nonpara1.PDF

Chapter 1 Introduction 1 Introduction (parameter) (assumption) (rank), (median) p-value distribution free, assumption free, statistical inference based on ranks 11 Nonparametric? John Arbuthnot (1710) 194Wolfowitz,, 111 advantage ( ), (rank) 11 disadvantage ( ) 1 statistical terms 11 descriptive and inferential ( ), (descriptive) (statistic) (parameter) (inferential) 1 population and sample ( ) (population) (sample), 000 1 [001 1 ] http://wolfpackhannamackr Nonparametric 1

Chapter 1 Introduction, (sample) (, n 0) 13 parameter and statistic ( ) (parameter) µ, σ, ρ (unknown) (statistic) (estimation) x, s (median) rank 14 random variable and probability density function ( ) (simple random sample),, (random sample) ( ) 70 10 60 70 70 ( ) 15 measurement and categorical ( ) (measurable numerical) (categorical) [001 1 ] http://wolfpackhannamackr Nonparametric

Chapter 1 Introduction,,,,,, 16 statistical hypothesis ( ) : (hypothesis),, =100, : ( ) ( ) 1 (type I error) 1 (significant level) α (1-α) (confidence level) ( ) ( ) (β) 1 (α) [001 1 ] http://wolfpackhannamackr Nonparametric 3

Chapter 1 Introduction p-value: (observed significant level) p- p-value 170 x : (1-β) (power), x u 10 u {x >10} Pr( < ) s / n s / n (µ) : 95% 5%, 95% (100, 110) H : µ 105 0 = 5% 95% 95% 95% [001 1 ] http://wolfpackhannamackr Nonparametric 4

Chapter 1 Introduction 13 order statistic [001 1 ] http://wolfpackhannamackr Nonparametric 5

Chapter 1 Introduction 13 order statistic ( ) 131 review : (random variable) X ( c) = x : c=, x= (real number) ( ) : X ( c) = i wherei = 1,, 3, 4, 5, 6 : (probability density function) ( 1) : f ( x) = 1/ 6 where x = 1,, 3, 4, 5, 6 ( ) f ( x ) = Γ( r 1 / ) r / x ( r / 1) x / ) e x : (cumulative distribution function) F ( x) = Pr( X x) = f ( x) dx : X 1, X,, X n (random sample) 13 definition n X 1, X,, X n X (1), X () X ( 1) < X () < < X ( n) X ( 1) X (),, (order statistic) 133 (minimum) X (1), (maximum) X (n) (range) X (n) - X (1), midrange: X + X ]/ [ ( 1) ( n) (median) m= [ X X ]/ (n ), m= X n + n ( ) ( + 1) n+ 1 ( ), X ( n) (n ) [001 1 ] http://wolfpackhannamackr Nonparametric 5

Chapter 1 Introduction 134 1) X ( 1), X (),, X ( n) (joint distribution function) ) X (1) marginal distribution function 3) X (n) marginal distribution function f ( x1, x,, xn ) = n! f ( x1 ) f ( x) L f ( xn ) n 1 f ( x1) = n[1 F( x1)] f ( x1) f ( x n ) = n[ F( x n )] n 1 f ( x 4) n (median) m marginal distribution function (k)! k k f m F m v 1 m F t 1 ( ) = [ ( )] [1 ( )] f (m t) f ( t) dt, n=k [( k 1)!] f ( x) = x, 0 < x < 1 x, x, x x x, x, x x ( 1) () (3), (4) n ) 1 3, 4 x x (1) (n) 14? ( ) one population + location parameter 60 f (x) [001 1 ] http://wolfpackhannamackr A 0 60 ) f (x 0 10, 9,, 18, 3, 3, 5, 14, 10, 9 6, 8, 1, 15, 1, 9, 4, 7, 8, 3 (elementary statistic) ( ) Location parameter: mean, median x, K x 1 x,, f (x), 0 parameter histogram, stem-leaf, box-plotf (x) Nonparametric 6

Chapter 1 Introduction f (x) skewed, outlier stem-leaf 60? population mean( µ ) median( M ) sample mean: (statistic) x = 0 xi / i= 1 sample median: x + ] / n [ ( 10) x(11) µˆ = x, Mˆ = x + ] / [ ( 10) x(11) t f ( x) dx = 1/ µ, M t 60 9? (null hypothesis): H : µ 9 H : M 9 0 = 0 = (alternative hypothesis): H : µ > 9 : M > 9 a H a f(x) µ 9 ( ) [001 1 ] http://wolfpackhannamackr Nonparametric 7

Chapter 1 Introduction ( ) f ( x; u, ο) σ / x u, n ( ) [Central Limit Theorem] n ( ) normal distribution f x ~ Normal( u, σ / n) µ x u : t = ~ normal(0,1) σ / n (σ) (s) Normal (0,1) ( ) 0 t *) [Student t-distribution] ( ) x u t = ~ t(n -1) s / n [001 1 ] http://wolfpackhannamackr Nonparametric 8

Chapter 1 Introduction 1) χ - ) 0 3) xc ~ χ (λ=), xc6 ~ χ (λ=10) 4) (xc, xc6) 5) ( ) )-4) 1000 6) [001 1 ] http://wolfpackhannamackr Nonparametric 9

Chapter One sample (statistic) (parameter) (estimate) (statistical hypothesis) (statistical inference) (location parameter) (mean) (median) Population f ( x : µ, σ ) (random sample): ( x, 1, x, K xn ) xi statistic: x n x ( n/ ) + x ( n/ + 1) =, m = or = x(( n+ 1) / ) = ( x x) /( n 1) s i 1 parameteric procedure ( ) (sample mean) E ( x) = µ, V ( x) = σ / n f ( x; µ, σ ) f ( x; µ, σ / n) 11 (large sample) 1) (central limit theorem) n x ) f (x ) (hypothesis testing) (null hypothesis): 0 (alternative hypothesis): 0 µ = µ ( ) µ µ (two-sided) µ > µ 0 (one-sided) (test statistic): [001 1 ] http://wolfpackhannamackr Nonparametric 10

Chapter (, p-value) T x µ 0 = s / n ~ Normal(0,1) µ 0 x (conclusion): T (critical region) (p- ) (1, α) ( ) two-sided ( ) (α) one-sided ( ) 3) (confidence interval) 100(1-α)% (lower limit): µ x z l α / x ± zα / s =, (upper limit): n s n µ = x + z u α / ( ) 95% ( ) 95%, 100 95 4) 100(1-α)% (µ) µ = x + z u α / µ = x z l α / s s n n s n x ( ) [001 1 ] http://wolfpackhannamackr Nonparametric 11

