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2 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), [001 1 ] Nonparametric 1

3 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) ( ) ( ) 15 measurement and categorical ( ) (measurable numerical) (categorical) [001 1 ] Nonparametric

4 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 ] Nonparametric 3

5 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 = 5% 95% 95% 95% [001 1 ] Nonparametric 4

6 Chapter 1 Introduction 13 order statistic [001 1 ] Nonparametric 5

7 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 ] Nonparametric 5

8 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 ] 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

9 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 ] Nonparametric 7

10 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 ] Nonparametric 8

11 Chapter 1 Introduction 1) χ - ) 0 3) xc ~ χ (λ=), xc6 ~ χ (λ=10) 4) (xc, xc6) 5) ( ) )-4) ) [001 1 ] Nonparametric 9

12 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 ] Nonparametric 10

13 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(1-α)% (µ) µ = x + z u α / µ = x z l α / s s n n s n x ( ) [001 1 ] Nonparametric 11

14 Chapter [001 1 ] Nonparametric 1

15 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 ] Nonparametric 13

16 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 ] Nonparametric 14

17 Chapter : : M 3 5 H, : M = H a : ( x i M 0) =1 k 1 p- = Pr( K 1 n = 10, p = 05) = =9, + =1, 0 p<005 ( α / ) ( ) 35 k ? ( : 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) 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 ] Nonparametric 15

18 Chapter 100(1-α)% α Pr( K 3 n = 16, p = 05) = Pr( K 4 n = 16, p = 05) = 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 / ) / 4 4 n / 4 95% ( X = 9, X 8 11 ( 4) 1 (13) = ) 95% α=00105*=001 95% 979% [001 1 ] Nonparametric 16

19 Chapter [001 1 ] Nonparametric 17

20 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 ) 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 ] Nonparametric 18

21 Chapter np,( n 1) p ) 9 min( 0 0 < α H : M 107 : M = + 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 ] Nonparametric 19

22 Chapter sampling distribution of T + : n=4 T + Wilcoxon + T + + T + + T + + T , 1,3 1, ,3,4 3,4 1,,3 Wilcoxon (n=4) ,,4 1,3,4,3,4 1,,3, (confidence intervals) Wilcoxon xi + x j 1) uij =, 1 i j n ) uij? C 10 = ) 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 ij( 8) = 3) n=10p=0044t=8 ( u 7 75, u % ij( 9) = ij( 47) = ) Large-sample approximation : 0 Wilcoxon n( n + 1) z 4 K α / n( n + 1)(n + 1) 4 [001 1 ] Nonparametric 0

23 Chapter [001 1 ] Nonparametric 1

24 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 ] Nonparametric

25 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 n 1 1 ) ( n + n 1) [001 1 ] Nonparametric 3

26 Chapter [001 1 ] Nonparametric 4

27 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), 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 ] Nonparametric 5

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

29 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 ] Nonparametric 7

30 Chapter (SAS v8 ) 5 1) Report [001 1 ] Nonparametric 8

31 Chapter (SAS v8 ) ) ( ) 3), [001 1 ] Nonparametric 9

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

33 Chapter (SAS v8 ) 53? 1)? [001 1 ] Nonparametric 31

34 Chapter (SAS v8 ) ) pop-up 3) 4) 5) SAS data [001 1 ] Nonparametric 3

35 Chapter (SAS v8 ) 6)? 7) box plot ( )? 8)?? [001 1 ] Nonparametric 33

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

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

38 Chapter (SAS v8 ) 4) p-value= (5%) 10 55?? (54 ) 1) data step univariate procedure [001 1 ] Nonparametric 36

39 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 ] Nonparametric 37

40 Chapter (SAS v8 ) 1)? [ ] ),,? [ ] [001 1 ] Nonparametric 38

41 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 ] Nonparametric 39

42 Chapter ) : 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 ] Nonparametric 40

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

44 Chapter 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 ] Nonparametric 4

45 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 Y ) : M = M x y ) : M M ( ) 3) : x y X Y total Total (1 / 3) (1 /16) T = = 45 p ˆ = (1 + 1) / 48 = (1 05)(1/ 3 + 1/16) 4) : ( < 196) X Y X (why?) [001 1 ] Nonparametric 43

