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International Journal of Information and Management Sciences 20 (2009), 459-467

Simultaneous Fiducial Generalized Confidence Intervals for All Pairwise Comparisons of Means Yi-Ping Chang Department of Financial Engineering and Actuarial Mathematics Soochow University R.O.C.

Wen-Tao Huang Department of Management Sciences and Decision Making Tamkang University R.O.C.

Yi-Hsuan Wong Department of Statistics National Chung-Hsing University R.O.C.

Abstract In this paper, we propose and study simultaneous fiducial generalized confidence intervals for pairwise comparisons of means in the one-way fixed-effects model. Our approach is based on fiducial generalized pivotal quantities for vector parameters proposed by Hannig, Iyer, and Patterson [4]. Simulation studies show that our proposed procedures have satisfactory performances.

Keywords: Pairwise Comparison, Generalized Pivotal Quantity, Simultaneous Generalized Confidence Interval. 1. Introduction Consider the usual one-way fixed-effects model: Xij = µi + εij ;

i = 1, . . . , k; j = 1, . . . , ni ,

where the εij are independent N (0, σi2 ) random variables. The means µi and variances σi2 are assumed to be unknown. When the variances are all equal, Hayter [6] proposed a simultaneous confidence intervals for pairwise comparisons of means. When the variances are unequal, there is no standard statistical procedure for pairwise comparisons of means. Received May 2008; Revised August 2008; Accepted September 2008. This research was supported by NSC95-2118-M-032-014.

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International Journal of Information and Management Sciences, Vol. 20, No. 3, September, 2009

In practice some approximate procedures are used, for example, GH procedure (Games and Howell [3]), T2 procedure (Tamhane [10] [11]), and T3 procedure (Dunnett [2]), among others. Weerahandi [13] [14] introduced the concepts of generalized pivotal quantities (GPQs) and demonstrated some applications for problem of generalized confidence intervals (GCIs) involving nuisance parameters. The GCIs are not based on conventional repeated sampling considerations, but rather on exact probability statements, see Weerahandi [14] [15]. Some GCIs have been constructed in many practical problems of several normal populations, see Chang and Huang [1], Krishnamoorthy and Lu [7], Liao and Iyer [8], Mathew and Krishnamoorthy [9], and Tian [12]. These intervals do not always have exact frequentist coverage. However, simulation results seem to support that coverage probabilities of GCIs are sufficiently close to their nominal value. Recently, Hannig, Iyer, and Patterson [4] proposed a subclass of GPQs, which is called fiducial generalized pivotal quantities (FGPQs), and showed that under fairly mild conditions, fiducial generalized confidence intervals (FGCIs) constructed using FGPQs have correct asymptotic frequentist coverage. Hannig, Iyer, and Patterson [4] also investigated the connection between the generalized procedures and fiducial inference. Hannig, Lidong, Abdel-Karim, and Iyer [5] also proposed a method for constructing simultaneous fiducial generalized confidence intervals (SFGCIs) for ratios of means of lognormal distributions based on FGPQs. In this paper, we construct SFGCIs for pairwise comparisons of means based on the concept of FGPQs in the one-way fixed-effects model. Some Monte Carlo results are also attached. 2. Simultaneous Fiducial Generalized Confidence Intervals For convenience, let X i = (Xi1 , . . . , Xini ) and let xi be the observation of X i with sample size ni , and X = (X 1 , . . . , X k ), x = (x1 , . . . , xk ), µ = (µ1 , . . . , µk ), σ 2 = (σ12 , . . . , σk2 ), ζ i = (µi , σi2 ), and ζ = (ζ 1 , . . . , ζ k ). Based on the observations x of X, our goal is to derive an approximate 100(1 − α)% two-sided simultaneous FGCIs (SFGCIs) for µi − µj , for all i 6= j. In the following, the idea of simultaneous FGPQs is based on Hannig, Iyer, and Patterson [4]. Let X ∗ represent an independent copy of X, Ri (X, X ∗ , ζ) is a function of X, X ∗ , and ζ, and R(X, X ∗ , ζ) = (R1 (X, X ∗ , ζ), . . . , Rk (X, X ∗ , ζ)). For real functions g1 , . . . , gℓ , the random quantities g1 (R(X, X ∗ , ζ)), . . . , gℓ (R(X, X ∗ , ζ)) are said to be simultaneous FGPQs for g1 (µ), . . . , gℓ (µ) if the following properties are satisfied: (FGPQ1) The conditional distribution of R(X, X ∗ , ζ), conditional on X = x, is free of ζ. (FGPQ2) For every x, R(x, x, ζ) = µ. Definition 2.1. Suppose that FGPQ1 and FGPQ2 hold and there exist subsets Ci (x), i = 1, . . . , ℓ, of sample space such that P (gi (R(X, X ∗ , ζ)) ∈ Ci (x), i = 1, . . . , ℓ|X = x) ≅ 1 − α,

