Agenda: Recap. Lecture. Chapter 12. Homework. Chapt 12 #1, 2, 3 SAS Problems 3 & 4 by hand. Marquette University MATH 4740/MSCS 5740

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1 Ageda: Recap. Lecture. Chapter Homework. Chapt #,, 3 SAS Problems 3 & 4 by had. Copyright 06 by D.B. Rowe

2 Recap.

3 6: Statistical Iferece: Procedures for μ -μ 6. Statistical Iferece Cocerig μ -μ Recall yes Variaces kow? Idepedet populatios? o Case 4 yes Case Z Case o F test Case 3 30 Z 30 t, 30, 30, 30, 30 Z t Z t 3

4 6: Statistical Iferece: Procedures for μ -μ 6. Statistical Iferece Cocerig μ -μ Depedet Populatios: Case 4: Test Statistic Cofidece Iterval & ukow 30 s X 30 Z d d d Xd Z / s / & ukow df t X d / d d s d X i d t / di X i X i Xd di sd ( d ) i X d s d i 4

5 : Noparametric Tests. The Sig Test (Two depedet Samples Test) If d i s are ot N(μ d,σ ), & <30 the we kow that X d is ot N(μ d,σ /), o CLT, ad that X d X d d μ d is ot N(0,σ ), o CLT, ad that / ( ) s X d d t= s / d d is ot N(0,), o CLT, ad that is ot χ (-), but approximately ~t(-)! Studies have show that if X i s are from moud shaped dist, the approximatio ot bad. This used i Chapt 6 Case 4 Robust to mild departures from ormality. 5

6 : Noparametric Tests. The Sig Test (Two depedet Samples Test) For the Sig Test, we still calculate differeces d i =X i X i, but ot the sample mea, X d. We determie whether the differeces are positive +, egative, or zero 0. If the medias of the two distributios are truly the same, the we should have a similar allocatio of + ad. d i i 6

7 : Noparametric Tests. The Sig Test (Two depedet Samples Test) H 0 : p=( or )0.5 (this is a media test). Let X be the umber of + sigs. If there is o differece i medias, i.e. H 0 true, the X has a biomial distributio with p=0.5. Sigificace from biomial. Assumptios For Test:. d i s are idepedet (each trial idepedet). d i s from same distributio (p ot chagig) 7

8 : Noparametric Tests. The Sig Test (Two depedet Samples Test) If the medias of the two distributios are truly the same, the we should have a similar allocatio of + ad. Example.: Pre/Post Blood Pressures H 0 : p 0.5, =0, x=7. Assume H 0 true. How likely is it to observe 7 or more successes out of 0 whe p=0.5? P(X 7)=0.79. Reject H 0 if p-value p-value 8

9 : Noparametric Tests. The Sig Test (Two depedet Samples Test) Example.: was H 0 : p 0.5 vs. H 0 : p>0.5. by calculatig P(X x), reject H 0 if p-value<α. p-value But we ca also test H 0 : p 0.5 vs. H 0 : p<0.5. by calculatig P(X x), reject H 0 if p-value<α. p-value But we ca also test H 0 : p=0.5 vs. H 0 : p 0.5. by calculatig P(X x)+p(x x), reject H 0 if p-value<α. p-value 9

10 : Noparametric Tests. The Wilcoxo Siged-Rak Test (Two Depedet Samples Test) Example.: Pre/Post Blood Pressures H 0 : p 0.5, =0, x=7. Assume H 0 true. T=40.5. The max possible T is 45. ad the media is.5. Sice T>37, reject H 0! 0

11 : Noparametric Tests. The Wilcoxo Siged-Rak Test (Two Depedet Samples Test) Geerally the exact critical value table is ot used, a approximate Z statistic is used, Z ( ) T 4 ( )( ) 4 ad the usual ormal Z table is used. ( ) ET ( ) ( )( ) var( T) 4 6

12 : Noparametric Tests. The Wilcoxo Siged-Rak Test Example.: Pre/Post Blood Pressures H 0 : p 0.5, =0, x=7. Assume H 0 true. T=40.5. The media is.5. (Two Depedet Samples Test) Z ( ) T 4 ( )( ) 4 Z (9 )( 9 ) 4.3 Sice Z>.645, reject H 0!

