Solving some behrens fisher problems using modified bartlett correction

SOLVING SOME BEHRENS-FISHER PROBLEMS USING MODIFIED BARTLETT CORRECTION LIU XUEFENG NATIONAL UNIVERSITY OF SINGAPORE 2013 SOLVING SOME BEHRENS-FISHER PROBLEMS USING MODIFIED BARTLETT CORRECTION LIU XUEFENG (B.Sc. University of Science and Technology of China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2013 ii ACKNOWLEDGEMENTS First of all, I would like to show my great thanks to my supervisor, Professor Zhang Jin-Ting. He is always nice to me and teach me a lot during the past four years. This thesis can never be done without his patient guidance. I would also like to thank all my dear friends in the Department of Statistics and Applied Probability. They made my life enjoyable as a graduate student. Finally, I want to thank the National University of Singapore and the Department of Statistics and Applied probability for providing the precious opportunity and financial support for me to study in Singapore. iii CONTENTS Acknowledgements Summary ii vii Chapter Introduction 1.1 The Behrens-Fisher Problems . . . . . . . . . . . . . . . . . . . . . 1.1.1 Heteroscedastic One-Way ANOVA . . . . . . . . . . . . . . 1.1.2 Heteroscedastic Multi-Way ANOVA . . . . . . . . . . . . . . 1.1.3 Heteroscedastic One-Way MANOVA . . . . . . . . . . . . . 1.1.4 Heteroscedastic Two-Way MANOVA . . . . . . . . . . . . . 1.1.5 Comparison of Regression Coefficients under Heteroscedasticity 1.2 Classifying the Approximate Solutions to the BF Problems . . . . . 1.2.1 Approximate Degree of Freedom Tests . . . . . . . . . . . . 10 11 CONTENTS iv 1.2.2 Series Expansion-Based Tests . . . . . . . . . . . . . . . . . 12 1.2.3 Simulation-Based Tests . . . . . . . . . . . . . . . . . . . . . 13 1.2.4 Transformation-Based Tests . . . . . . . . . . . . . . . . . . 14 1.3 Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 15 Chapter MB Test for One-Way ANOVA 19 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1 The MB test . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.2 Properties of the MB Test . . . . . . . . . . . . . . . . . . . 25 2.2.3 MB Test for One-Way Random-effect Models . . . . . . . . 26 2.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.4 Applications to the PTSD Data . . . . . . . . . . . . . . . . . . . . 38 2.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.6 Technical Proofs 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter MB Test for Multi-Way ANOVA 48 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2.1 Main and Interaction Effects in Multi-Way ANOVA Models 51 3.2.2 Wald-type Statistic and χ2 Test . . . . . . . . . . . . . . . . 56 3.2.3 Bartlett Correction and Bartlett Test . . . . . . . . . . . . . 58 3.2.4 Modified Bartlett Correction and MB Test . . . . . . . . . . 59 3.2.5 Properties of the MB Test . . . . . . . . . . . . . . . . . . . 61 3.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 CONTENTS v 3.4 A Real Data Example . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.5 Technical Proofs 70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter MB Test for One-Way MANOVA 76 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.1 The MB Test . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.2 Some Desirable Properties of the MB Test . . . . . . . . . . 84 4.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Application to the Egyptian Skull Data . . . . . . . . . . . . . . . . 91 4.5 Technical Proofs 93 . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter MB Test for Two-Way MANOVA 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 99 5.2 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.1 Main and Interaction Effects . . . . . . . . . . . . . . . . . . 101 5.2.2 Wald-Type Test Statistic . . . . . . . . . . . . . . . . . . . . 105 5.2.3 The MB Test . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.2.4 Some Desirable Properties of the MB Test . . . . . . . . . . 110 5.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.4 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 5.5 MB Test for Multi-Way MANOVA . . . . . . . . . . . . . . . . . . 124 5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.7 Technical Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Chapter MB Test for Regression Coefficient Comparison 137 CONTENTS vi 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.2 Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.2.1 Wald-Type Test Statistic . . . . . . . . . . . . . . . . . . . . 138 6.2.2 χ2 , Bartlett and Modified Bartlett Tests . . . . . . . . . . . 141 6.2.3 Some Desirable Properties of the MB Test . . . . . . . . . . 144 6.3 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.3.1 Simulation A: Two-Sample Cases . . . . . . . . . . . . . . . 146 6.3.2 Simulation B: Multi-Sample Cases . . . . . . . . . . . . . . . 147 6.4 Real Data Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4.1 A Two-Sample Example . . . . . . . . . . . . . . . . . . . . 154 6.4.2 A Multi-Sample Example 6.5 Technical Proofs . . . . . . . . . . . . . . . . . . . 156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 Chapter Summary and Discussion 163 vii SUMMARY The Behrens-Fisher (BF) problems refer to compare the means or mean vectors of several normal populations without assuming the equality of the variances or covariance matrices of those normal populations. These BF problems are challenging and caught much attention for decades since the standard testing procedures such as the t-test, F -test, Hotelling T -test, or the Lawley-Hotelling trace test may fail for these BF problems. In this thesis, we solve various BF problems by applying the modified Bartlett correction of Fujikoshi (2000). These BF problems include heterogenous one-way ANOVA, multi-way ANOVA, one-way MANOVA, two-way MANOVA, and regression coefficient comparison under heteroscedasticity. For each BF problem, we show that the asymptotic distribution of the test statistic is χ2 with some known degrees Summary of freedom and we find out the expressions of the asymptotic mean and variance of the test statistic which allow us to apply the modified Bartlett correction. In each of these BF problems, by simulation studies and real data applications, we find that the resulting modified Bartlett test works well compared with the existing approximate solutions to the associated Behrens-Fisher problem. viii CHAPTER Introduction In this chapter, we first give a brief review of various Behrens-Fisher problems in Section 1.1. We then give a classification of the various existing approximate solutions to the Behrens-Fisher problems in Section 1.2. An overview of the thesis is outlined in Section 1.3. 1.1 The Behrens-Fisher Problems In this section, we review various Behrens-Fisher problems and their approximation solutions scattered in the literature. 