Chapter [001 1 ] http://wolfpackhannamackr Nonparametric 1

Chapter 1 (small sample) 1) : n ) f (x ) t- : ) f (x W S Gosset Student x µ ~ t( n 1) s / n ; =0, =n/(n-) f ( x) ~ Normal(0,1) f ( x) ~ t( n) [ t- ] 3) x µ T = ~ t( n 1) s / n 100(1-α)% : x ± t( n 1; α / ) s n Nonparametric procedure I: Sign Test n (sign) 1 (hypothesis testing) x x,, x, 1) (assumption): 1 K n f ( x : M ) (random sample) ( ) [001 1 ] http://wolfpackhannamackr Nonparametric 13

Chapter ) (statistical hypothesis): H : M = M 0 H a : M M ( ) H a : M > M 0 H a : M < M 0 ( ) : 0 : 0 3) (test statistic): x i = M 0 x i M ( 0 ) M 0 +, - + ( ) 4) (decision rule): ( x i M 0 ) +, - ( ) p ( ; + ) (Bernoulli) + ( - ) K ~ B( n,05) ( K ) ( n, p) (Binomial Dist; B ( n, p) + k p- i) k 0 5n : p-value= Pr( K k n,05) ii) k 0 5n : p-value= Pr( K k n,05) ) + k sign test p- p-value= Pr( K k n,05) p- Sign test (population ratio) : p ) ( 0 p0 H = np,( n 1) p ) 9 min( 0 0 < α [001 1 ] http://wolfpackhannamackr Nonparametric 14

Chapter 18 33 565 5 5 35 75 35 31 7 30 : : M 3 5 H, : M 3 5 0 = H a : ( x i M 0) =1 k 1 p- = Pr( K 1 n = 10, p = 05) = 0 0108 - =9, + =1, 0 p<005 ( α / ) ( ) 35 k + 35 35? ( : M < 3 5 H a ) p- 005 Large-sample approximation: 1 p-value (normal approximation to the binomial) ( K + 05) 05n z = ~ Normal(0,1) 05 n p-, z=-1 Pr( K 1 n = 10, p = 05) 00136 n=0 n z- SAS : UNIVARIATE procedure (confidence intervals) ( : point estimate) ( ) ( : interval estimate) Sign Test 100(1-α)% Pr( K K * α ) M l = X ( k * +1) lower limit= M u * + ( k 1 ) upper limit [001 1 ] http://wolfpackhannamackr Nonparametric 15

Chapter 100(1-α)% α 174 41 401 371 811 83 007 307 Pr( K 3 n = 16, p = 05) = 0 0105 Pr( K 4 n = 16, p = 05) = 0 0383 α X = 99, X 4 01 ( 5) 1 (1) = Large-sample approximation : 1 (normal approximation to the binomial) * * ( K + 1) ( n / ) + zα / ( K + 1) (16 / ) + 196 16 / 4 4 n / 4 95% ( X = 9, X 8 11 ( 4) 1 (13) = ) 95% α=00105*=001 95% 979% [001 1 ] http://wolfpackhannamackr Nonparametric 16

Chapter [001 1 ] http://wolfpackhannamackr Nonparametric 17

Chapter 3 Nonparametric procedure II: Wilcoxon Signed-ranks test sign test 31 (hypothesis testing) x x,, x, x i M ( 0 ) 1) (assumption): 1 K n f ( x : M ) (random sample) ( ) sign test ) (statistical hypothesis): H : M = M 0 ( ), H 0 : M M 0 H : M M 0 0 ( ) H a : M M ( ), H a : M > M 0 H a : M < M 0 ( ) : 0 : 0 3) (test statistic): D = x M ) D = 0 i i ( i 0 i i M 0 D D D 3 Di 1,, 3 Di (1=+3)/3=, 3 D i T, + T T = [ n( n + 1) / ] T + T T T T+ H a : M M = min( T, T ) 0 + + i T H a : M > M 0 T = T H a : M < M 0 T = T + 4) (decision rule): Wilcoxon n (critical region) α + Sign test (population ratio) : p ) ( 0 p0 H = [001 1 ] http://wolfpackhannamackr Nonparametric 18

Chapter np,( n 1) p ) 9 min( 0 0 < α H : M 107 : M 107 0 = + H a T = 64 5 T = 40 5 T = min( 645,405) = 40 5 T 1 T = 40 5 α H : M 107 : M < 107 T 5 T 0 = T = 405 H a Large-sample approximation: 0 Wilcoxon T n( n + 1) / 4 T* = ~ Normal(0,1) n( n + 1)(n + 1) / 4 * T T, T T + z- [001 1 ] http://wolfpackhannamackr Nonparametric 19

Chapter sampling distribution of T + : n=4 T + Wilcoxon + T + + T + + T + + T + 1 3 0 1 3 4 1, 1,3 1,4 4 3 4 5,3,4 3,4 1,,3 Wilcoxon (n=4) 5 6 7 6 1,,4 1,3,4,3,4 1,,3,4 7 8 9 10 3 (confidence intervals) Wilcoxon xi + x j 1) uij =, 1 i j n ) uij? C 10 =55 10 + 3) Wilcoxon n P T K(=T+1) uij (lower limit) K dl (upper limit) xi + x j 1) uij =, 1 i j n ) uij u 31 45 ij( 8) = 3) n=10p=0044t=8 ( u 7 75, u 35 05 951% ij( 9) = ij( 47) = ) Large-sample approximation : 0 Wilcoxon n( n + 1) z 4 K α / n( n + 1)(n + 1) 4 [001 1 ] http://wolfpackhannamackr Nonparametric 0

Chapter [001 1 ] http://wolfpackhannamackr Nonparametric 1

Chapter 4 One-sample runs test for randomness (random sample) randomness Run 40 Example 1) p control chart: control limit randomness pattern control ) 41 0) : randomness runs( )run MFMFMFMFMF runs 10 pattern MMMMMFFFFF runs 5 1) n, n 1, n n=n 1 +n ) (null hypothesis): randomness (alternative hypothesis): randomness 3) : runs (r) 4) (decision rule): r (n 1, n ) (critical values of r in the runs test) * ) ( ) [001 1 ] http://wolfpackhannamackr Nonparametric

Chapter n 17 n 13 n = 30 1 = = { r 10} { r } r = 8 Large-sample approximation: n 1, n 0 r {[( n1n) /( n1 + n )] + 1} T* = ~ Normal(0,1) n n (n n n n ) ( n 1 1 1 + n 1 1 ) ( n + n 1) [001 1 ] http://wolfpackhannamackr Nonparametric 3

