Column 2016/08/12 Brunner-Munzel 150 KEY WORDS Student t Welch Brunner-Munzel ( ) 1)2) 3)4) 5) 2 6) Big data Deep learning 7) 1
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1 Vol.2 No.16
2 Column 2016/08/12 Brunner-Munzel 150 KEY WORDS Student t Welch Brunner-Munzel ( ) 1)2) 3)4) 5) 2 6) Big data Deep learning 7) 1
3 8) i.i.d. 9) Gosset 10) Student 11) t t (unpaired data) 12)13)14) [ ] m X 1,, X m n Y 1,, Y n 15) X, Y 16) X i N(µ 1, σ 2 1 ) i.i.d. (i = 1,, m) Y j N(µ 2, σ 2 2 ) i.i.d. ( j = 1,, n) 17)18)19)20) µ 1 µ 2 21)22) H 0 : µ 1 = µ 2 H 1 : µ 1 µ 2 2
4 23) X = 1 m X i N(µ 1, σ2 1 m m ) i=1 Y = 1 n Y j N(µ 2, σ2 2 n n ) j=1 i.e., X Y N(µ 1 µ 2, σ2 1 m + σ2 2 n ) 24) Z X N(µ 1, σ2 1 m ) 25) ae(x) = E(aX), av(x) = V(a 2 X)(a R) 26)27)28) 29)30) (stadardization) 31) Z : Z = (X Y) (µ 1 µ 2 ) σ 2 1 m + σ2 2 n N(0, 1) σ 2 1, σ2 2 Z 32) 33) (estimator) 34) [ ]. t 35)36) σ 2 1 = σ2 2 = σ2 X Y N(µ 1 µ 2, σ2 m + σ2 n ) σ 2 37)38) σˆ 2 1 = 1 m 1 σˆ 2 2 = 1 n 1 σ 2 ˆ σ 2 39) m (X i X) 2 i=1 n (Y j Y) 2 j=1 σˆ 2 1 = {(m 1) σˆ 2 m + n (n 1) σˆ 2 2 } 3
5 m + n 2 t T 40) T = (X Y) (µ 1 µ 2 ) σ ˆ 2 1 m + σ ˆ2 2 n. Welch 41) 42) T ϕ 43) t 44)45) ϕ 46)47) t m+n 2 T = (X Y) (µ 1 µ 2 ) t ϕ σ ˆ 2 1 m + σ ˆ2 2 n σ ( ˆ2 1 m + σ ˆ2 2 n )2 ϕ = ( σ ˆ2 1 m )2 ˆ σ 2 2 n )2 m 1 + ( n 1 t t Welch Welch 48)49) Welch t R Excel Welch Cochran-Cox 50) 51) 52) t 53) t t 4
6 [ ].. W δ(x) 54)55) δ(x) W H 1 δ(x) W c H 0 56)57) (accept) 58) H 1 H 1 H 0 H 0 [ (Type I error)] 59) H 0 60) H 1 H 0 [ (Type II error)] 61) H 1 62) H 0 α β α 63)64) 1 β 65)66) )68) 69) (accept) 70) α β α β δ(x) W H 0 H 1 H 0 α H 1 H 1 H 0 δ(x) W c H 0 H 0 H 1 β 71) H 0 H 0 72) 5
7 α β α + β = 1 α β H 0 H 1 β 73) α α n 1 α 74) n 1 (1 α) n 75) α = 0.05, n = (=14%) t β α 76) F α 1 = 0.2 F α 2 = 0.05 t 0.24(=24%) 77) 78) t 79) 80) [ 2010] 81) [ ] n α 1, α 2,, α n n 1 (1 α i ) i=1 6
8 82) 0.05(=5%) Fisher ) Fisher 5% 84) Fisher 20 1 (1 0.05) (1 0.05) 0.64(= 64%) } {{ } 20 85) 86) [ 1997] [ 1998] 51 Web [ 2004] H 0 : σ 2 1 = σ2 2 H 1 : σ 2 1 σ2 2 F σ 2 1 = σ2 2 t [Zimmerman2004] [Zar2010] Kolmogorov-Smirnov Kolmogorov-Smirnov Kolmogorov-Smirnov t Mann-Whitney U U 87) U t U t t Welch 7
9 [t (robustness) 88) ] 89) t 90) α (central limit theorem) 91) t 92) t 93) t Welch [ 1996] t Welch Welch [Welch ]. α.. t Welch Welch t t t Welch t t U t U 8
10 [ (parametric test)] 94)95) [ (non-parametric test)] 96) 97) 98)99)100) 101) (lognormal distribution) 102)103) Wilcoxon Wilcoxon W 104) x i (i = 1,, n) x 1,, x n x i (rank) R i n W= ε i R i i=1 1 (x i > 0) ε i = 0 (x i 0) W n 105) α 106) Cochran Q Mantel-Haenszel 107) 108) 9
11 U (median) 109) U U t 110)111) U t 112) QUESTION [ ] 113) QUESTION 114) 115) t U 116) 10