Simultaneous Fiducial Generalized Confidence Intervals for all Pairwise Comparisons of Means

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then (C1 (x), . . . , Cℓ (x)) is said to be an approximate 100(1− α)% SFGCIs for (g1 (µ), . . ., gℓ (µ)). This definition is essentially the same as that of Hannig, Iyer, and Patterson (2006), but we use a slightly different notation to interpret the concepts of SFGCIs. For the one-way fixed-effects model, N=

k X

ni ,

i=1

ni X ¯i = 1 X Xij , ni j=1

n

Si2

i 1 X ¯i )2 , = (Xij − X ni − 1

j=1

Sp2

k ni 1 XX ¯ i )2 , = (Xij − X N −k i=1 j=1

¯ i , S 2 , and S 2 , respectively. Also, let and let x ¯, s2i , and s2p be the observed values of X p i ni X ∗ ¯∗ = 1 Xij , X i ni j=1

n

Si∗2

i 1 X ∗ ¯ i∗ )2 , = (Xij −X ni − 1

j=1

Sp∗2

k ni 1 XX ∗ ¯ i∗ )2 . = (Xij −X N −k i=1 j=1

In the following, we present SFGCIs for all pairwise comparisons of means when the variances are equal or unequal. 2.1. When σ12 = · · · = σk2 = σ2 Let ∗

Yi (X, X , ζ) = =

r r

¯∗ (N − k)Sp∗2 −1/2 √ N −k Xi − µi Sp × n i × + µi ni σ σ2 N − k Sp Zi∗ × √ + µi , ni V∗

i = 1, . . . , k,

where ¯∗ √ Xi − µi ni ∼ N (0, 1), σ (N − k)Sp∗2 V∗= ∼ χ2N −k , σ2 Zi∗ =

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and Z1∗ , . . . , Zk∗ , and V ∗ are independent. Define Ri (X, X ∗ , ζ) = µi − (Yi (X, X ∗ , ζ) − Yi (X, X, ζ)) r N − k Sp Zi∗ = Yi (X, X, ζ) − × √ , i = 1, . . . , k. ni V∗ Observe that, for any fixed j, the random quantities Ri (X, X ∗ , ζ) − Rj (X, X ∗ , ζ) for all i, i 6= j, are indeed FGPQs for µi − µj for all i, i 6= j. Again, we have λij (X) = V ar(Ri (X, X ∗ , ζ) − Rj (X, X ∗ , ζ)|X) (N − k)Sp2 1 1 + = . N − k − 2 ni nj Define Wij (X, X ∗ , ζ) =

(Ri (X, X ∗ , ζ) − Rj (X, X ∗ , ζ)) − (Yi (X, X, ζ) − Yj (X, X, ζ)) p . λij (X)

Then the approximate 100(1 − α)% SFGCIs for µi − µj , for all i 6= j, can be obtained by the following theorem. Theorem 2.1. For any x, let Dα (x) be the solution of the equation s ( r r ) k Z ∞Z ∞ Y X nj nj nj λij (x)v Φ z −Φ z − Dα (x) ni ni (N − k)s2p 0 −∞ i=1

j6=i

×φ(z)pχ2

N−k

(v)dzdv = 1 − α,

where Φ(·) and φ(·) are respectively the cumulative distribution function and probability density function of N (0, 1), pχ2 (·) is the density function of χ2N −k . Then, N−k

P (|Wij (X, X ∗ , ζ)| ≤ Dα (x) for all i 6= j|X = x) ≅ 1 − α, with equality holds if and only if n1 = · · · = nk . Proof. It can be shown that P (|Wij (X, X ∗ , ζ)| ≤ Dα (x) for all i 6= j|X = x) s k q N − k Sp Z ∗ r N − k S Z ∗ X j p i ≅ P ×√ − × √ ≤ Dα (x) λij (x) for all j 6= i, nj ni V∗ V∗ i=1 ! r s N − k Sp Zj∗ N − k Sp Zj∗ ×√ = min × √ X=x 1≤j≤k ni nj V∗ V∗ s r k q ∗ X N − k sp Zj N − k sp Zi∗ = P − Dα (x) λij (x) ≤ ×√ − ×√ ≤ 0 for all j 6= i nj ni V∗ V∗ i=1