6.5 Technical Proofs 6.5 157 Technical Proofs Proof of Theorem 6.1 Under the given conditions, we have d σ ˆl2 = σl2 χ2nl −1 /(nl − p), l = 1, 2, · · · , k, (6.19) d where X = Y means X and Y have the same distribution. It follows that (ˆ σl2 − −1/2 σl2 ) = Op (nl ˆ β − Σβ = ), l = 1, 2, · · · , k. Since X Tl X l = O(nmin ), we have Σ −3/2 Op (nmin ). Noticing that Σβ = O(n−1 ), we further have ˆ β − Σβ )H T = Op (n−1/2 ), R = H(Σ (6.20) 1/2 ˆ where H is defined in (6.7) and H = O(nmin ). This implies that W = I q +H(Σ− −1/2 Σ)H T = I q + R = I q + Op (nmin ). Theorem 6.1 follows by Slutsky’s theorem and noticing that under H0 , z T z ∼ χ2q . Proof of Theorem 6.2 Notice that under H0 , we have z ∼ N (0, I q ). Applying the conditional expectation rule and some simple algebra leads to E(T ) = E tr(W −1 ) and E(T ) = 2E tr(W −2 ) + E tr2 (W −1 ). (6.21) where and throughout, E2 (X) = [E(X)]2 , tr2 (Y ) = [tr(Y )]2 and tr(Y ) denotes the trace of Y , i.e., the sum of the diagonal entries of Y . From the proof of −1/2 Theorem 6.1, we have that W = I q + R with R = Op (nmin ); see (6.20). Then we −2 have W −1 = (I q + R)−1 = I q − R + R2 − R3 + Op (n−2 = (I q + R)−2 = ) and W 6.5 Technical Proofs 158 I q − 2R + 3R2 − 4R3 + Op (n−2 ). It is easy to see from (6.20) that E(R) = and Etr(R) = 0. Thus Etr(W −1 ) = q + Etr(R2 ) − Etr(R3 ) + O(n−2 ), Etr(W −2 ) = q + 3Etr(R2 ) − 4Etr(R3 ) + O(n−2 ), Etr (W −1 (6.22) ) = q + Etr (R) + 2qEtr(R ) − 2qEtr(R ) −2Etr(R)tr(R2 ) + O(n−2 ). We now find Etr(R2 ) and Etr2 (R) among others. ∑k l=1 By (6.20), we have R = Ωl ul , where ul = σ ˆl2 /σl2 − 1, for l = 1, 2, · · · , k. By (6.19), we have E(ul ) = 0, E(u2l ) = , and E(u3l ) = , l = 1, 2, · · · , k, nl − p (nl − p)2 (6.23) where we use the fact E(χ2d /d − 1)3 = 8/d2 . Noticing that u1 , u2 , · · · , uk are independent and by (6.23), we have Etr(R2 ) = Etr2 (R) = Etr(R ) = Etr(R)tr(R2 ) = ∑k 2 l=1 tr(Ωl )E(ul ) = ∑k l=1 ∑k tr2 (Ωl )E(u2l ) = 3 l=1 E(ul )tr(Ωl ) ∑k l=1 ∑k = l=1 ∑k l=1 tr(Ω2l ) = 2∆ , nl −p tr2 (Ωl ) = 2∆ , nl −p ∑k (6.24) l=1 (nl −p)2 tr(Ωl ), E(u3l )tr(Ωl )tr(Ω2l ) = ∑k l=1 (nl −p)2 tr(Ωl )tr(Ωl ). Later we will show that the eigenvalues of Ωl , l = 1, 2, · · · , k are all less than 1. Hence, ≤ tr(Ω3l ) ≤ tr(Ω2l ) ≤ tr(Ωl ) ≤ p, l = 1, 2, · · · , k. −2 Combining (6.22) and (6.24) gives that Etr(W −1 ) = q+2∆1 +O(n−2 )= ), Etr(W 6.5 Technical Proofs 159 −1 q + 6∆1 + O(n−2 ) = q + 4q∆1 + 2∆2 + O(n−2 ), and Etr (W ). These, to[ ] α1 gether with (6.21), yield that E(T ) = q + nmin + O(n−2 ) and E(T ) = q(q + ] [ (4q+12)∆1 +2∆2 2∆1 2) + nαmin + O(n−2 nmin as desired. ) where α1 = q nmin and α2 = q(q+2) We now find the lower and upper bounds of ∆1 and ∆2 as given in (6.10). For l = 1, 2, · · · , k, set B l = σl H l (X Tl X l )−1/2 , a q × p full rank matrix so that Ωl = B l B Tl . It follows that Ωl are nonnegative, so are their eigenvalues. Notice that Ωl and Ql = B Tl B l : p × p have the same nonzero eigenvalues. Thus, Ωl has at most p nonzero eigenvalues. Denote the largest p eigenvalues of Ωl by λl,r , r = 1, 2, · · · , p which include all the nonzero eigenvalues of Ωl . It is easy to verify that I q . Therefore, we have ∑k l=1 tr(Ωl ) = q and I q − Ωl = ∑ r̸=l ∑k l=1 Ωl = Ωr . Therefore I q − Ωl is nonnegative, showing that the eigenvalues of Ωl are less than 1. It follows that tr(Ω2l ) = together with and ∆2 = ∑p r=1 λ2l,r ≤ ∑p r=1 λl,r = tr(Ωl ) and tr(Ωl ) = ∑k ∑k l=1 l=1 tr(Ωl ) = q, imply that ∆1 = ∑k l=1 ∑p r=1 λl,r ≤ p. These, tr(Ω2l )/(nl −p) ≤ q/(nmin −p) tr2 (Ωl )/(nl − p) ≤ pq/(nmin − p). Notice that for any nonnegative numbers a1 , a2 , · · · , am , we always have m ∑ a2l m ∑ ≥( al )2 /m. l=1 It follows that tr(Ω2l ) = ∆1 ≥ p−1 ∑k l=1 ∑k l=1 ∑p r=1 (6.25) l=1 ∑ λ2l,r ≥ ( pr=1 λl,r )2 /p = tr2 (Ωl )/p. It follows that tr2 (Ωl )/(nl − p) = ∆2 /p. Using (6.25) again and the fact that tr(Ωl ) = q, we have ∆2 ≥ [ ∑k l=1 tr(Ωl )]2 /[(nmax − p)k] = q2 . (nmax −p)k It follows 6.5 Technical Proofs that ∆1 ≥ q2 . (nmax −p)kp 160 The theorem is then proved. Proof of Theorem 6.3 From the definition of T (6.5), it is easy to see that T is invariant under the transformation (6.16). Then by (6.15), we only need to show ˆ and ∆ ˆ are invariant under the transformation that θˆ1 and θˆ2 , or equivalently, ∆ (6.16). ˜ = (P ⊗ I p )C. Define C , · · · , C k be the k matriUnder (6.16), we have C ˜ 1, · · · , C ˜ k similarly so that ces of size q × p so that C = [C , · · · , C k ]. Define C ˜ = [C ˜ 1, · · · , C ˜ k ]. It follows that C ˜ l = (P ⊗ I p )C l , l = 1, 2, · · · , k. Set Gl = C ˆ β C T )−1 C l , l = 1, 2, · · · , k. Then it follows that G ˜l = C ˜ T (C ˜Σ ˆ βC ˜ T )−1 C ˜l = C Tl (C Σ l ˆ β C T )−1 C l = Gl , l = 1, 2, · · · , k. Therefore, Gl , l = 1, 2, · · · , k are inC Tl (C Σ variant under (6.16). p) = ∑k l=1 ∑k l=1 ˆ )/(nl − ˆ = ∑k tr(Ω By (??) and (6.14), we have ∆ l l=1 ˆ l )/(nl − p) = ˆ = ∑k tr2 (Ω σ ˆl4 tr([(X Tl X)−1 Gl ]2 )/(nl − p) and ∆ l=1 ˆ and ∆ ˆ are also invariant σ ˆl4 tr2 ((X Tl X l )−1 Gl )/(nl − p), showing that ∆ under (6.16). Theorem 6.3 is then proved. Proof of Theorem 6.4 The theorem will be proved if we can prove T (6.5) ˆ 1, ∆ ˆ (6.14) are invariant to transform (6.17). Denote the unbiased esand ∆ timator of the regression coefficients and variance of the error term of the l-th ˜ˆ ˜ˆ ˆ˜ 2l respectively, l = 1, 2, · · · , k. Then β transformed sample with β l and σ l = ˆ + ve1,p , σ ˆ˜ 2l = u2 σ uβ ˆl2 where er,k denotes a unit vector of length k with r-th l 6.5 Technical Proofs 161 ˜ˆ ˜ˆ T ˜ˆ T T ˜ˆ T ˆ entry being and others 0. It follows β = [β , · · · , β k ] = uβ + v1k ⊗ e1,p , Σβ = ˜TX ˜ )−1 , · · · , σ ˜TX ˜ k )−1 ] = u2 Σ ˆ β . So the transformed test statistic T˜ = ˆ˜ 21 (X ˆ˜ 2k (X diag[σ k ˜ˆ ˜ˆ ˜ˆ T −1 T −1 T T ˆ ˆ ˆ (C β − c)T (C Σ β C ) (C β − c) = (C(uβ + v1k ⊗ e1,p )) (u C Σβ C ) (C(uβ + v1Tk ⊗ e1,p )) = T, where for coefficients comparison, c is set equal to and we use the fact that C(1Tk ⊗ e1,p ) = (Q ⊗ I k )(1Tk ⊗ e1,p ) = (Q1Tk ) ⊗ (I k e1,p ) = 0. ˆ and ∆ ˆ are invariant to transform (6.17). Notice We now turn to show ∆ ˆ β C T )−1 C l ) ˆ l) = σ tr(Ω ˆ tr((X T X)−1 C Tl (C Σ ˆ 2) = σ tr(Ω ˆ tr([(X l T (6.26) ˆ β C T )−1 C l ]2 ). X)−1 C Tl (C Σ ˜ˆ 2ˆ ˆ ˆ2 Recall σ ˜ˆ l = u2 σ ˆl2 and Σ β = u Σβ , it is easy to see that tr(Ωl ), tr(Ωl ) are invariant ˆ 1, ∆ ˆ 2. under transformation (6.17), so are ∆ ˆ 1, ∆ ˆ (6.14) are inProof of Theorem 6.5 The theorem is proved if T (6.5) and ∆ ˜ l = X l B, l = 1, 2, · · · , k. variant to transform (6.18). When X l is replaced with X The estimator of the regression coefficient and the variance of the error term ˜ˆ −1 ˆ ˆ and σ ˆ˜ 2l = σ are β ˆl2 , where β ˆl2 are the corresponding estimal = B β l and σ l ˜ˆ ˆ˜ T , · · · , β ˆ˜ T ]T = (I k ⊗ tors based on the original data. It follows that β = [β k ˜ˆ ˆ Σ ˜TX ˜ k )−1 ] = (I k ⊗ B −1 )Σ ˆ β (I k ⊗ (B T )−1 ). ˜ )−1 , · · · , σ ˜TX ˆ˜ 2k (X ˆ˜ 21 (X B −1 )β, β = diag[σ k ˜ˆ ˜ˆ ˜ˆ T −1 So the transformed test statistic T˜ = (C β − c)T (C Σ β C ) (C β − c) = (C(I k ⊗ ˆ T (C(I k ⊗ B −1 )Σ ˆ where for regression ˆ β (I k ⊗ B −1 )T C T )−1 (C(I k ⊗ B −1 )β), B −1 )β) coefficient comparison, c is set equal to 0. Notice C(I k ⊗ B −1 ) = (Q ⊗ I p )(I k ⊗ B −1 ) = (I q ⊗ B −1 )(Q ⊗ I p ) = (I q ⊗ B −1 )C and I q ⊗ B −1 is a full rank matrix, 6.5 Technical Proofs it is easy to show T˜ = T . ˆ and ∆ ˆ are invariant to transform (6.18). Using (6.26) We now turn to show ∆ and the fact that C(I k ⊗ B −1 ) = (I q ⊗ B −1 )C, (I q ⊗)C l = C l B, one can show ˆ 1, ∆ ˆ 2. ˆ l ), tr(Ω ˆ ) are invariant under transformation (6.17), so are ∆ tr(Ω l 162 163 CHAPTER Summary and Discussion In this thesis, we have discussed the MB test for various heterogenous ANOVA and MANOVA models. In all these models, we found that the MB test works well compared with the existing approximate solutions for the associated BehrensFisher problem. This shows that the modified Bartlett correction is very powerful in solving the various BF problems. As mentioned in the previous chapters, the key idea of the MB test includes three main steps: (1) show that the asymptotic distribution of the test statistic is a χ2 -distribution with some known degrees of freedom; (2) find the expressions of the asymptotic mean and variance of the test statistic and show that they are in 164 the forms required by the modified Bartlett correction (Fujikoshi 2000); and (3) apply the modified Bartlett correction. In statistical literature, it is often the case that the asymptotic distribution of a test statistic is χ2 with some degrees of freedom. Examples include the empirical likelihood ratio tests and the usual likelihood ratio tests, among others. 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Technometics, 53, 426-439. Zhang, J. T. (2012a). Tests of linear hypotheses in the ANOVA under heteroscedasticity. Manuscript. Zhang, J. T. (2012b). An approximate degrees of freedom test for heterogenous two-way ANOVA. J. Statist. Plann. & Infer., 142, 336-346. Zhang, J. T. and Liu, X. (2012). A modified Bartlett test for linear hypotheses in heterogenous one-way ANOVA. Statist. & its Interface, 5, 253-162. Zhang, J.T. and Liu, X. (2013). A modified Bartlett test for heterogenous one-way MANOVA. Metrika, 76, 135-152. [...]... The Behrens- Fisher Problems 1.1.1 Heteroscedastic One-Way ANOVA For several decades, much attention has been paid to comparing k normal means under heteroscedasticity (Welch 1947, 1951; James 1951, 1954; Krutchkoff 1988; Wilcox 1988, 1989; Krishnamoorthy, Lu, and Mathew 2007 etc) When only two normal means are involved, this problem is referred to as the Behrens- Fisher (BF) problem (Behrens 1929, Fisher. .. The Behrens- Fisher Problems power, their PB test generally performs the best, followed by James’ (1954) second order test while the Welch and generalized F tests are sometimes very liberal when k is large Since the PB test is time-consuming and James’ (1954) second-order test has a very complicated form which prevents it from being widely used in real data analysis, it is still worthwhile to develop some. .. 