12 [Mann and Whitney 1947] U [Siegel and Castellan1988] [Sokal and Rohlf2012] 117) [Kasuya2001] Siegel-Tukey t F t Weich U Brunner-Munzel (Brunner-Munzel test) 118) [Brunner and Munzel2000] Brunner-Munzel Brunner-Munzel n 1, n 2 N = n 1 + n t t ϕ σ ( ˆ2 1 n 1 σ + ˆ2 2 n 2 ) 2 ϕ = ( σ ˆ2 1 n 1 ) 2 ˆ σ 2 2 n 2 ) 2 n ( n 2 1 Brunner-Munzel ˆ σ 2 1, ˆ σ 2 2 Welch t Welch-Satterthwaite (Welch-Satterthwaite equation) Brunner-Munzel (permuted Brunner-Munzel test) )120) Brunner-Munzel R Brunner-Munzel Brunner-Munzel t Welch U Brunner-Munzel Web [ 2007] [ 2015] [ 2015] Welch Brunner-Munzel Mann-Whitney U Brunner-Munzel 11
13 GONE WITH STATISTICS (DONE WRONG) 121) 122) 123) 124) 125) 126)127) α 128) 129)130) (experimental group) (control group) µ = 9, 800, 003 [ ] µ = 9, 800, 000 [ ] ) 3 132)133) 134)135) 136) 12
14 p 137) 138) 139) 140) 141) 142) p 143) 144) 145) 146) Notes 1) 2) ( ) 3) [ 2012] 4) p 5) MBA p 6) 7) [ 2015] 8) 9) stochastic.fractal@gmail.com 10) 1=2 11) Student Gosset, William Sealy Gosset K. Pearson Student 12) (paired data) 13) 1 [ 2015] 13
15 14) [ 2015] (ratio) (proportion) (rate) [ 2003] 15) m, n 16) 17) X B X B 18) N(µ, σ 2 ) µ σ 2 19) i.i.d.=independently identically distributed 20) (characterization) [ 2010] Helmert Cochran 21) null hypothesis, alternative hypothesis 22) [ 1994] 23) (quasi-population) 24) m n parameter N(µ, σ 2 ) µ σ 2 25) X i N(µ i, σ 2 i ) i X i N( i µ i, i σ 2 i ) 26) X E(X) V(X) 27) a A a A A B 28) R (real number) Z (Zahlen[ ]) N (natural number) Q (quotient) C (complex number) {0,1,2, } N {0} 29) E( n i=1 X i ) = E(X 1 ) + E(X n ), V( n i=1 X i ) = V(X 1 ) + + V(X n ) X 30) V( n i=1 X i ) = V(X 1 ) + + V(X n ) X i (i = 1,, n) 31) z ae(x) = E(aX), av(x) = V(a 2 X)(a R) 32) z 33) µ 1 = µ 2 Z µ 1 µ 2 = 0 34) θ {x 1, x 2, } ˆθ 35) Student t t 36) t 37) ˆθ E(ˆθ) = θ (unbiased estimator) X E(X) 38) (efficient estimator) Craḿer-Rao V( θ) I(θ) 1 θ I(θ) = E[{ log f (x) θ } 2 ] Fisher [ 1992] θ = Const.( ) V(θ) = 0 θ 39) m + n 2 = (m 1) + (n 1) ˆ σ 2 40) t n n t 41) Welch t Aspin-Welch Sattethwaite Welch 14
16 42) 2016 Behrens - Fisher s problem 43) 44) [Welch1938] 45) Welch t 46) ϕ t ϕ 47) ϕ Welch Welch t 48) Welch 49) t T Welch [Zar2010] 50) Cochran-Cox t 51) Welch Cochran-Cox Welch 52) [ 2010] 53) preliminary test 54) Z T T 55) (statistical inference and decision) δ (decision function) (observation space) (action space) [ 2003] 56) A (complement) A c 57) H 0 H 0 58) 59) error of the first kind 60) 61) error of the second kind 62) 63) level of significance 64) 1 α confidence coefficient confidence level 65) power stastical power power of test 66) 1 β H 0 67) 68) 69) (power analysis) [ 2009] 70) [ 2003] 71) 1 β 72) [ 2009] 73) β β (well-defined ) 74) α P(X) = α P(X c ) = 1 P(X) 75) 1 α n (1 α) n n n 1 (1 α) n 76) [Altman1990] [ 1990] [ 1992] 15