Chapter [001 1 ] http://wolfpackhannamackr Nonparametric 4

Chapter 5 Cox-Stuart Test for Trend trend Sign test Cox, D R and A Stuart, Some Quick Tests for trend in Location and dispersion, Biometrika, 4 (1955), 80-95 50 Example ( ) 51 1) ) (null hypothesis): trend (alternative hypothesis)1: upward trend (alternative hypothesis): downward trend (alternative hypothesis)3: trend 3) : ( x i, x c + i ) ( xc+ i xi ) (+, -) trend n c = n / n c = ( n +1) / 4) (decision rule): 0 + sign test [001 1 ] http://wolfpackhannamackr Nonparametric 5

Chapter : trend : trend ( ) : (07, 7), (3,13), ( n ' = 40 6 (sign test ) p-value P ( K 6 n = 0, p = 05) ) + =6 =14, =00577 005 trend Large-sample approximation: sign test [001 1 ] http://wolfpackhannamackr Nonparametric 6

Chapter (SAS v8 ) 5 SAS 80 SAS (Excel, ASC format, DB) SAS data DATA step ( 1) DATA one; input var1 var; cards; ---- run; ( ) DATA one; infile Text file ; input var1 var; run; SAS version 8 spreadsheet 51 (spreadsheet ) 1) SAS Solution SAS SAS data Work Library ) (Analyst) (project) (1) spreadsheet () ( #3, 7 ) Explorer (3) SAS data Work Library [001 1 ] http://wolfpackhannamackr Nonparametric 7

Chapter (SAS v8 ) 5 1) Report [001 1 ] http://wolfpackhannamackr Nonparametric 8

Chapter (SAS v8 ) ) ( ) 3), [001 1 ] http://wolfpackhannamackr Nonparametric 9

Chapter (SAS v8 ) 4) 5) ( SK) (Analyst) Default Report Temporary SAS data 6)? SAS? Tabulate procedure? [001 1 ] http://wolfpackhannamackr Nonparametric 30

Chapter (SAS v8 ) 53? 1)? [001 1 ] http://wolfpackhannamackr Nonparametric 31

Chapter (SAS v8 ) ) pop-up 3) 4) 5) SAS data [001 1 ] http://wolfpackhannamackr Nonparametric 3

Chapter (SAS v8 ) 6)? 7) box plot ( )? 8)?? [001 1 ] http://wolfpackhannamackr Nonparametric 33

Chapter (SAS v8 ) 9)? 10), SAS?? (program editor) 11) ( ) F8 1) OUTPUT( ) [001 1 ] http://wolfpackhannamackr Nonparametric 34

Chapter (SAS v8 ) 54 ( ) 10? 1)??, t ) (variable) 10 3) (interval ) (significant level) [001 1 ] http://wolfpackhannamackr Nonparametric 35

Chapter (SAS v8 ) 4) p-value=03796 005 (5%) 10 55?? (54 ) 1) 10 10 data step univariate procedure [001 1 ] http://wolfpackhannamackr Nonparametric 36

Chapter (SAS v8 ) ) Sign Test( ) Wilcoxon Ranks-sum Test ( ) p-value M K n / =, S = T + n( n + 1) / 4 3) Wilcoxon ranks-sum pairwise( ) n + C pairwise n? Point Estimator ( ): sign test Wilcoxon ranks-sum test [001 1 ] http://wolfpackhannamackr Nonparametric 37

Chapter (SAS v8 ) 1)? [ ] ),,? [ ] [001 1 ] http://wolfpackhannamackr Nonparametric 38

Chapter 3 3 (independent samples) (paired samples) 4 30 (location parameter) ( ) ( ) t- 301 µ x µ y 1) : σ = x σ y ) : σ σ F n 1; n ) x y ( max( sx, s y ) 3) : T = ~ F ( n1 1; n 1) min( s, s ) x y (α/) n 1 =, n =, 4) : T (critical region) (cf) Hartley Test ( 3 ) [001 1 ] http://wolfpackhannamackr Nonparametric 39

Chapter 3 30 1) : µ = x µ y ) : µ ( ) 3) : x µ y ( x y) ( µ x µ T = s 1/ n + 1/ n p x y y ) ~ N (0,1) where s p = ( n x 1) s ( n x x + ( n + n y y ) 1) s y Why? 4) : T (critical region) 100(1-α)% α/ α/ Normal (0,1) z α / z α / 5) 100(1-α)% x y) ± z s 1/ n 1/ n : ( α / p x + y 30 1) : ) : µ = x µ y 3) : µ ( ) 4) : x µ y ( x y) ( µ x µ y ) T = ~ t( nx + ny s 1/ n + 1/ n p x y 5) : T (critical region) ) t( α / ; n x n ) + y 6) 100(1-α)% ( x y) ± t( α / ; n + n ) s 1/ n + 1/ n : x y p x y [001 1 ] http://wolfpackhannamackr Nonparametric 40

Chapter 3 303 SAS 19 (pound) (SAS JMP class ) *) t- t- (Satterthwaite ) HOMEWORK #6 [due 4 10 ] 31 homework #6(3) ( =005) [001 1 ] http://wolfpackhannamackr Nonparametric 41

Chapter 3 31 31 Median Test ( ) 1) (assumption) ( x1, x, K, xn 1 ) M x (random sample) y, y, K, y ) M y (random sample) ( 1 n (grand median) ) (hypothesis) M = M x = M : y M M (median ) : x y 3) (test statistic) (M) (M) ( n 1 + n ) 1 (X) (Y) total A B A+B C D C+D Total A+C=n 1 A+C=n N=n 1 +n A = C = /, B = D = / n 1 n n 1 + n ( ) Hyper-geometric distribution N M (N-M) K (without replacement: ) x x M N M ( )( ) x K x H ( x N, M, K ) =, x = 0,1,, K, K, N ( ) K KM KM ( N M )( N K ) E ( x) =, V ( x) = ( ) N N N ( N 1) [001 1 ] http://wolfpackhannamackr Nonparametric 4

Chapter 3 Median Test : N, n 1, n, Ax, (A+B) K n1 n ( )( ) A B H ( A N, M, K ) =, A = 0,1,, K, ( A + B) N ( ) A + B Binomial approximation to Hyper-geometric distribution (N ) min( np, npq) >5 H ( x N, M, K ) ~ Binomial( x n = K, p = M / N) ~ Nomal( np, npq) ( A / n1) ( B / n ) T = ~ Normal(0,1) : pˆ(1 pˆ)(1/ n + 1/ n ) 1 EXAMPLE (X, Y) ( =005) X 5 5 17 6 18 30 4 1 13 30 0 3 6 1 0 37 9 17 37 0 11 3 16 31 46 0 5 17 36 54 8 6 Y 31 1 38 19 38 41 68 8 43 4 30 0 9 13 3 30 1) : M = M x y ) : M M ( ) 3) : x y X Y total 1 1 4 0 4 4 Total 3 16 48 (1 / 3) (1 /16) T = = 45 p ˆ = (1 + 1) / 48 = 0 5 05(1 05)(1/ 3 + 1/16) 4) : ( < 196) X Y X (why?) [001 1 ] http://wolfpackhannamackr Nonparametric 43

