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�� Nonparametric Statistics

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Email: [email protected]

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- ��<�ก ��'(�� #��$�� (truly nonparametric procedures) �=E+��<�ก �%�� *��+'$� � � ��%�=��( ก� �(+ก ��*�) ก ��*�) run, goodness of fit -��<�ก �#$ก#$�#))�� (distribution-free procedures)

ก ��*�) Run Test '(�'+ก ��*�)ก ��K� (randomness) �(+ก ��Lก! ��%��&�=M��%� � )ก ��$ ��Lก�&�=M��+ก�� %��&��&=#))� ก!"� M F M F M F M F #�*��&=#))%��&�#))�� (mixed) �� M M M M F F F F �=E+�&=#))ก�K� (cluster) QL�� ก!"�� R��'(�&=#))��ก�*S*��K�

H0 : ก ��V * )%��&��=E+� ก!"��K�HA : ก ��V * )%��&��=E+� ก!"��K�

. list

sex1 sex2 sex3

1. 1 1 1

2. 0 1 0

3. 1 1 0

4. 0 1 1

5. 1 0 0

6. 0 0 1

7. 1 0 1

8. 0 0 0

. runtest sex1

N(sex1 <= .5) = 4

N(sex1 > .5) = 4

obs = 8

N(runs) = 8

z = 2.29

Prob>|z| = .02

. runtest sex2

N(sex2 <= .5) = 4

N(sex2 > .5) = 4

obs = 8

N(runs) = 2

z = -2.29

Prob>|z| = .02

. runtest sex3

N(sex3 <= .5) = 4

N(sex3 > .5) = 4

obs = 8

N(runs) = 6

z = .76

Prob>|z| = .45

ก �'(�� '(�� -%��ก��=E+�=� �� (+ ��* )ก �� * (�� +��ก�#� ก �#$ก#$�#))��=ก�� #=�=��+�� ก +-%��&��* )ก �� * + ��ก� +* )��ก�-%��&��$V +�+�� ก

��#<��=ก��%<ก%< >?��'��-A/B�<��+�<�ก.B��C&��(� D. su chol,de

chol

-------------------------------------------------------------

Percentiles Smallest

1% 150 150

5% 150 155

10% 152.5 155 Obs 10

25% 155 160 Sum of Wgt. 10

50% 165 Mean 169

Largest Std. Dev. 16.46545

. su hei,de

hei

-------------------------------------------------------------

Percentiles Smallest

1% 150 150

5% 150 160

10% 155 162 Obs 10

25% 162 163.5 Sum of Wgt. 10

50% 165 Mean 177.55

Largest Std. Dev. 30.15925

chol hei

155 150.0

155 163.5

150 170.0

200 210.0

180 180.0

170 165.0

160 165.0

160 162.0

170 160.0

190 250.0

. swilk chol hei

Shapiro-Wilk W test for normal data

Variable | Obs W V z Prob>z

-------------+-------------------------------------------------

chol | 10 0.92386 1.173 0.279 0.39027

hei | 10 0.75492 3.777 2.644 0.00410

. pwcorr chol hei, sig

| chol hei

-------------+------------------

chol | 1.0000

|

|

hei | 0.7838 1.0000

| 0.0073

|

. spearman chol hei

Number of obs = 10

Spearman's rho = 0.5890

Test of Ho: chol and hei are independent

Prob > |t| = 0.0732

��$�) %��&��)� � �#=� Cholesterol =ก��#� hei ��ก �#$ก#$��=ก��

Paired t-test Wilcoxon Match paired sign rank testindependent Wilcoxon rank sum testt-test (Mann-Whithany test)Pearson Spearman rankCorrlelation Correlationone-way Kruskal-WallisANOVA ANOVA

ก$%&'()*(++,-./$0

1230 45$+6-ก-3$7%8953$7ก:,3+Parametric : <2=43$1>:?@ANonparametric: <2=43$+BCA/$01DE043$&?@ <2=<0ก$%&'()*1230ก$%&'()*45$+6-ก-3$7F)7%8'B*cholesterol %8953$7 sex

ก$%&'()*(++,-./$0T-test H0 : 43$1>:?@AF)7%8'B* cholesterol 1JK2$A6:89L.7M+36-ก-3$7กB0 H0 : 43$1>:?@AF)7%8'B* cholesterol 1JK2$A6:89L.76-ก-3$7กB0Wilcoxon Rank sum Test H0 : +BCA/$0F)7%8'B* cholesterol 1JK2$A6:89L.7M+36-ก-3$7กB0 H0 : +BCA/$0F)7%8'B* cholesterol 1JK2$A6:89L.76-ก-3$7กB0