17 77) 1 (1 α 1 )(1 α 2 ) = = ) [ 2010] 79) 80) [ 1998] 81) [ 2010] 82) Σ Π n i=1 x i = x 1 x 2 x n 1 x n 83) 20 1 = ) 5% [Michael and Caroline 1982] 5% 85) p [Lang2004] 86) 87) Wilcoxon (Wilcoxon rank sum test) Wilcoxon (Wilcoxon signed-rank test) t Mann-Whitney U Wilcoxon Wilcoxon t 88) 89) [ 1984] 90) ) X i (i = 1,, n) Lindeberg 92) 93) [Glass and Hopkins1996] 94) (nominal scale) (ordinal scale) (interval scale) (ratio scale) [Stevens1946] 95) [ 2015] (Column) 96) f (x) = λe λx (λ > 0, x > 0) 97) f (x) = ce ax (a > 0, x > 0) 98) Boltzmann f (ε) = λe kt ϵ [ 2014] 99) [ 2014] 100) (Gutenberg Richter Law) [ 2004] [Buchanan2009] 101) 102) (log x µ)2 1 (Gibrat s distribution) f (x) = e 2σ 2 x logx 2πσ 2 103) x logx
18 ) [ 1991] 105) n W 106) Wilcoxon (Wilcoxon signed rank sum test) Wilcoxon (Wilcoxon rank sum test) 107) [ 2012] 108) U Brunner-Munzel 109) [Mann and Whitney 1947] 110) 111) [ 2015] GPA(Grade Point Average) (= ) 112) (trimmed mean) (Huber s estimator) 113) 114) 115) Rubin 116) t t U 0.95 [Mood1954] 117) [Murphy1976] [Kasuya2001] 118) [Brunner and Munzel2000] [Wilcox2012] 119) [Neubert and Brunner2007] 120) Brunner Munzel [Brunner and Munzel2000] 10 Brunner-Munzel Brunner-Munzel 121) 122) analysis of covariance ANCOVA 123) (analysis of variance ANOVA) 124) Twitter :10 125) [Lang2004] [Lang2004] 126) 127) 0 128) α β α + β = 1 129) [ 1994] 130) GIGO(garbage in garbage out) 131) ) 3 µ = 13[ ], µ = 10[ ] 17
19 133) 134) 135) ) p *(star) * (statistics) star-tistics 137) 1% 5% 138) [ 2015] ph 139) 140) [ 2009] [ 2015] 141) X x 142) Basic and Applied Social Psychology 143) p [Johnson1999] p [Kline2004] 144) 145) 146) [Welch1938] B.L. Welch(1938). The significance of the difference between two means when the population variances are unequal, Biometrika, 29: [Stevens1946] S.S. Stevens(1946). On the Theory of Scale of Measurement, Science, 103: [Mann and Whitney 1947] H.B. Mann and D.R. Whitney (1947). On a test of whether one of two random variables is stochastically larger than the other, Ann. Math. Statist., 18: [Mood1954] A.M. Mood(1954). On the asymptotic efficiency of certain nonparametric two-sample tests, Ann. Math. Statist., 25: [Murphy1976] Murphy BP(1976). Comparison of some two sample means tests by simulation, Commun Statist Simula Computa, 5: [Michael and Caroline 1982] Michael Cowles and Caroline Davis(1982). On the Origins of the.05 Level of Statistical Significance, AMERICAN PSYCHOLOGIST, 37: [Siegel and Castellan1988] Siegel S and Castellan NJ Jr(1988). Nonparametric Statistics for the Behavioral Sci- 18
20 ences(2nd ed.). McGraw-Hill, New York [Altman1990] D.G. Altman(1990). Practical statistics for medical research. Chapman and Hall/CRC [Glass and Hopkins1996] G.V. Glass and K.D. Hopkins(1996). Statistical methods in education and psychology(3rd ed.). Pearson [Johnson1999] Douglas H. Johnson(1999). The Insignificance of Statistical Significance Testing, The Journal of Wildlife Management, 63: [Brunner and Munzel2000] Edgar Brunner and Ullrich Munzel(2000). The nonparametric Behrens-Fisher problem: Asymptotic theory and a small-sample approximation, Biometrical Journal, 42: [Kasuya2001] Kasuya(2001). Mann-Whitney U test when variances are unequal. Animal