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s r r k X nj ∗ nj λij (x)V ∗ nj ∗ ∗ = P Z − Dα (x) ≤ Zj ≤ Z for all j 6= i ni i (N − k)s2p ni i i=1 s ( r r ) k Z ∞Z ∞ Y X nj nj nj λij (x)v = Φ z −Φ z − Dα (x) ni ni (N − k)s2p 0 −∞ i=1

j6=i

×φ(z)pχ2

N−k

(v)dzdv

= 1 − α,

and the first equality holds if and only if n1 = · · · = nk . The proof of Theorem 2.1 is completed. Therefore, by Theorem 2.1, a set of approximate (1 − α)% SFGCIs for µi − µj , for all i 6= j, is given by q q x ¯i − x ¯j − Dα (x) λij (x), x ¯i − x ¯j + Dα (x) λij (x) , for all i 6= j. 2.2. When σi2 are unequal Let Y˜i (X, X , ζ) = ∗

= where

r r

¯∗ √ ni − 1 Xi − µi (ni − 1)Si∗2 −1/2 Si × n i × + µi ni σi σi2 Si Z ∗ ni − 1 × p i∗ + µi , ni Vi

i = 1, . . . , k,

¯∗ √ Xi − µi ni ∼ N (0, 1), σi (ni − 1)Si∗2 Vi∗ = ∼ χ2ni −1 , σi2 Zi∗ =

and Zi∗ and Vi∗ , i = 1, . . . , k, are independent. Define ˜ i (X, X ∗ , ζ) = µi − (Y˜i (X, X ∗ , ζ) − Y˜i (X, X, ζ)) R r ni − 1 Si Z ∗ ˜ = Yi (X, X, ζ) − × p i∗ , i = 1, . . . , k. ni Vi

˜ i (X, X ∗ , ζ)− R ˜ j (X, X ∗ , ζ) for all i, i 6= j, Then, for any fixed j, the random quantities R are indeed FGPQs for µi − µj for all i, i 6= j. Again, we have ˜ ij (X) = V ar(R ˜ i (X, X ∗ , ζ) − R ˜ j (X, X ∗ , ζ)|X) λ (ni − 1)Si2 (nj − 1)Sj2 = + . ni (ni − 3) nj (nj − 3)

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Define ∗ ∗ ˜ ˜ , ζ)) − (Y˜i (X, X, ζ) − Y˜j (X, X, ζ)) ˜ ij (X, X ∗ , ζ) = (Ri (X, X , ζ) − Rj (X, X q W . ˜ ij (X) λ

Then an approximate 100(1 − α)% SFGCIs for µi − µj , for all i 6= j, can be obtained by the following theorem. ˜ α (x) is the solution of the equation Theorem 2.2. For any x, let D s ( ) s s k Z ∞ Y X ˜ ij (x) nj s2i nj s2i n λ j ˜ Gnj −1 pTni −1 (t)dt 2 t − Gnj −1 2 t − Dα (x) 2 n s n s s i i −∞ j j j i=1 j6=i

=1−α

(2.1)

where pTni −1 (·) and Gni −1 (·) are respectively the cumulative distribution function and probability density of T distribution with ni − 1 degrees of freedom. Then, ˜ ∗ ˜ (X, X , ζ) ≤ D (x) for all i = 6 j = x ≅ 1 − α, P W X ij α ˜ i1 (x) = · · · = λ ˜ ik (x) for all i. with equality holds if and only if λ