4O$05P43$: <2=:O$'B*&?@<0ก$%4O$05P ก%P?Q:%5+ rank 1&3$R กB0 1230 %8'B* cholesterol %8953$71JK2$A 1JK9L.7 1JK2$A 1JK9L.7 200 (1) 220 (2) 230 (4) 225 (3) 255 (6) 240 (5) %5+ 11 10Danial W.W. 1978 (applied nonparametic statistics)

0O$MD<2=4O$05P p-value -3)MD

Two-sample Wilcoxon rank-sum (Mann-Whitney) test

sex | obs rank sum expected

---------+---------------------------------

1 | 3 11 10.5

2 | 3 10 10.5

---------+---------------------------------

combined | 6 21 21

unadjusted variance 5.25

adjustment for ties 0.00

----------

adjusted variance 5.25

Ho: chol(sex==1) = chol(sex==2)

z = 0.218

Prob > |z| = 0.8273

�� Rank�ก� ��ก��

�� p-value

4O$05P43$: <2=:O$'B*&?@<0ก$%4O$05P ก%P?Q:%5+6-ก-3$7กB0 1230 %8'B* cholesterol %8953$71JK2$A 1JK9L.7 1JK2$A 1JK9L.7 250 (5) 195 (1) 260 (6) 200 (2) 270 (7) 205 (3)

280 (8) 210 (4) %5+ 26 10

0O$MD<2=4O$05P p-value -3)MD



---------+---------------------------------

1 | 4 26 18

2 | 4 10 18

---------+---------------------------------

combined | 8 36 36



----------



z = 2.309

Prob > |z| = 0.0209

�� Rank ��ก��

(_.-. Wilcoxon sign rank test F=)+̀:2,'1'?A5

�� T = minimum(T+, T-)

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nnTZ

ก��

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T+T- rankdi yi xi

T- = 55, T+ = 0

T=minimum(55, 0) = 0

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-4.5 4.5 -15120135

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T+T- rankdi yi xi

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80.2

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∑ ∑−−++

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ก��

�� : ก�"�%��&� 2 (K*Mann-Whitney Test

'(��*�)%��&� 2 (K* ��=E+��ก +

�� กก ��*�)�� <�[ +%�%��&� 2 ก�K�� ก + ��%�

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2

1)2

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n

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(n1

n

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1n

minu

Mann-Whitney Test��

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��

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2

1)n(nTu 1

+−=

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++=

�� ก��ก� ��ก�� cholesterol ��!��"#$%�&�!'(� ��

%8'B* cholesterol %8953$71JK2$A 1JK9L.7 1JK2$A 1JK9L.7 200 220 230 225 255 240

ก ��*�) (manual)1. � R��K��[ +2. '��V * )��%��&� $ ก+��=� ก �V * )�� ก +'��V * )��`��3. � ��%��V * )�� R� 2 ก�K�4. �V +�"� %�� Mann-Whitney

n1 = $V +�+(K*%��&�(K*�� 1n2 = $V +�+(K*%��&�(K*�� 2�� n1 < n2

5. �=g*� � �� p-value /��K=��

Mann-Whitney U Test

ก��ก)!��!* M1 "-.��/%0��*��&$1�234 1 M2 "-.��/%0��*��&$1�234 21. �� !"�# Two Tailed Ho : M1 = M2

HA : M1 < M2

2. ก%�&#��'��#��%�(�) 0.05

-<9=:O$'B*&?@ /%5+Q:%5+F)7 rank %8'B* cholesterol %8953$71JK2$A 1JK9L.7 1JK2$A 1JK9L.7 200 (1) 220 (2) 230 (4) 225 (3) 255 (6) 240 (5)

%5+ 11 10

Mann-Whitney U Test(%�#�,(��-!�!

−+

+

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n

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5

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102

1)3(33(3)

112

1)3(33(3)

minu

2

1)n(nTu 1

+−=

410 =+

−=2

1)3(3u

./0�� k .�23� n1=n2=3 ; p-value >.10 [non significant]

.ranksum chol, by(sex)



---------+---------------------------------

1 | 3 11 10.5

2 | 3 10 10.5

---------+---------------------------------

combined | 6 21 21



----------



z = 0.218

Prob > |z| = 0.8273

+BCA/$0%8'B* cholesterol %8953$71JK2$A6:81JK9L.7M+36-ก-3$7กB0 (p=.8273)

Wilcoxon Match paired sign rank test!"#$%��&�'��($�( $�)� 2 +,"!�-��#��.��&/ก�� -Pretest-Post test (repeated measure) -Twins, litter mates -match pair

Wilcoxon Matched-pair Signed rank Test(̀-% T = minimum(T+ or T- ) T+ = Q:%5+F)7:O$'B*&?@&?@+?14%e@)79+$A*5ก