Behaviour, 61: [Salsburg2002] David Salsburg(2002). The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century. Henry Holt & Co [Lang2004] Tom Lang(2004). Twenty Statistical Errors Even YOU Can Find in Biomedical Research Articles, CROATIAN MEDICAL JOURNAL, 45: [Kline2004] Rex B. Kline(2004). Beyond Significance Testing: Reforming Data Analysis Methods in Behavioral Research. Amer Psychological Assn [Zimmerman2004] D.W.Zimmerman(2004). Inflation of Type I Error Rates by Unequal Variances Associated with Parametric, Nonparametric, and Rank-Transformation Tests, Psicologica, 25: [Ruxton2006] Graeme D. Ruxton(2006). The unequal variance t-test is an underused alternative to Student s t-test and the Mann-Whitney U test, Behavioral Ecology, 17: [Neubert and Brunner2007] Karin Neubert and Edgar Brunner(2007). A studentized permutation test for the nonparametric Behrens-Fisher problem, Computational Statistics and Data Analysis, 51: [Zar2010] Jerrold H. Zar(2010). Biostatistical Analysis(5th ed.). Pearson [Rasch, Kubinger, Moder2011] Rasch D, Kubinger KD, Moder K(2011). The two-sample t test: Pre-testing its assumptions does not pay off, Stat Papers, 52: [Sokal and Rohlf2012] Sokal RR, Rohlf FJ(2012). Biometry(4th ed.). W. H. Freeman and Company, New York [Wilcox2012] Wilcox R(2012). Modern Statistics for the Social and Behavioral Sciences. A Practical Introduction.. CRC Press, Boca Raton [Reinhart2015] Alex Reinhart(2015). Statistics Done Wrong. No Starch Pr [ 1973] (1973)., [ 1976] (1976)., [ 1984] (1984)., [ 1990] (1990)., [ 1991] (1991)., [ 1991] ( ) (1991)., [ 1992] ( ) (1992)., [ 1992] (1992)., [ 1992] (1992)., [ 1994] ( ) (1992)., [ 1994] (1994). (1) 30 19
21 ,, 21: [ 1995] (1995)., [ 1996] (1996)., [ 1996] (1996)., [ 1997] (1997)., [ 1998] (1998)., [ 1998] (1998)., [ 2003] (2003)., [ 2003] (2003)., [ 2003] (2003)., [ 2004] (2004)., [ 2004] (2004)., [ 2005] (2005). Mann-Whitney U,, 61: 1-6 [Salsburg2006] ( )David S. Salsburg, ( ), (2006)., [ 2006] (2006)., [ 2008] (2008). Q&A, [ 2007] (2007)., [ 2007] ( ) (2007)., [Buchanan2009] ( )Mark Buchanan, ( ) (2009)., [ 2009] (2009)., [ 2009] (2009)., [ 2009] (2009)., [ 2010] (2010). 2010, [ 2010] (2010). [ 2010] (2010).,, 6: [ 2012] (2012)., [ 2012] (2012). MCMC [ 2012] (2012)., [ 2014] (2014)., [ 2014] (2014). U,, 30: [ 2015] (2015)., [ 2015] (2015)., 20
22 [ 2015] (2015)., [ 2015] (2015)., [ 2016] (2016)., [ 2004] 51 (2004)., kubo/ce/ 2004/ [ 2007]., [ 2010]., chino/welcome_news/contents.html# test-for-means [ 2015]. Brunner-Munzel, p1 [ 2016]., ac.jp/ okumura/stat/ 21
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email: takahash@sci.u-hyogo.ac.jp May 14, 2009 Outline 1. 2. 3. 4. 5. 6. 2 / 262 Today s Lecture: Mode-mode Coupling Theory 100 / 262 Part I Effects of Non-linear Mode-Mode Coupling Effects of Non-linear
N cos s s cos ψ e e e e 3 3 e e 3 e 3 e
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