Chapter 3 : n = n 1 + n ) Median test (homogeneity) χ - ( ( ) How? HOMEWORK #6 [due 4 10 ] The quality control manager with a drug manufacturer wishes to know whether two methods of producing a particular tablets result in a difference between the median thick-nesses A random sample of tablets is drawn from batches produced by the two methods The following table shows the results, which have been coded for computational convenience Do these data provide sufficient evidence to indicate that the two population medians are different? Let α = 005 Method Thickness A 51 4 45 48 5 44 58 41 5 44 45 5 61 60 41 B 40 47 36 39 37 46 43 55 53 56 [001 1 ] http://wolfpackhannamackr Nonparametric 44

Chapter 3 3 Mann-Whitney Test 1) (assumption) ( x1, x, K, xn1) M x ( y1, y, K, yn ) M y ) (hypothesis) : M = M x = M y : M x M y ( ) M x < M y M x > M y ( ) 3) (test statistic) x, x, K, x ) y, y, K, y ) (rank) ( 1 n1 ( 1 n X Y x, x, K, x ) y, y, K, y ) ( 1 n1 n ( 1) 1 n 1 + T = S, S= 1 ( 1 n 4) T C (Mann-Whitney : w α / ) wα / w 1 α / w 1 α / = n1n wα / M x < M y T w α M > M T w x y 1 α Large sample Approximation: n1 n 0 C z = 1 T n n 1 1 n n ( n + n / + 1) /1 [001 1 ] http://wolfpackhannamackr Nonparametric 45

Chapter 3 EXAMPLE (X, Y) ( =005) X 119 117 95 94 87 8 77 74 74 71 69 68 63 5 4 41 Y 66 58 54 51 5 43 39 33 4 17 1) : M = x M y ) : M M ( ) 3) : x y T = 96 5 17(17 + 1) / = 1435 4) : C n 1 = 17, n = 10 w α / = w0 05 =46 & w1 α / = w0975 = 17*10 46 =14 X (Why?) p-value?: Yes, but approximation n 1 = 17, n = 10 T =143 5 ( 17)(10) 6 = 144 ( 17)(10) 35 = 135 0005 > p > 0 001 SAS [001 1 ] http://wolfpackhannamackr Nonparametric 46

Chapter 3 [001 1 ] http://wolfpackhannamackr Nonparametric 47

Chapter 3 HOMEWORK #6 [due 4 17 ] [001 1 ] http://wolfpackhannamackr Nonparametric 48

Chapter 3 33 ( M = M ) x y α 100(1-α )% Median Test Median Approximation T = ( A/ n ) ( B / n 1 pˆ(1 pˆ)(1/ n 1 ) + 1/ n ) ~ Normal(0,1) M M ) 100(1-α )% ( x y A / n ) ( B / n ) ± zα pˆ(1 pˆ)(1/ n + 1/ ), ( 1 / 1 n A + B pˆ = n + n 1 EXAMPLE Median Test (X, Y) 95% (43 page) ( 1/ 3) (1/16) ± 196 (4/ 48)(1 4 / 48)(1/ 3 + 1/16) Mann-Whitney Test M-W 1) (X, Y) x y ) ( i i n 1,n n 1n ) ( = 3) Mann-Whitney ) ( 4) (100-α)% w α / w w α / (lower bound) α / ), (upper bound) [001 1 ] http://wolfpackhannamackr Nonparametric 49

Chapter 3 EXAMPLE 95% X 77 8 85 86 86 86 89 91 9 93 100 Y 65 65 73 75 77 78 83 85 90 97 1) (X, Y) x y ) ( i i n 1n ) ( = 3) Mann-Whitney ( 11, 10 ) w 7 α / ) 4) 95% (lower bound) 7 1 (upper bound) 7 17 [001 1 ] http://wolfpackhannamackr Nonparametric 50

Chapter 3 34 341 (dispersion) : F- 1) : σ = ) : max( s x, s y ) 3) : F = ~ F( n1, n ) min( s, s ) x y x σ y x σ y σ ( ) 4) : HOMEWORK #7 [due 5 8 ] ( : $) ( =005) n 50 / x = 50 / s = 3 1 n 30 / x = 10 / s = 87 x = x y = x 34 : Ansari Bradley 1) : σ = ) : x σ y x σ y σ ( ) 3) : X Ansari Bradley ( n 1 =X, n =Y) EXAMPLE (X, Y) ( =005) X 384 6 119 Y 397 5 7 336 3 1) : ) : ( ) 3) : 119 3 5 6 7 336 384 397 (group) X X Y Y X Y Y X Y 1 3 4 5 4 3 1 T=1++5+=10 ( n 1 =4, n =5) 005 005 0975 T T=16(00159) 8(09603) 8 16 [001 1 ] http://wolfpackhannamackr Nonparametric 51

Chapter 3 4) : 10 ( 16 T T 8 ) 5% Large sample Approximation: Ansari-Bradley n + n 0 1 T T * * HOMEWORK #7 [ ] T [ n1( n1 + n + ) / 4] = ~ N(0,1) if n 1 + n n n ( n + n + )( n + n ) /[48( n + n 1)] 1 1 1 1 T [ n1( n1 + n + 1) / 4( n1 + n )] = ~ N (0,1) if n 1 + n n n ( n + n + 1)(3+ ( n + n ) ) / 48 ( n + n ) 1 1 1 Dopamine ( =01) 433 347 38 607 478 48 37 434 45 336 SAS 1 [001 1 ] http://wolfpackhannamackr Nonparametric 5

Chapter 4 4 (paired) before and after, pre and post (treatment effect) (paired) 3 (0 ) u x = u y??? x y X Y = 0??? di = xi yi x, y ), ( x, y ), K, ( x n, y ) ( 1 1 n 40 (paired t-test) d = x y ) 1) (hypothesis) i ( i i : µ = 0 (, ) d : µ 0 ( ) µ > 0 µ < 0 ( ) ) (test statistic) d d = x y ) i d ( i i d i ( d ) D T = ~ Normal (0,1) ( ) ~ t ( n 1) ( : s n d / ) 3) (decision rule): t- p-value d HOMEWORK #8 [due 5 10 ] 10 ( =005) SAS 1 3 4 5 6 7 8 9 10 50 5 30 50 60 80 45 30 65 70 53 7 38 55 61 85 45 31 7 18 [001 1 ] http://wolfpackhannamackr Nonparametric 53