46 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 B [001 1 ] Nonparametric 44

47 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 ] Nonparametric 45

48 Chapter 3 EXAMPLE (X, Y) ( =005) X Y ) : M = x M y ) : M M ( ) 3) : x y T = (17 + 1) / = ) : 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 = > p > SAS [001 1 ] Nonparametric 46

49 Chapter 3 [001 1 ] Nonparametric 47

50 Chapter 3 HOMEWORK #6 [due 4 17 ] [001 1 ] Nonparametric 48

51 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 ] Nonparametric 49

52 Chapter 3 EXAMPLE 95% X Y ) (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 ] Nonparametric 50

53 Chapter (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 Y ) : ) : ( ) 3) : (group) X X Y Y X Y Y X Y T=1++5+=10 ( n 1 =4, n =5) T T=16(00159) 8(09603) 8 16 [001 1 ] Nonparametric 51

54 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)] 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 ) Dopamine ( =01) SAS 1 [001 1 ] Nonparametric 5

55 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 [001 1 ] Nonparametric 53

56 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 ) : M = 0 ) : M < 0 ( ) 3) : K = 4) : p- = Pr( K n = 10, p = 05) = d d [001 1 ] Nonparametric 54

57 Chapter 4 HOMEWORK #8 [due 5 10 ] Solve the following problems [001 1 ] Nonparametric 55

58 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 ) : M = 0 d [001 1 ] Nonparametric 56

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

60 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 ] Nonparametric 58

61 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 No Total ) (hypothesis) : p 1 = p ( ) p < ( ) : 1 p 37 6 ) (test statistic): z = = p-value= pr ( z 138) = [001 1 ] 59 Nonparametric

62 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 ] 60 Nonparametric

63 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 ] Nonparametric 61

64 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 ] Nonparametric 6

65 Chapter 5, 53 Chi-Square (associate) A 1 30 (3 ) A BC Total Total n ( ) : : n n i j Pr( nij ) = Eij = ( )( ) n n n Pr( A ) = Pr( ) Pr(A ) n1 n 1 1 E 11 = ( )( ) n = / 30 = 65 7 n n 3 E = / = [001 1 ] Nonparametric 63

66 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 (approximate) ( df = ( 1)(3 1) = ) χ (critical region) 5% } {> 599? SAS A, B A [001 1 ] Nonparametric 64

67 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 ] Nonparametric 65

68 Chapter 5, EXAMPLE 764 5% 1) : ( ) ) : 3) : E = / = ( ) ( ) (59 41) 4) : T = + + K+ = 47 9, χ - {> 15507} HOMEWORK #10 (due May 17) 1) SAS ( =01) ) SAS [001 1 ] Nonparametric 66

69 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 ] Nonparametric 67

70 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 B ( ) T = = 338 ~ χ ( df = 1) χ ( df = 1) (0 + 6)( )( )(6 + 16) {> 384} AIDS HOMEWORK #11 (due May ) Fourfold Chi-square (hand calculation) ( =005) [001 1 ] Nonparametric 68

71 Chapter 5, RXC ( ) n E = =( )x( ) ij j ( ) ni n HOMEWORK #11 ( due May ) SAS ( =005) A B C D (Fisher exact test) Chi-square 5 ( 10 ) Cochran 5 0% Chi-square Cochran SAS % [001 1 ] Nonparametric 69

72 Chapter 5, Chi-Square? X 0 1 (,, ) X Exact test Fisher x RxC Fisher s Exact test (x ) ( ) : [001 1 ] Nonparametric 70

73 Chapter 5, p 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! !6!8!5! 7!6!8!5! 7!6!8!5! p = = 0363, p = = , p = = !!3!3! 13! 61!!!4! 13! 7!01!!5! 13! p ( ) p- p- p- SAS [001 1 ] Nonparametric 71

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

75 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 ] Nonparametric 73

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

77 Chapter 6 One-way layout ( ) 3 1m (ppm) 10 Lake Observation 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 ] Nonparametric 75

78 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 ] Nonparametric 76

79 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 u [001 1 ] Nonparametric 77