1.1 The Behrens- Fisher Problems experimental sciences, e.g., biology, psychology, physics, among others; examples may be found in Johnson and Wichern (2002), Xu and Cui (2008), and Tsai and Chen (2009), among others As for one-way MANOVA, when the cell covariance matrices are known to be the same, this problem can be solved using the Wilks likelihood ratio, Lawley-Hotelling trace (LHT), Pillai -Bartlett. .. that the Wald-type test statistic follows an asymptotic χ2 -distribution with some known degrees of freedom but with a slow convergence rate To apply the modified Bartlett correction to the test statistic, we first find out the asymptotic expressions of the mean and variance of the test statistic We then apply the modified Bartlett correction of Fujikoshi (2000) to the test statistic Simulation studies are... (2005), the modified Bartlett correction of Fujikoshi (2000) is applied to improve the convergence rate of T , resulting in the so-called modified Bartlett (MB) test The MB test considered by Yanagihara and Yuan (2005) is for a multivariate two-sample BF problem Let hl = (CΣC T )−1/2 cl , l = 1, 2, · · · , k where c1 , · · · , ck are the k columns of C To apply the modified Bartlett correction in the current... works well as long as at least one of the sample sizes is large But when error variances 8 1.1 The Behrens- Fisher Problems between the two models differ and sample sizes are small, this procedure becomes inadequate So Toyoda (1974) modified the Chow’s test by approximating the distribution of Chow’s statistic using an F distribution Schmidt and Sickles (1977) calculated the exact distribution of the Chow... real data analysis Thus, some further study is worthwhile 1.2 Classifying the Approximate Solutions to the BF Problems In the previous section, we have reviewed various BF problems and their approximation solutions proposed in the literature In this section, we give a brief classification of these approximated solutions 10 1.2 Classifying the Approximate Solutions to the BF Problems 1.2.1 Approximate... and Yuan (2005) called the resulting test a modified Bartlett (MB) test The MB test of Yanagihara and Yuan (2005) has several merits It maintains the type-I error well and has good power It is simple in form and fast in computation Therefore, it is worthwhile to further investigate the MB test for other 14 1.3 Overview of the Thesis Behrens- Fisher problems mentioned in the previous section 1.3 Overview... include James (1954), Johansen (1980) and Gamage, Mathew, and Weerahandi (2004), among others Tang and Algina (1993) compared James’s first- and second-order tests, Johansen’s test, and 5 1.1 The Behrens- Fisher Problems Bartlett- Nanda-Pillai’s trace test and concluded that none of them is satisfactory for all sample sizes and parameter configurations Overall, they recommended James’ (1954) second-order test... GLHT 15 1.3 Overview of the Thesis problem, we again use the Wald-type test and show its asymptotical distribution is χ2 with some known degrees of freedom We then find the associated asymptotic mean and variance of the test statistic and apply the modified Bartlett correction Some simulation studies are conducted under heterogenous two-way ANOVA and a real data example illustrates the methodologies In . SOLVING SOME BEHRENS- FISHER PROBLEMS USING MODIFIED BARTLETT CORRECTION LIU XUEFENG NATIONAL UNIVERSITY OF SINGAPORE 2013 SOLVING SOME BEHRENS- FISHER PROBLEMS USING MODIFIED BARTLETT CORRECTION LIU. to the Behrens- Fisher problems in Section 1.2. An overview of the thesis is outlined in Section 1.3. 1.1 The Behrens- Fisher Problems In this section, we review various Behrens- Fisher problems. trace test may fail for these BF problems. In this thesis, we solve various BF problems by applying the modified Bartlett correction of Fujikoshi (2000). These BF problems include heterogenous one-way ANOVA,