Proof. It can be shown that ˜ ∗ ˜ P W (X, X , ζ) ≤ D (x) for all i = 6 j X = x ij α s q k n − 1 Sj Z ∗ r n − 1 ∗ X S Z j j i i ˜ ij (x) for all j 6= i, ˜ α (x) λ ≅ P ×q − × p i∗ ≤ D ∗ nj n V i Vj i i=1 s ! r nj − 1 Sj Zj∗ ni − 1 Si Zi∗ × p ∗ = min ×q X = x 1≤j≤k ni nj Vi Vj∗ s (s k X ˜ ij (x) Zj∗ nj s2i Zi∗ nj λ ˜ p q = P × − Dα (x) ≤ ni s2j s2j Vi∗ /(ni − 1) Vj∗ /(nj − 1) i=1 s ) nj s2i Zi∗ ≤ ×p ∗ for all j 6= i ni s2j Vi /(ni − 1) s s s ! k 2 X ˜ ij (x) nj s2i ∗ n s n λ j j i ∗ ˜ = P ≤ Tj∗ ≤ 2 Ti − Dα (x) 2 2 Ti for all j 6= i n s s n s i i j j j i=1 s ( ) s s k Z ∞ Y X ˜ ij (x) nj s2i nj s2i n λ j ˜ = Gnj −1 pTni −1 (t)dt 2 t −Gnj −1 2 t− Dα (x) 2 n s n s s i i −∞ j j j i=1 j6=i

= 1 − α,


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q ˜ i1 (x) = · · · = λ ˜ ik (x) for all i, where T ∗ = Z ∗ / V ∗ /(nj − 1) with equality if and only if λ j j j are independent T distribution with nj − 1 degrees of freedom, respectively. The proof of Theorem 2.2 is completed. By Theorem 2.2, a set of approximate (1 − α)% SFGCIs for µi − µj , for all i 6= j, is given by q q h i ˜ ˜ ij (x) for all i 6= j. ˜ ˜ x ¯i − x ¯j − Dα (x) λij (x), x ¯i − x ¯j + Dα (x) λ 3. Simulations From the simulation study of Dunnett [2], T2 (Tamhane [10] [11] and T3 (Dunnett [2]) procedures are conservative whereas GH (Games and Howell [3]) is not in some situations. Also, T3 always has shorter interval length than that of T2. Accordingly, we compare the SFGCIs against T3 procedure by Monte Carlo simulation. The 100(1 − α)% simultaneous confidence intervals of T3 procedure (Dunnett, 1980) is defined by s s s2j s2j s2i s2i ¯j − SMMα,k(k−1)/2,ˆvij + , x ¯i − x ¯j + SMMα,k(k−1)/2,ˆvij + , x ¯i − x ni nj ni nj where SMMα,k(k−1)/2,ˆvij denotes the (1 − α)-quantile of the studentized maximum modulus distribution of k(k − 1)/2 uncorrelated normal variates with vˆij degrees of freedom, where 2 s2j 2 si + ni nj vˆij = . 4 s4j si + n2 (n −1) n2 (n −1) i

i

j

j

To compare the empirical coverage probability performances of the proposed SFGCIs and the T3 procedure, suppose that k = 3, (µ1 , µ2 , µ3 ) = (1, 1, 1), and (σ1 , σ2 , σ3 ) =(1, 2, 3), (3, 2, 1), (1, 2, 4), and (4, 2, 1), respectively. We use the IMSL subroutine RNNOA to generate the simulated data. The integral and the T distribution function involved in (2.1) are evaluated by using the IMSL subroutines QDAGI and TDF, respectively. For each combination of (n1 , n2 , n3 ) and (σ1 , σ2 , σ3 ), the empirical coverage probability of each procedure calculated using 10000 simulated samples. Here, we consider the volume formed by each confidence interval from some procedure. Based on the average volume, we consider efficiency between two procedures. The efficiency is then defined by eff =

average volume of SFGCIs . average volume of T3

Table 1 presents the empirical coverage probabilities of the SFGCIs, T3 procedure and the efficiency of SFGCIs with respect to T3. It is evident from this table that the empirical coverage probabilities of the SFGCIs and T3 procedure for pairwise comparisons of means almost attain the confidence level. When the model is unbalanced and population

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Table 1. Empirical coverage probabilities of T3 procedure and SFGCIs and its efficiency. n1

n2

n3

σ1

σ2

σ3

7 7 9 9 9 9 10 10 15 15 15 15

7 7 7 7 7 7 10 10 10 10 10 10

7 7 5 5 5 5 10 10 5 5 5 5

1 1 1 3 1 4 1 1 1 3 1 4

2 2 2 2 2 2 2 2 2 2 2 2

3 4 3 1 4 1 3 4 3 1 4 1

Empirical Coverage Probability T3 0.9525 0.9535 0.9486 0.9532 0.9490 0.9523 0.9535 0.9547 0.9523 0.9495 0.9525 0.9526