Chapter 4 41 Sign Test + ( ) 1) (assumption) ( x 1, y 1), ( x, y ), K, ( x n, y n ) di = yi xi d i ) (hypothesis) : M = 0 ( 0 ) d : M 0 ( ) M > 0 M < 0 ( ) 3) (test statistic) d i i i d = y x d = 0 K d i d + p-value : Pr( K k n,05), Pr( K k n,05) 4) p- α EXAMPLE 10 ( =005) W Daniel Applied Nonparametric Statistics 1 3 4 5 6 7 8 9 10 463 46 46 456 450 46 418 415 409 40 53 494 461 535 476 454 448 408 470 437 - - + - - - - + - - 1) : M = 0 ) : M < 0 ( ) 3) : K = 4) : p- = Pr( K n = 10, p = 05) = 0 0547 d d [001 1 ] http://wolfpackhannamackr Nonparametric 54

Chapter 4 HOMEWORK #8 [due 5 10 ] Solve the following problems [001 1 ] http://wolfpackhannamackr Nonparametric 55

Chapter 4 4 Wilcoxon Matched pairs Singed Ranks Test Sign test ( ) WSR WRS 1) (assumption) ( x 1, y 1), ( x, y ), K, ( x n, y n ) di = yi xi i d ) (hypothesis) : M = 0 ( 0 ) d : M 0 ( ) M > 0 M < 0 ( ) 3) (test statistic) d i i i d = y x d = 0 i d d d = y x d i i i i = yi xi + ( T + ) ( T ) T = min( T, T ) M d > 0 T = T M d < 0 T = T + 4) + Wilcoxon (WRS ) n (critical region) α α/ EXAMPLE 9 ( =005) W Daniel Applied Nonparametric Statistics 1 3 4 5 6 7 8 9 33 17 30 5 36 5 31 0 18 1 17 13 33 0 19 13 9 1) : M = 0 d [001 1 ] http://wolfpackhannamackr Nonparametric 56

Chapter 4 ) : M < 0 ( ) 3) : d 4) : n=8 T = 0 + WRS p- 00039 ( ) 005 HOMEWORK #9 [due 5 15 ] [001 1 ] http://wolfpackhannamackr Nonparametric 57

Chapter 4 43 d = x y ) i ( i i Sign test, WRS 3 HOMEWORK #9 [due 5 15 ] 11 95% Sign WRS 44 (McNemar )? A (, (panel) ),, ( ) McNemar 441 McNemar After Yes No Total Before Yes A B A+B No C D C+D Total A+C B+D N [001 1 ] http://wolfpackhannamackr Nonparametric 58

Chapter 4 N= A= YES D= NO 44 McNemar B=YES, NO D=No, YES McNemar (Yes, No) Bennett & Underwood 3 1) (hypothesis) p = ( yes yes ) : 1 p : p1 p ( ) p 1 > p p 1 < p ( ) ) (test statistic) A + B : pˆ 1 =, N ( B C) / N = 0 A + C pˆ = : pˆ N p 1 ˆ B C = N McNemar (B+C) 10 z = B C B + C ~ Normal(0,1) EXAMPLE 85 ( =005) Yes No Total Yes 7 37 41 No 6 15 44 Total 33 5 85 1) (hypothesis) : p 1 = p ( ) p < ( ) : 1 p 37 6 ) (test statistic): z = = 1 385 p-value= pr ( z 138) =0084 37 + 6 [001 1 ] http://wolfpackhannamackr 59 Nonparametric

Chapter 4 43 SAS (McNemar ) order=data nopercent norow McNemar B C SAS z = χ = B + C ) Chi-square?? ( B C B + C HOMEWORK #9 [due 5 15 ] McNemar [001 1 ] http://wolfpackhannamackr 60 Nonparametric

Chapter 5, 5, (associate), (independence) ( ) (likeness) (homogeneity) ( ) H 0 : p 1 = p z- t- χ - (Chi-Square) (cross-tabulation) (contingency table) 51 Chi-Square ( = µ, = σ ) X Z ( =0, =1) X µ Z = σ ~ Normal(0,1) Z 1 χ X µ Z = ( ) ~ χ ( df = 1) σ k Chi-Square 1 ( k / ) 1 x / f ( x) = x e, 0 x < k / Γ( k / ) k Chi-Square k k χ ( df = 1) W 1, W, K, Wk k Chi-square Y = Wi ~ χ ( df = k) W ~ χ ( df = m1) V ~ χ ( df = m) W V F- H = / ~ F( m1, m) m1 m k = Chi-Square (exponential dist: β = ) [001 1 ] http://wolfpackhannamackr Nonparametric 61

Chapter 5, Chi-Square (df)= df=4 df=6 5 Contingency Table 1, ( ) 1 1 r, c 1 1 c total 1 O 11 O 1 O 1c n 1 O 1 O O c n : : : : : : r O r1 O r O rc n r Total n 1 n n c n Chi-Square??? ( ) Pr( AB ) = pr( A) Pr( B) n n i j Eij = ( )( ) n n n ( O T = ij E ) E ij ij??? [001 1 ] http://wolfpackhannamackr Nonparametric 6

Chapter 5, 53 Chi-Square (associate) A 1 30 (3 ) A BC Total 75 46 3 144 30 3 4 86 Total 105 78 47 30 n ( ) : : n n i j Pr( nij ) = Eij = ( )( ) n n n Pr( A ) = Pr( ) Pr(A ) n1 n 1 1 E 11 = ( )( ) n = 105 144 / 30 = 65 7 n n 3 E = 86 47 / 30 17 6 3 = [001 1 ] http://wolfpackhannamackr Nonparametric 63

Chapter 5, ) O E ) ( ( O T = ij E ) E ij ij ( ij ij ( r 1)( c 1) χ (75 657) (4 176) T = + + = 7 68 ~ χ 657 176 0 - (approximate) ( df = ( 1)(3 1) = ) χ - 599 (critical region) 5% 768 768} {> 599? SAS A, B A [001 1 ] http://wolfpackhannamackr Nonparametric 64

Chapter 5, SAS NOCOL: NOPERCENT: CHISQ: EXPECTED: Chi-Square = χ ij ij ij χ Likelihood Ratio = O ln( O / E ) ~ ( df = ( r 1)( c 1)) M-H Q MH = ( n 1) r ~ χ ( df = 1) r Person M-H (ordinal) Phi, Contingency, Cramer s coefficient [001 1 ] http://wolfpackhannamackr Nonparametric 65