13 Chapter : Noparametric Tests Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece 3

14 6: Statistical Iferece: Procedures for μ -μ Itroductio Recall X Four Cases: Case. Two idepedet populatios populatio variaces are kow, ( & kow & ). Case. Two idepedet populatios populatio variaces are ukow but assumed to be equal, ( & ukow & =). Case 3. Two idepedet populatios populatio variaces ukow ad possibly uequal, ( & ukow & ). Case 4. Two depedet populatios data are assumed to be matched or paired ( & kow or ukow). 4

15 6: Statistical Iferece: Procedures for μ -μ 6. Statistical Iferece Cocerig μ -μ Recall yes Variaces kow? Idepedet populatios? o Case 4 yes Case Z Case o F test Case 3 30 Z 30 t, 30, 30, 30, 30 Z t Z t 5

16 6: Statistical Iferece: Procedures for μ -μ 6. Statistical Iferece Cocerig μ -μ Recall Idepedet Populatios: Case : Test Statistic & ukow X X Z S p & ukow ( ) ( ) ( X X) ( ) t S p S df ( ) s ( ) s P 6

17 6: Statistical Iferece: Procedures for μ -μ 6. Statistical Iferece Cocerig μ -μ Recall Idepedet Populatios: Case 3: Test Statistic & ukow X X 30 Z s s 30 & ukow ( ) ( ) ( X X) ( ) t s s Next larger umber s s df s / s / 7

18 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) We kow that if X i s~n(μ, ) & X i s~n(μ, ), the we kow X ~N(μ, / ) & X~N(μ, / ), ad Case :, 30 ( X X ) ~ N(( ), / / ), ad ( X X ) ( ) ~ N(0, / / ), ad ( X X) ( ) ~ N (0,), ad 8

19 : Noparametric Tests.3 The Wilcoxo Rak Sum Test idepedet that ( X X) ( ) ~ N (0,) s ( ) ( ) s (Two Idepedet Samples Test) ~ ( ), ad ~ ( ), ad, ad Case : ( ) s ( ) s ~ ( ), 30 9

20 : Noparametric Tests.3 The Wilcoxo Rak Sum Test that (Two Idepedet Samples Test) ( X X) ( ) ~ N (0,) Case :, ad, 30 ( ) s ( ) s ( X X) ( ) ~ ( ) t t Z S p ~ ( ) S if large. ( ) s ( ) s P 0

21 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) If X i s ot~n(μ, ) & X i s ot~n(μ, ), the X Case : ot~n(μ, / ) & X ot~n(μ, / ), ad, 30 ( X X ) ot ~ N(( ), / / ), ad ( X X ) ( ) ot ~ N(0, / / ), ad ( X X ) ( ) ot ~ N(0,), ad

22 : Noparametric Tests.3 The Wilcoxo Rak Sum Test idepedet (Two Idepedet Samples Test) ( X X ) ( ) that ot ~ N(0,), ad ( ) s ( ) s ot ( ) s ( ) s ~ ( ), ad ot ~ ( ), ad ot Case : ~ ( ), 30

23 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) ( X X ) ( ) that ot ~ N(0,), ad, 30 ( ) s ( ) s ot ~ ( ), but approximately ( X X) ( ) t ~ t ( ) S p Case : Studies have show that if X i s are from moud shaped dist, the approximatio ot bad. This used i Chapt 6 Case 3

24 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) Whe & <30 ad we are cocered that ( X X) ( ) t does ot have a Studet-t distributio, S p the we eed to resort to a oparametric test. H 0 : MD MD =0 ot H 0 : µ -µ =0 Case :, 30 4

25 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) For the Wilcoxo Rak Sum Test with ad, Rak all the observatios i icreasig order. Assig ties their average rak. Let S=the sum of the raks for the sample of size. The sum of all raks is (+)/, = +. If there is o differece, the we expect some small ad some large raks so S (+)/4! 5

26 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) S=the sum of the raks for the sample of size. Table Not i Book. 6

27 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) Example.3: Health Status H 0 : MD MD =0 H : MD MD 0 α=0.05 S =6 S =0 If there is o differece, the we expect to see some low ad some high raks i each group. Sum of all raks=36. Expect S 8. S=6. Accordig to the Table, sice S 5, do ot reject. 7

28 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) Geerally the exact critical value table is ot used, a approximate Z statistic is used, Z ( ) S ( ) ad the usual ormal Z table is used. E( S) ( ) ( ) var( S) 4 3 8

29 : Noparametric Tests.3 The Wilcoxo Rak Sum Test (Two Idepedet Samples Test) Example.3: Health Status H 0 : MD MD =0 H : MD MD 0 α=0.05 4(4 4 ) 6 Z (4 4 ) We kow that sice Z <.96, do ot reject. Z ( ) S ( ) 9