SFGCIs 0.9591 0.9577 0.9548 0.9620 0.9514 0.9589 0.9523 0.9505 0.9510 0.9649 0.9497 0.9628

eff 1.0177 1.0149 0.9572 1.1317 0.8876 1.0624 0.9610 0.9114 0.8681 1.2268 0.8110 1.1599

associate with larger variance is drawn with smaller sample size, the efficiency is less then 1. That is, the SFGCIs perform better than that of the T3 procedure under these situations. It is to be noted that the empirical coverage probabilities of the SFGCIs, T3 procedure and the proposed efficiency are free from values of (µ1 , µ2 , µ3 ). 4. Conclusions In this article constructions of the SFGCIs for all of the pairwise differences of means have been proposed and studied. Simulation results show that the empirical coverage probabilities of the SFGCIs almost attain the confidence level. When the model is unbalanced and population associate with larger variance is drawn with smaller sample size, the SFGCIs perform better than that of the T3 procedure in Dunnett [2]. References [1] Chang, Y. P. and Huang, W. T., Generalized confidence intervals for the largest value of some functions of parameters under normality, Statistica Sinica, Vol. 10, pp.1369-1383, 2000. [2] Dunnett, C. W., Pairwise multiple comparisons in the unequal variance case, Journal of the American Statistical Association, Vol. 75, pp.796-800, 1980. [3] Games, P. A. and Howell, J. F., Pairwise multiple comparison procedures with unequal n’s and/or variances: a Monte Carlo study, Journal of Education Statistics, Vol. 1, pp.113-125, 1976. [4] Hannig, J., Iyer, H. and Patterson, P., Fiducial generalized confidence intervals, Journal of the American Statistical Association, Vol. 101, pp.254-269, 2006. [5] Hannig, J., Lidong, E., Abdel-Karimand, A. and Iyer, H., Simultaneous fiducial generalized confidence intervals for ratios of means of lognormal distributions, Austrian Journal of Statistics, Vol. 35, pp.261-269, 2006. [6] Hayter, A. J., Pairwise comparisons of generally correlated means, Journal of the American Statistical Association, Vol. 84, pp.208-213, 1989. [7] Krishnamoorthy, K. and Lu, Y., Inferences on the common mean of several normal populations based on the generalized variable method, Biometrics, Vol. 59, pp.237-247, 2003.


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[8] Liao, C. T. and Iyer, H. K., A tolerance interval for the normal distribution with several variance components, Statistica Sinica, Vol. 14, pp.217-229, 2004. [9] Mathew, T. and Krishnamoorthy, K., One-sided tolerance limits in balanced and 10 unbalanced oneway random models based on generalized confidence limits, Technometrics, Vol. 46, pp.44-52, 2004. [10] Tamhane, A. C., Multiple comparisons in model I: one way ANOVA with unequal variances, Communication in Statistics: Theory and Methods, Vol. 6, pp.15-32, 1977. [11] Tamhane, A. C., A comparison of procedures for multiple comparisons of means with unequal variances, Journal of the American Statistical Association, Vol. 74, pp.471- 480, 1979. [12] Tian, L., Inferences on the common coefficient of variation, Statistics in Medicine, Vol. 24, pp.22132220, 2005. [13] Weerahandi, S., Generalized confidence intervals, Journal of the American Statistical Association, Vol. 88, pp.899-905, 1993, correction in Vol. 89, pp.726, 1994. [14] Weerahandi, S., Exact Statistical Methods for Data Analysis, Springer-Verlag, New York, 1995. [15] Weerahandi, S., Generalized Inference in Repeated Measures, Wiley, New York, 2004.

Authors’ Information Yi-Ping Chang is currently a full professor in Department of Financial Engineering and Actuarial Mathematics, Soochow University, Taiwan. His research interests are financial risk management, quantitative finance and statistical inference. Department of Financial Engineering and Actuarial Mathematics, Soochow University, Taiwan. E-mail: [email protected] Wen-Tao Huang is currently a full professor in Department of Management Sciences and Decision Making, Tamkang University, Taiwan. He got his PhD in Statistics from Purdue University. His research interest includes Multiple Comparisons, Empirical Bayes method and Censoring Scheme for Inferences. Department of Management Sciences and Decision Making, Tamkang University, Taiwan. E-mail: [email protected] Yi-Hsuan Wong received his Master Degree from Department of Statistics, National Chung-Hsin University. His research interest is statistical inferences. Degree from Department of Statistics, National Chung-Hsin University, Taiwan.