80 Chapter 6 One-way layout ( ) SAS [001 1 ] Nonparametric 78

81 Chapter 6 One-way layout ( ) HOMEWORK#1 (due May 4) (Research and Development ) (High, Moderate, Low) (10 ) (SAS ) Low Moderate High Box-Plot Bar pairwise (Tukey ) High, (Low + Moderate) [001 1 ] Nonparametric 79

82 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 ] Nonparametric 80

83 Chapter 6 One-way layout ( ) χ ( df = k 1) (critical value: ) EXAMPLE 1 ( 05 α = 0 ) ) : H 0 : M 1 = M = M ) : 3) : > (3 4) T = 4 (5 4) + 4 (0 4) ) : (13) χ ( df =, α = 005) = 5 99 = 13 HOMEWORK#13 (due June 5) 4 Median ( : hand calculation) =005 A B C D [001 1 ] Nonparametric 81

84 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 : 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 ] Nonparametric 8

85 Chapter 6 One-way layout ( ) EXAMPLE 3 Kruskal-Wallis ( =005) ) : H 0 : M 1 = M = M ) : 3) : Ri T = 3( N + 1) = [ + + ] 3( + 1) = 93 N( N + 1) n ( + 1) i 4) : (13) χ ( df =, α = 005) = 5 99 HOMEWORK#13 ( due June 5) 3 ( : ) Kruskal-Wallis ( : hand calculation) =005 Control LSD UML [001 1 ] Nonparametric 83

86 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 ] Nonparametric 84

87 Chapter 6 One-way layout ( ) EXAMPLE? ( =005) ) : H 0 : M 1 = M = M 3 ) : H 0 : M 1 M M 3) : 3 U =, U = 56, U 49 J = = = U ij i< j 4) : J-T n = 7, n = 7, n 8), 05 α = ( 1 3 = p ) J = 159, N = =, n = i= 1 i i = 16, n (n + 3) = 88 i i T = 159 [( 16) / 4] [ ( + 3) 88 ] / 7 = 473 p J-T HOMEWORK#14 (due June 7) 1 3 ( =005) p- J-T [001 1 ] Nonparametric 85

88 Chapter 6 One-way layout ( ) 65 Multiple Comparison ( ) Median Kruskal-Wallis (pairwise), (, contrast) post-hoc test ( multiple comparison) pairwise 3 pairwise (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 ] Nonparametric 86

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

90 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 ] Nonparametric 88

91 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) [001 1 ] Nonparametric 89

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

93 Chapter 7 Goodness of Fit test 5 x 5 x p( x) = ( )014 (1 014), x = 0,1,, K, 5 x : ( ) (69 107) (1 003) T = + + K = 4499 : χ ( =5-1-1=3, α = 005) =781 3 EXAMPLE3 Poisson Poisson ( =005) : Poisson : Poisson Poisson (λ) e λ p( x) =, x = 0,1,, K x! p (λˆ) λ x λ ˆ = ( K+ 7 4) / 300 = 67 [001 1 ] Nonparametric 91

94 Chapter 7 Goodness of Fit test Poisson e p( x) = x! x : (0 07) (54 555) (4 39) T = + + K = 034 : χ ( =8-1-1=6, α = 005) =155 Poisson HOMEWORK#15 (due JUNE 1) (uniformly distributed) ( =005) HOMEWORK#15 ( : due JUNE 1) Poisson 8 ( ) λˆ ( =005) [001 1 ] Nonparametric 9

95 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 ] Nonparametric 93

96 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 ] Nonparametric 94

97 Chapter 7 Goodness of Fit test sup : D = S( x) F0 ( x) = x : K-M n=36 01 ( µ = 85, σ = 15) HOMEWORK#15 ( : due JUNE 1) ( =005) [001 1 ] Nonparametric 95

98 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 ] Nonparametric 96

99 Chapter 8 EXAMPLE ( =005) : ρ = 0 : ρ 0 : t = (1 r r = ) /( n ) 0878 = 41 ( ) /(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 ] Nonparametric 97

100 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 ] Nonparametric 97

101 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 ] Nonparametric 98

102 Chapter 8 Spearman p IQ Kendall 03596p IQ [001 1 ] Nonparametric 99