30 9: Aalysis of Variace Recall ANOVA is a geeralizatio of these methods ad is for testig for differeces i three or more populatio meas. i.e. H 0 : μ = μ = μ 3 H : meas ot all equal (at least two are differet) (Ca be ad ; or ad 3 ; or ad 3 ; or,, ad 3 ) More geerally, H 0 : μ = μ = = μ k, k is # of populatio meas comparig H : meas ot all equal (at least two are differet) I ANOVA there are o iequality hypotheses. No vs > or vs <. 30

31 9: Aalysis of Variace X X Assumptios for valid applicatio of ANOVA ) k idepedet populatios ) Radom samples from each of k > populatios 3) Large samples ( i 30) or ormal populatios 4) Equal populatio variaces k 3

32 : Noparametric Tests.4 The Kruskal-Wallis Test (k Idepedet Samples Test) The K-W Test is a oparametric alterative to ANOVA. Rak all the observatios ad replace by raks, R ij. Assig ties their average rak. rak i i populatio j Let R i =the sum of the raks for the ith sample of size i. k R. ( ) i k ( ) H i s Rij s i i 4 i j 4 Reject H 0 for large H. If there ( ) s are o ties 3

33 : Noparametric Tests.4 The Kruskal-Wallis Test (k Idepedet Samples Test) The K-W Test is a oparametric alterative to ANOVA. There are exact tables available that deped o k ad i s. 33

34 : Noparametric Tests.4 The Kruskal-Wallis Test (k Idepedet Samples Test) The K-W Test is a oparametric alterative to ANOVA. H 0 : MD =MD =MD 3 =MD 4, H : At least two MDs. H CV 7.38 Reject H 0, at least two MDs. R ( ) k. i H s i i 4 s k i k i Rij i j s R. ( ) i 349. i 4 ( )

35 : Noparametric Tests.4 The Kruskal-Wallis Test (k Idepedet Samples Test) Geerally the exact critical value table is ot used. If the i are 5, the H has a approximate χ distributio uder the ull hypothesis. H k R. ( ) i s i i 4 ad the usual χ table is used. Reject H 0 if H> ( k ). ~ ( k ) 35

36 : Noparametric Tests.4 The Kruskal-Wallis Test (k Idepedet Samples Test) The K-W Test is a oparametric alterative to ANOVA. R ( ) k. i H s i i 4 H 0 : MD =MD =MD 3 =MD 4, H : At least two. H ( k ) 7.8 Reject H 0, at least two MDs. s

37 0: Correlatio ad Regressio 0. Correlatio Aalysis 0.. The Sample Correlatio Coefficiet Recall The sample correlatio is r also give by r cov( XY, ) var( X) var( Y) ( x y ) ( x )( y ) i i i i i i i [ ( xi ) ( xi ) ][ ( yi ) ( yi ) ] i i i i cov( X, Y) ( X i X )( Yi Y ) i var( X ) ( X i X ) i var( Y) ( Yi Y ) i 37

38 : Noparametric Tests.5 Spearma Correlatio (Correlatio Betwee Variables) The Spearma rak correlatio betwee X ad Y is to covert the actual observed values to raks R X ad R y, the calculate the correlatio betwee R X ad R y. r s cov( R, R ) x var( R ) var( R ) x y y cov( R, R ) ( R R )( R R ) x y xi x yi y i var( R ) ( R R ) x xi x i var( R ) ( R R ) y yi y i 38

39 : Noparametric Tests.5 Spearma Correlatio (Correlatio Betwee Variables) We ca perform a hypothesis test o r s. H 0 : ρ s =0 vs. ρ s 0. r s cov( R, R ) x var( R ) var( R ) x y y Reject H 0 if r s large. 39

40 : Noparametric Tests.5 Spearma Correlatio There is a exact table. r s cov( R, R ) x var( R ) var( R ) x y y 40

41 : Noparametric Tests.5 Spearma Correlatio But the exact table is ot geerally used. t r ~ ( ) s t r s H 0 : ρ s = 0 vs. H : ρ s 0, α=0.05 Reject if t >t.05 (df-) 4

42 : Noparametric Tests Questios? Homework: Chapter #,, 3, SAS Problems 3 & 4 by had. 4

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700

Class 23. Daniel B. Rowe, Ph.D. Department of Mathematics, Statistics, and Computer Science. Marquette University MATH 1700 Class 23 Daiel B. Rowe, Ph.D. Departmet of Mathematics, Statistics, ad Computer Sciece Copyright 2017 by D.B. Rowe 1 Ageda: Recap Chapter 9.1 Lecture Chapter 9.2 Review Exam 6 Problem Solvig Sessio. 2