T- = Q:%5+F)7:O$'B*&?@&?@+?14%e@)79+$A:*

1)/241)(2nn(n

4

1)n(n-T

z++

+

=

48

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24

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∑ ∑−−++

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ttnnn

nnTZ

ก��

ก��!� Wilcoxon sum rank test ��!��A��!ก.�

Wilcoxon Matched-pair Signedrank Test-B5)A3$7 ก��?@กA��B-$�ก��C�,"#)%%,C��-�� $�� FEV1 � ก ��ก�%ก�$��Hก#)%C�B$I�� ??@กA��) #)%%,C��-KL�� 10 �� "��FEV1 2 ��N� ��!�-�� K#��OP�.�$" ��QP��C��K�ก�� K��N��ก6 �"B$�

idno fev1 fev1p1 83 802 76 763 80 774 76 745 75 736 78 707 77 728 85 799 80 7510 77 71

ก ��*�) (manual)1. � R��K��[ +2. � �� %�%��&� 2 (K* (di)2. '��V * )�� (di) $ ก+��=� ก �V * )�� ก +'��V * )��`��

3. � ��%��V * )�� (T+ or T-)4. �V +�"� %��5. �=g*� � �� p-value /��K=��

idno fev1p fev1 di T- T+1 80 83 3 - 3.52 76 76 0 - ��3 77 80 3 - 3.54 74 76 2 - 1.55 73 75 2 - 1.56 70 78 8 - 97 72 77 5 - 5.58 79 85 6 - 7.59 75 80 5 - 5.510 71 77 6 - 7.5� ! 0 45

1. (++,-./$0 H0 : Mdi = 0 HA : Mdi < 02. กO$90'%8'B* 0BA(O$4BL 0.053. (̀-%T = min(T+ or T- ) = min(0, 45) = 0

4. P-Value 1Dh'-$%$7 T=0,n = 9 ; p-value<.0001.signrank fev1= fev1pWilcoxon signed-rank test

sign | obs sum ranks expected

---------+---------------------------------

positive | 9 54 27

negative | 0 0 27

zero | 1 1 1

---------+---------------------------------

all | 10 55 55


adjustment for ties -0.50

adjustment for zeros -0.25

----------


Ho: fev1 = fev1p

z = 2.763

Prob > |z| = 0.0057

+BCA/$0%8'B* FEV1 ก3)06-ก-3$7กB*9:B79A,'(`**,9%?@ )A3$7+?0BA(O$4BL&$7(_.-. (p-value = 0.0057)

Kruskal-Wallis ANOVA&'()*45$+6-ก-3$743$+BCA/$0F)7F=)+:̀ >22,' &?@1DE0).(%8-3)กB0

-ordinal scale-Interval/ratio Scale (non normal)-Unequal Variance

ก ��*�) (manual)1. � R��K��[ +2. '��V * )��%��&� $ ก+��=� ก �V * )�� ก +'��V * )��`��3. � ��%��V * )��Kกก�K� [R]4. �V +�"� %��

5. �=g*� � �� p-value /��K=�� k >3 [or] ni >5 '(�� Chi-Square

∑=

+−+

=k

1i

1)3(N

in

2

iR

1)N(N

12H

�� ก��ก� ��ก�� cholesterol ��!��$3#��@ ��

%8'B* cholesterol F)7)$2?J-3$7R 1กp-%ก%%+ %B*%$2ก$% 4=$F$A

200 240 250 300 270 280 210 260 290

-<9=:O$'B*&?@ /%5+Q:%5+F)7 rank

%8'B* cholesterol F)7)$2?J-3$7R 1กp-%ก%%+ %B*%$2ก$% 4=$F$A 200 (1) 240 (3) 250 (4) 300 (9) 270 (6) 280 (7) 210 (2) 260 (5) 290 (8)

%5+ 12 14 19

(%�#�,(��-!�!

∑=

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3

14

3

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1)9(9

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222

++++

=

= 1.15 ./0�� H=1.15 n1=n2=n3=3 ;p-value>0.10+BCA/$0F)7%8'B* cholesterol %8953$7ก:,3+)$2?J-3$7R M+36-ก-3$7กB0

. kwallis chol,by(occ)Test: Equality of populations (Kruskal-Wallis Test)

occ _Obs _RankSum

1 3 12.00

2 3 14.00

3 3 19.00

chi-squared = 1.156 with 2 d.f.

probability = 0.5611

_=$ก$%5.14%$89q Kruskal-Wallis ANOVA Q:ก$%&'()* 6-ก-3$7 <9=&'()*45$+ 6-ก-3$7'=5A Multiple comparison-1230 Dunn , Wilcoxon ranksum test r'A&'()*&?@:84̀3

0.05

Chi-Square disribution d.f. = 2

Critical Values of Chi-square

df .50 .25 .10 .05 .025 .01 .001

1 0.45 1.32 2.71 3.84 5.02 6.63 10.83

2 1.39 2.77 4.61 5.99 7.38 9.21 13.82

3 2.37 4.11 6.25 7.81 9.35 11.34 16.27

...