Chapter 5, EXAMPLE 764 5% 1) : ( ) ) : 3) : E = 56 1640 / 764 153 68 11 = (186 15368) (7 19343) (59 41) 4) : T = + + K+ = 47 9, 15368 19343 41 8 χ - {> 15507} HOMEWORK #10 (due May 17) 1) SAS ( =01) ) SAS 15 15-19 0 38 6 8 93 15 116 10 131 37 178 [001 1 ] http://wolfpackhannamackr Nonparametric 66

Chapter 5, 54 Chi-Square (, )? 1 1 c 1 O 11 O 1 O 1c n 1 O 1 O O c n : : : : : : r O r1 O r O rc n r n 1 n n c n x ( fourfold ) Total A B A+B C D C+D Total A+C B+D N ( )=( ), ( )=( ) x N( AD BC) T = ~ χ ( df = 1) ( A + C)( B + D)( C + D)( A + B) [001 1 ] http://wolfpackhannamackr Nonparametric 67

Chapter 5, : ( p 1 ) ( p ) T = pˆ pˆ 1 pˆ(1 pˆ)(1/ n 1 + 1/ n ) appro z(0,1) B pˆ 1 =, A + B pˆ D B + D =, pˆ = C + D N EXAMPLE AIDS A B ( =005) AIDS yes no A 0 6 6 B 16 14 30 36 0 56 56(0 14 16 6) T = = 338 ~ χ ( df = 1) χ ( df = 1) (0 + 6)(16 + 14)( 0 + 14)(6 + 16) {> 384} AIDS HOMEWORK #11 (due May ) Fourfold Chi-square (hand calculation) ( =005) 30 30 31 19 50 17 33 50 48 5 100 [001 1 ] http://wolfpackhannamackr Nonparametric 68

Chapter 5, RXC ( ) n E = =( )x( ) ij j ( ) ni n HOMEWORK #11 ( due May ) SAS ( =005) A 50 0 70 B 1 5 37 C 6 8 14 D 1 1 69 74 143 55 (Fisher exact test) Chi-square 5 ( 10 ) Cochran 5 0% Chi-square Cochran SAS 5 6 5 33% [001 1 ] http://wolfpackhannamackr Nonparametric 69

Chapter 5, Chi-Square? X 0 1 (,, ) X 0 1 1 Exact test Fisher x RxC Fisher s Exact test (x ) ( ) : 5 7 3 3 6 8 5 13 [001 1 ] http://wolfpackhannamackr Nonparametric 70

Chapter 5, p- 6 1 7 4 6 8 5 13 7 0 7 1 5 6 8 5 13 3 p- N 11 N 1 N 1 N 1 N N N 1 N N n1! n! n1! n!? p = n! n! n! n! n! 11 1 1 7!6!8!5! 7!6!8!5! 7!6!8!5! p = = 0363, p = = 0 0816, p = = 0 047 5!!3!3! 13! 61!!!4! 13! 7!01!!5! 13! p- 0416 ( ) p- p- p- SAS [001 1 ] http://wolfpackhannamackr Nonparametric 71

Chapter 5, SAS left sided 1 1 5, 4, 3,, 1, 0 HOMEWORK #11 (due May ) Fisher Exact test ( =005), SAS 1 5 6 6 0 6 7 5 1 [001 1 ] http://wolfpackhannamackr Nonparametric 7

Chapter 6 One-way layout ( ) 6 One-way layout 3 (oneway layout) k k 1 1 y 11 1 y 1 y y 1n1 n 1 y11 y1 K y1n 1 y y n n 1 y1 y K yn k y k1 y k yknk n k yk1 yk K y1 nk (grand mean) (SST) (SStr: ) (SSE= SST-SStr), ( 39 ) ( )(ratio) F- (normality assumption), Median, Kruskal-Wallis ( ) [001 1 ] http://wolfpackhannamackr Nonparametric 73

Chapter 6 One-way layout ( ) 61 ( : Analysis of Variance: ANOVA) t n t 1-n (unit) (CRD: Completely Randomized Design) (A, B, C) 4 1 1 CRD 1 1, 5,, 7, 1, 10, 3 CRD ( :block) (randomized) Randomized Block Design B A C B C A B A [001 1 ] http://wolfpackhannamackr Nonparametric 74

Chapter 6 One-way layout ( ) 3 1m (ppm) 10 Lake Observation 1 0 1 3 1 3 4 1 5 1 3 4 6 8 7 5 3 4 5 3 14 6 5 18 19 1 16 0 30 y = u + + e = µ + e ij τ i ij i ij i = 1,, K k j = 1,, K, ni : ( y y) = ( y y ) + ( y y) ij ij i i Y( ) : y y) = ( y y ) + ( ( y y) ij SST = SSE + SStr ij i i 1 3 ~ Normal(0, σ ) e ij : Hartley s test 0 : 1 t : H σ = σ = K = σ max( si ) : Fmax = ~ F ( : 1, 1) min( s ) i (Homoscadicity)dl (Heteroscadicity) σ = ku y * = y σ = ku y * = log( y) [001 1 ] http://wolfpackhannamackr Nonparametric 75

Chapter 6 One-way layout ( ) ( ) Source DF SS MS Treatment t-1 = y SStr ( i y) MStr = SStr /( t 1) MStr F = Error n-t SSE = SST SStr MSE = SSE /( n t) MSE Total n-1 = ( y y) SST ij ~ F( t 1, n t) (Post-hoc test) (multiple comparison) F- H 0 : u1 = u = = ut (pairwise: : H 0 : u 1 = u 3 ) (contrast: : H 0 : u 1 u + u3 = ) ( ) F- ( ) (controlled experimental error rate) 1 (1 α) c = t( t 1) / Fisher s Least Significant Difference c c pairwise o pairwise ( ) o LSD Tukey W procedure o studentized range distribution: W = max( y i ) min( y ) s q = w / o Student-Newman-Keuls procedure o Tukey (critical value) Tukey Duncan Multiple range test o Tukey 1 (1 α) r o i r [001 1 ] http://wolfpackhannamackr Nonparametric 76

Chapter 6 One-way layout ( ) Scheffe s S method o (contrast) Dunnett s procedure o control ( ) ( : placebo,, ) pairwise (contrast) o 1, 3? : Q = u ( u + 3 ) /? Q = y ( y + 3 ) / 1 u 1 y ( Q) : F = ~ F( 1, n t) where c c ci MSE( ) n i = i c i o 4? Q = u + u ) ( u + ) c =, c = 1, c = 1, c = 1 ( 1 4 u3 1 1 3 4 [001 1 ] http://wolfpackhannamackr Nonparametric 77

