a) and (b) are two selected trals shown n Table of Gould (99). Here, we supposed that tral (a) s a prevous tral and tral (b) s the present tral, and =, ndcate the placebo and drug A group, respectvely. Therefore, we utlzed the data of tral (a) to specfy the condtonal power pror. We suppose that α = α = α, and take α =0.0,0.,0.5,0.8,.0. The pror dstrbutons of (µ,σ ) were derved from (7) wth the followng data n 0, = 47, x 0, =3.04, S0, = n 0, = 44, x 0, =8.43, S0, = , and are shown n Table 4 for each α. Next, usng the followng data of tral (b) n = 53, x =3.75, S = n = 54, x = 0.0, S = , Table 3. Hypertenton data Placebo ( = ) Drug A ( = ) Tral n mean SD n mean SD (a) (b) Jpn J Bomet Vol. 38, No., 07

13 Bayesan Index and p-value for Normal Varances 3 Table 4. Pror dstrbutons of (µ,σ ) α Placebo ( = ) Drug A ( = ) 0.0 f(µ,σ) (σ) f(µ,σ) (σ) 0. NIG(3.04, 9.4, 4., ) NIG(8.43, 8.8, 3.9, 87.0) 0.5 NIG(3.04, 3.5,.3, ) NIG(8.43,.0, 0.5, 77.55) 0.8 NIG(3.04, 37.6, 8.3, ) NIG(8.43, 35., 7., 48.08).0 NIG(3.04, 47.0, 3.0, 946.7) NIG(8.43, 44.0,.5, 435.0) Table 5. Posteror dstrbutons of (µ,σ ). α Placebo ( = ) Drug A ( = ) 0.0 NIG(3.75, 53.0, 6.0, 99.6) NIG(0.0, 54.0, 6.5, ) 0. NIG(3.64, 6.4, 30.7, 69.98) NIG(9.95, 6.8, 30.9, 647.9) 0.5 NIG(3.53, 76.5, 37.8, 8.8) NIG(9.69, 76.0, 37.5, ) 0.8 NIG(3.46, 90.6, 44.8, ) NIG(9.50, 89., 44., ).0 NIG(3.4, 00.0, 49.5, ) NIG(9.4, 98.0, 48.5, ) Table 6. Margnal posteror dstrbutons of σ. α Placebo ( = ) Drug A ( = ) 0.0 Inv-Ga(6.0, 99.6) Inv-Ga(6.5, ) 0. Inv-Ga(30.7, 69.98) Inv-Ga(30.9, 647.9) 0.5 Inv-Ga(37.8, 8.8) Inv-Ga(37.5, ) 0.8 Inv-Ga(44.8, ) Inv-Ga(44., ).0 Inv-Ga(49.5, ) Inv-Ga(48.5, ) Table 7. Bayesan ndex of superorty and equvalence. Superorty Equvalence α θ θ θ e(.0) θ e(.5) θ e(.50) θ e(.00) we derved the posteror dstrbutons. The posteror dstrbutons of (µ,σ ) and the margnal posteror dstrbutons of σ are shown n Table 5 and Table 6, respectvely. Fnally, the Bayesan ndexes are shown n Table 7. For tral (a), the placebo group ( = ) showed a larger standard devaton than the drug A group ( = ). By contrast, for tral (b), the drug A group showed a larger standard devaton. Accordng to the present data (tral (b)) only, that s, when α = 0, θ s qute small, whch makes the varance of the placebo group seem greater. However, as α ncreases,.e., the weght of the nformaton of tral (a) ncreases, θ ncreases monotoncally, and s no longer small. Furthermore, the p-value of the F -test wth H 0 : σ = σ versus H : σ >σ s 0.979, and, as shown n remark 4, s equal to θ wth α =0.0. Next, we consder the stuaton Jpn J Bomet Vol. 38, No., 07

14 4 Do et al. of θ e( ). Based on the result of the smulaton n secton 4., we assume that θ e(.50) 0.95 shows the equvalence, because the sample sze s about 50 for both groups and for both the hstorcal and present data. Then, when usng only the present data (α = 0.0) and α = 0., the equvalence s not shown. By contrast, when α =0.5,0.8,.0, that s, when the weght of the hstorcal data s moderate to large, the equvalence s shown. In order to apply these ndexes to the real clncal trals, we have to consder whether α, and γ can be pre-specfed based on suffcently relable nformaton. If we can pre-specfy them sutably, we can determne the statstcal method for comparng the means based on whether θ e( ) γ holds or not. On the other hand, f we cannot pre-specfy them, t may be hard to determne the statstcal method for comparng the means based on whether θ e( ) γ or not because t depends on the choce of α, and γ. In such case, we have to determne the statstcal method based only on the present tral data, and we can utlze θ e( ) s for several α s to scrutnze the approprateness of the method. Dependng on the values of θ e( ) s, we may conduct senstvty analyss by changng the statstcal method for comparng the means. 6. Concluson We have proposed the Bayesan ndex of superorty to make a more effcent decson for comparng the varances between two groups than possble wth the conventonal F -test. Ths ndex was expressed by the hypergeometrc seres and the cumulatve dstrbuton functons of well-known dstrbutons. Furthermore, we showed that as the amount of pror nformaton decreases, the Bayesan ndex of superorty approaches the ( p) value of the F -test wth H 0 : σ = σ versus H : σ >σ. Moreover, f the pror dstrbuton of (µ,σ ) s f(µ,σ ) (σ ) for =,, then θ = p holds. Ths ndcates that the Bayesan ndex wth a nonnformatve pror or zero pror effectve sample sze can have the same statstcal propertes as the F -test; however, wth ncorporaton of sutable hstorcal data, the Bayesan ndex can potentally be used to make a more effcent decson. In addton, we proposed the Bayesan ndex of equvalence θ e( ), whch was evaluated wth a Monte Carlo smulaton. The results showed that the percentage satsfyng θ e( ) γ heavly depends on the sample sze. Therefore, the approprate values of and γ must be decded on a case-by-case bass. If we manly focus on comparng the varances, we can utlze the ndex of superorty and equvalence based on the objectves of trals. If we want to check the assumpton about the varances n some statstcal method, we can utlze the ndex of equvalence. In any case, n order to use these ndexes for the confrmatory purpose, t s crucal to pre-specfy α,α,, and γ sutably based on the suffcently relable nformaton because θ and whether θ e( ) γ or not depend on them. Therefore, the mportant future work s to develop a sutable method for constructng the pror dstrbutons, ncludng selectng sutable hstorcal data, and decdng α,α for the condtonal power pror. Jpn J Bomet Vol. 38, No., 07

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