100 99.33 109.14 118.50 124.34 129.56 135.81 149.45

EF!0F��=0 ?&'�C� %&' (� != 0 !=�� ก1=p-value A/�� 0��ก1=

k =<I��#�ก&-��

$��กI�/�F k= 3 alpha=.15

2

k

α

05.

)!23(!2

!3

15.0;

α;15.0 =

−

=

2

3α

Multiple comparison

�� ก��ก� ��ก�� cholesterol ��!��$3#��@ ��

%8'B* cholesterol F)7)$2?J-3$7R 1กp-%ก%%+ %B*%$2ก$% 4=$F$A

200 240 300 250 270 280 210 260 290

. kwallis chol,by(gr)

Test: Equality of populations (Kruskal-Wallis test)

+---------------------+

| gr | Obs | Rank Sum |

|----+-----+----------|

| 1 | 3 | 7.00 |

| 2 | 3 | 14.00 |

| 3 | 3 | 24.00 |

+---------------------+

chi-squared = 6.489 with 2 d.f.


chi-squared with ties = 6.489 with 2 d.f.


. ranksum chol if (occ==1 | occ==2) , by(occ)


occ | obs rank sum expected

-------------+---------------------------------

1 | 3 7 10.5

2 | 3 14 10.5

-------------+---------------------------------

combined | 6 21 21



----------


Ho: chol(occ==1) = chol(occ==2)

z = -1.528

Prob > |z| = 0.1266




-------------+---------------------------------

1 | 3 6 10.5

3 | 3 15 10.5

-------------+---------------------------------

combined | 6 21 21



----------



z = -1.964

Prob > |z| = 0.0495




-------------+---------------------------------

2 | 3 6 10.5

3 | 3 15 10.5

-------------+---------------------------------

combined | 6 21 21



----------



z = -1.964

Prob > |z| = 0.0495

�� Spearman Rank correlation-#1F�#��1�1�,��'/#��1#%(� 2 �1#%(�-ordinal scale, interval & ratio scale(non-normal)

yxi

d

1)2n(n

2i

d61

sr

−=

−

∑−=

�� Spearman Rank correlationก�+ � ��UDI�ก1�

∑ ∑

∑ ∑ ∑−+=

22

22

2 yx

dy i2x

sr

∑ ∑−−

= xTnn

x12

32

∑ ∑−−

= yTnn

y12

32

12

3tt

T ii

−=∑

�>?� i = x, y AFX

�1#�!�� A/B#� ��'/��#��1�1�,��'/#��'F1=�#�� !FA�ก��0I��ก1=�'F1=�#��Y�A<A�ก��0I�� stress sat

1 52 43 34 25 2

ก��0F��= (manual)1. �1D��-��Z��2. กI�/�F�'F1=�1!�I��1[3. A/B&I�F1=0 ?.B��C&<�ก�B�!�(��กA�%��&'ก&-�� &I�F1=0 ? 0��ก1�A/B&I�F1=0 ? \& ?!3. /��#��%�ก��'/#�� rank dj = R(x)-(R(y)4. �I��#+��.��5. (_F��/� p-value /��-(`&

30;2,1

2

30;

2≤−=

−

−=

>=

nndfr

nrt

n

s

1-nrZ s

0ρ:H0;ρ:0H s0s ≠=

stress sat rx ry d d21 5 1 5 -4 162 4 2 4 -2 43 3 3 3 0 04 2 4 1.5 2.5 6.255 2 5 1.5 3.5 12.25

∑ ∑

∑ ∑ ∑−+=

22

22

2 yx

dy i2x

sr

5.912

22

12

55 332 =

−−

−=∑ x 10

12

5532 =

−=∑ y

12

3tt

T ii

−=

∑ ∑

∑ ∑ ∑−+=

22

22

2 yx

dy i2x

sr

974679.)10(5.92

5.38105.9−=

−+=

sr

(_F��/� p-value <.005

2,1

22

−=−

−= ndf

r

nrt

s

25,167.6)9747.(1

259747.

2−=−=

−−

−−= dft

. spearman stress sat

Number of obs = 5

Spearman's rho = -0.9747

Test of Ho: stress and sat are independent

Prob > |t| = 0.0048

��-(`& �#�� !F� �#��1�1�,�0�&=ก1=�#��Y�A<A�� !�� 1!�I��1[0�� (rs = -.9747,p-value =.0048)

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