Chapter 6 One-way layout ( ) SAS [001 1 ] http://wolfpackhannamackr Nonparametric 78

Chapter 6 One-way layout ( ) HOMEWORK#1 (due May 4) (Research and Development ) (High, Moderate, Low) (10 ) (SAS ) Low 76 8 68 58 69 66 77 6 Moderate 67 81 94 86 78 77 89 79 83 87 71 84 High 85 97 101 78 96 95 Box-Plot Bar pairwise (Tukey ) High, (Low + Moderate) [001 1 ] http://wolfpackhannamackr Nonparametric 79

Chapter 6 One-way layout ( ) 6 Median Median? 1 + ( ) + 3 k M1, M, K M k, : H M = M = K = 0 : 1 M k : 1 1 k > O 11 O 1 O 1k A O 1 O O k B n 1 n n k N 3 {> } { } ( Oij Eij ) 4 T = ~ χ ( df = k 1) ( ) 11??? E E = ij [001 1 ] http://wolfpackhannamackr Nonparametric 80

Chapter 6 One-way layout ( ) χ ( df = k 1) (critical value: ) EXAMPLE 1 ( 05 α = 0 ) 1 185 187 09 194 175 197 188 185 189 193 176 195 169 183 185 179 19 04 19 34 33 194 09 195 1) : H 0 : M 1 = M = M ) : 3) : 1935 3 1 > 3 1 8 1 5 7 0 1 8 8 8 4 (3 4) T = 4 (5 4) + 4 (0 4) + + 4 4) : (13) χ ( df =, α = 005) = 5 99 = 13 HOMEWORK#13 (due June 5) 4 Median ( : hand calculation) =005 A 7 16 19 4 16 30 9 16 B 44 34 43 47 35 51 37 9 C 17 45 8 13 36 3 4 41 15 D 51 9 30 50 47 40 43 44 54 [001 1 ] http://wolfpackhannamackr Nonparametric 81

Chapter 6 One-way layout ( ) 63 Kruskal-Wallis 3 ( ) Mann-Whitney k M1, M, K M k, : H M = M = K = 0 : 1 M k : 1 3 1 Ri 4 : T = 3( N + 1) N( N + 1) n N = R i = i n i = i i Kruskal-Wallis K-W 5 3 5, K-W χ ( df = k 1) [001 1 ] http://wolfpackhannamackr Nonparametric 8

Chapter 6 One-way layout ( ) EXAMPLE 3 Kruskal-Wallis ( =005) 1 6 307 11 33 454 339 304 154 87 356 465 501 455 355 468 36 3 343 77 07 1048 838 687 1) : H 0 : M 1 = M = M ) : 3) : 3 1 4 7 3 8 14 9 6 1 5 1 69 16 18 15 11 17 13 90 3 10 0 1 19 94 1 Ri 1 69 90 94 T = 3( N + 1) = [ + + ] 3( + 1) = 93 N( N + 1) n ( + 1) 10 6 6 i 4) : (13) χ ( df =, α = 005) = 5 99 HOMEWORK#13 ( due June 5) 3 ( : ) Kruskal-Wallis ( : hand calculation) =005 Control 340 340 356 386 386 40 40 417 433 495 557 LSD 94 35 35 340 356 371 385 40 UML 63 309 340 356 371 371 40 417 [001 1 ] http://wolfpackhannamackr Nonparametric 83

Chapter 6 One-way layout ( ) 64 JONCKHEERE-TERPSTRA test Median Kruskal-Wallis, H : M < M < K < M (ordered) a 1 JONCKHEERE-TERPSTRA k k M1, M, K M k, : H M = M = K = 0 : 1 M k : H : M M K M < a 1 k : J = Uij U ij i< j i j i j JONCKHEERE -TERPSTRA J-T 3 n 1 < n < n3 J-T T = [ N J [( N n k (N + 3) n i k i i i ) / 4] (n i + 3] / 7 ~ N (0,1) [001 1 ] http://wolfpackhannamackr Nonparametric 84

Chapter 6 One-way layout ( ) EXAMPLE? ( =005) 54 67 47 71 6 44 67 80 79 8 88 79 85 81 88 98 99 95 93 98 91 94 1) : H 0 : M 1 = M = M 3 ) : H 0 : M 1 M M 3) : 3 U =, U = 56, U 49 J = = 159 1 54 13 3 = U ij i< j 4) : J-T n = 7, n = 7, n 8), 05 α = 0 108 ( 1 3 = 109 159 134 000443 p- 00443 5) J = 159, N = 8+ 7 + 7 =, n = 8 + 7 3 i= 1 i + 7 3 i = 16, n (n + 3) = 88 i i T = 159 [( 16) / 4] [ ( + 3) 88 ] / 7 = 473 p- 0001 J-T HOMEWORK#14 (due June 7) 1 3 ( =005) p- J-T 1 71 57 85 67 66 79 76 94 61 36 4 49 3 80 104 81 90 93 85 101 83 [001 1 ] http://wolfpackhannamackr Nonparametric 85

Chapter 6 One-way layout ( ) 65 Multiple Comparison ( ) Median Kruskal-Wallis (pairwise), (, contrast) post-hoc test ( multiple comparison) pairwise 3 pairwise 3 3 1 (1 α) 3 ( ) 1 (1 α) ) c (c 1) ) R Kruskal-Wallis ) ) k ( Ri R j ) ( ) N( N + 1) 1 1 z (1 [ α / k( k 1)]) ( + ) 1 n n i j ( i ( R = (i )/(i ) / N = i n = i i k ( k 1) / k ( k 1) [001 1 ] http://wolfpackhannamackr Nonparametric 86

Chapter 6 One-way layout ( ) EXAMPLE Kruskal-Wallis ( : page 83) : ( =015) k = 3 α / k( k 1) = 015 / 3() = 0 05 1 pairwise R1 R = 69 /10 90 / 6 = ( + 1) 1 1 1 96 ( + ) = 657 1 10 6 81 657 ( 1 ) (SAS ) HOMEWORK#14 ( due June 7) Kruskal-Wallis (page 83) pairwise ( =01) : 6 ( ) 1 ~ : :, Open Book Exam 70 A A [001 1 ] http://wolfpackhannamackr Nonparametric 87

Chapter 7 Goodness of Fit test 7 Goodness-of-fit test (, )?? ( ) (fit) (Goodness-of-fit test) 1)? )? ( ) 71 χ Goodness-of-fits test, χ (chi-square) (expected) (observed) x, x, K, x ) ( : random sample) ( 1 n (non-overlapping) 1 3 r E 1 E E 3 E r n O 1 O O 3 O r n p 1 p p 3 p r [001 1 ] http://wolfpackhannamackr Nonparametric 88

Chapter 7 Goodness of Fit test : XX : XX : r ( Oi Ei ) T = ~ χ ( df = r 1) E i= 1 i 5 χ - 5 0% χ - (Cochran) χ ( =r-1) ( ) (parameter)? p, p, K p ) ( 1 r (µ), ) (σ, χ - ( 1) g r ( r g 1) EXAMPLE1 36 ( =005) 13 6 0 3 11 3 [001 1 ] http://wolfpackhannamackr Nonparametric 89

Chapter 7 Goodness of Fit test : (Equally distributed) : 6 6 13 6 0 3 11 3 6 6 6 6 6 6 : (13 6) (6 6) (3 6) T = + + K+ 6 6 6 = 133 : χ ( =6-1=5, α = 005) =111 EXAMPLE 5 80 (binomial) ( =005) 0 1 3 4 5 157 69 35 17 1 1 : : (p) 5 x 5 x p( x) = ( ) p (1 p), x = 0,1,, K, 5 x p ( pˆ) p ˆ = (0 157 + 1 69 + 35 + K + 5 1) /(80 5) = 014 [001 1 ] http://wolfpackhannamackr Nonparametric 90

Chapter 7 Goodness of Fit test 5 x 5 x p( x) = ( )014 (1 014), x = 0,1,, K, 5 x 0 1 3 4 5 157 69 35 17 1 1 04047 0389 0147 0003 00017 00001 1317 107 349 57 05 003 : (157 1317) (69 107) (1 003) T = + + K + 1317 107 003 = 4499 : χ ( =5-1-1=3, α = 005) =781 3 EXAMPLE3 Poisson 30 300 Poisson ( =005) 0 1 3 4 5 6 7 0 54 74 67 45 5 11 4 : Poisson : Poisson Poisson (λ) e λ p( x) =, x = 0,1,, K x! p (λˆ) λ x λ ˆ = (0 0 + 1 54 + K+ 7 4) / 300 = 67 [001 1 ] http://wolfpackhannamackr Nonparametric 91

Chapter 7 Goodness of Fit test Poisson e p( x) = 67 67 x! 0 1 3 4 5 6 7 0 54 74 67 45 5 11 4 0069 0185 047 0 0147 0078 0035 0013 07 555 741 66 441 34 105 39 x : (0 07) (54 555) (4 39) T = + + K + 07 555 39 = 034 : χ ( =8-1-1=6, α = 005) =155 Poisson HOMEWORK#15 (due JUNE 1) 196-1967 - (uniformly distributed) ( =005) 196 1963 1964 1965 1966 1967 7 1 6 10 1 6 HOMEWORK#15 ( : due JUNE 1) Poisson 8 ( 8 + 9 + 10 + 11 + 1) 0-7 1 λˆ ( =005) 0 1 3 4 5 6 7 8 9 10 11 1 4 16 16 18 15 9 6 5 3 4 3 0 1 [001 1 ] http://wolfpackhannamackr Nonparametric 9

Chapter 7 Goodness of Fit test 7 KOLMOGOROV-SMIRNOV one-sample test χ (chi-square) ( ) 8-10 ( ) χ (chi-square) (page 88 ) K-S K-S (cumulative distribution) F( x) = P( X x) (Theorem) x, x, K, x ) ( : random sample) ( 1 n : F ( x ) = F 0 ( x ) F : F ( x ) F 0 ( x ) F 0 0 : n S ( x) sup D = S( x) F 0 ( x) x K-M =(x )/(n) K-M Massey 1 [001 1 ] http://wolfpackhannamackr Nonparametric 93

Chapter 7 Goodness of Fit test EXAMPLE ( =85, =15) ( =005) : Normal ( µ = 85, σ = 15) : Normal( µ = 85, σ = 15) : S ( x) = (# of x) / n 36 x -113 [001 1 ] http://wolfpackhannamackr Nonparametric 94

Chapter 7 Goodness of Fit test sup : D = S( x) F0 ( x) = 0 1485 x : K-M n=36 01 ( µ = 85, σ = 15) HOMEWORK#15 ( : due JUNE 1) ( =005) 111 1435 1789 1489 1805 1738 193 750 513 301 105 935 085 988 450 [001 1 ] http://wolfpackhannamackr Nonparametric 95

Chapter 8 8 (association) 5 χ - (nominal) 81 Pearson ) x, y ( i i : ( x, y) : ρ 0 0 ρ = 0 : t = r ~ t( df = n ) (1 r ) /( n ) H ρ = ρ 0 0 : 0 1+ r 1+ ρ T = 05 ln ~ Normal(05ln,1/( n 3)) 1 r 1 ρ, r =, n= 1+ ρ 0 T 05ln 1 ρ 0 Z = ~ Normal(0,1) 1/( n 3) t ( =n-), [001 1 ] http://wolfpackhannamackr Nonparametric 96

Chapter 8 EXAMPLE 19 0878 ( =005) : ρ = 0 : ρ 0 : t = (1 r r = ) /( n ) 0878 = 41 (1 0878 ) /(19 ) : t( df = 17, α / = 005) = 11 8 Spearman rank correlation coefficient ( ) ) x, y ( i i : ( x, y) : : R( x i ) 6 d n i rs = 1 where d i = [ R( xi ) R( yi )] n( n 1) i= 1 x i R y ) y i ( i [001 1 ] http://wolfpackhannamackr Nonparametric 97

Chapter 8 (x, y) d i 0 rs = 1 (x,y) d 1 r = [ R ( x) = 1, R( y) = n], [ R ( x) =, R( y) = n 1],, [ R ( x) = n, R( y) = 1] Spearman ( SAS ) i S Large Sample approximation z = rs n 1 ~ Normal(0,1) 83 Kendall s Tau x, y ) ( i i : ( x, y) : : n = ti, where = (tied) X u = (tied) Y i Kendall ( SAS ) [001 1 ] http://wolfpackhannamackr Nonparametric 97

Chapter 8 Large Sample approximation 3ˆ n( n 1) z = τ (n + 5) ~ Normal(0,1) 84 Kendall s Tau Spearman Spearman n Kendall s Tau Spearman p- Kendall s Tau 85 5 IQ ( =005) KENDALL, SPEARMAN SAS PEARSON [001 1 ] http://wolfpackhannamackr Nonparametric 98

Chapter 8 Spearman 04975 p- 0056 005 IQ Kendall 03596p- 00348005 IQ [001 1 ] http://wolfpackhannamackr Nonparametric 99