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��������������� ����Nonparametric Statistics
��. ���� ���� �� !��"�#�#�"�����$%&'�&#�(�)�����
�*&��+��*�,-)����� �.�"�/ �%0 -��$(��Web: http://home.kku.ac.th/nikom
Email: [email protected]
��������������� �����������
��������� ���������: ������-������������������� !"���#$�#$�%��&�-��������'(���)%��&����������*�"*�) +�������+*�)����-Interval/Ratio Scale %��&�� Normal
2
������������������������������ ��������������
- ��������� !"�������� ���������� ������ #��$�� (truly nonparametric procedures)�<=+��>����%�������������� ��������� �����*��+'$�� ���������%��<�"(����(+����*��) ����*��) run, goodness of fit
-�������#$�#$#������� (distribution-free procedures)
�������� Run Test '(�'+����*��)����E (randomness) �(+����H� �� �%���&�<I������%����)������$
����H��&�<I��+�����%��&���&<#))��� !" M F M F M F M F #�*��&<#))%��&�#))�� (mixed) ��M� M M M M F F F F �<=+�&<#))��E (cluster)NH����� !"��O�����'(�&<#))������*P*��E
H0 : ���������S�*�)%��&��<=+��� !"�EHA : ���������S�*�)%��&���<=+��� !"�E
3
. 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
���!"���������������� ����!"���������������� ���� �� ���-%��������<=+�<���������� ����������(+ ��"*�)�����* (�� �������+����#�
���#$�#$�#))�<��� ���#<�<��+������+
-%��&���"*�)�����* +����� ��+*�)����-%��&��$S�+�+���
4
�������������� �������-��� ������������������� . su chol,de
chol
-------------------------------------------------------------
Percentiles Smallest
1% 150 150
5% 150 155
10% 152.5 155 Obs 10
25% 155 160 Sum of Wgt. 10
50% 165 Mean 169
Largest Std. Dev. 16.46545
. su hei,de
hei
-------------------------------------------------------------
Percentiles Smallest
1% 150 150
5% 150 160
10% 155 162 Obs 10
25% 162 163.5 Sum of Wgt. 10
50% 165 Mean 177.55
Largest Std. Dev. 30.15925
chol hei
155 150.0
155 163.5
150 170.0
200 210.0
180 180.0
170 165.0
160 165.0
160 162.0
170 160.0
190 250.0
. swilk chol hei
Shapiro-Wilk W test for normal data
Variable | Obs W V z Prob>z
-------------+-------------------------------------------------
chol | 10 0.92386 1.173 0.279 0.39027
hei | 10 0.75492 3.777 2.644 0.00410
. pwcorr chol hei, sig
| chol hei
-------------+------------------
chol | 1.0000
|
|
hei | 0.7838 1.0000
| 0.0073
|
. spearman chol hei
Number of obs = 10
Spearman's rho = 0.5890
Test of Ho: chol and hei are independent
Prob > |t| = 0.0732
���$��) %��&� )�� ���#<� Cholesterol <��� #� hei ����#$�#$��<���
5
Paired t-test Wilcoxon Match pairedsign rank test
independent Wilcoxon rank sum testt-test (Mann-Whithany test)Pearson Spearman rankCorrlelation Correlationone-way Kruskal-WallisANOVA ANOVA
#$%&'()*(++,-./$0
1230 45$+6-#-3$7%8953$7#:,3+Parametric : �� ���������Nonparametric: �� ����������� !����"��
�� ��#�$"%&'(����#�$"%&'(�)��*+#+��,-',$.%�(cholesterol $.2)��, sex
6
#$%&'()*(++,-./$0T-test H0 : 43$1?:@ABC)7%8'D* cholesterol
1EF2$B6:89G.7H+36-#-3$7#D0H0 : 43$1?:@ABC)7%8'D* cholesterol
1EF2$B6:89G.76-#-3$7#D0Wilcoxon Rank sum Test
H0 : +DJB/$0C)7%8'D* cholesterol 1EF2$B6:89G.7H+36-#-3$7#D0
H0 : +DJB/$0C)7%8'D* cholesterol 1EF2$B6:89G.76-#-3$7#D0
4K$05L43$: M2N:K$'D*&@AM0#$%4K$05L #%L@O:%5+ rank 1&3$P #D0 1230%8'D* cholesterol %8953$71EF2$B 1EF9G.7
1EF2$B 1EF9G.7200 (1) 220 (2)230 (4) 225 (3)255 (6) 240 (5)
%5+ 11 10Danial W.W. 1978 (applied nonparametic statistics)
0K$H\M2N4K$05L p-value -3)H\
7
Two-sample Wilcoxon rank-sum (Mann-Whitney) test
sex | obs rank sum expected
---------+---------------------------------
1 | 3 11 10.5
2 | 3 10 10.5
---------+---------------------------------
combined | 6 21 21
unadjusted variance 5.25
adjustment for ties 0.00
----------
adjusted variance 5.25
Ho: chol(sex==1) = chol(sex==2)
z = 0.218
Prob > |z| = 0.8273
4�$)� Rank�#� ����,#��
* �,� !���� p-value
4K$05L43$: M2N:K$'D*&@AM0#$%4K$05L#%L@O:%5+6-#-3$7#D0 1230%8'D* cholesterol %8953$71EF2$B 1EF9G.7
1EF2$B 1EF9G.7250 (5) 195 (1)260 (6) 200 (2)270 (7) 205 (3)280 (8) 210 (4)
%5+ 26 10
0K$H\M2N4K$05L p-value -3)H\
8
Two-sample Wilcoxon rank-sum (Mann-Whitney) test
sex | obs rank sum expected
---------+---------------------------------
1 | 4 26 18
2 | 4 10 18
---------+---------------------------------
combined | 8 36 36
unadjusted variance 12.00
adjustment for ties 0.00
----------
adjusted variance 12.00
Ho: chol(sex==1) = chol(sex==2)
z = 2.309
Prob > |z| = 0.0209
4�$)� Rank+��,#��
( .-. Wilcoxon sign rank test CN)+_:2,'1'@B5
�<�� T = minimum(T+, T-)
"�+��=��"*- .��"��$�(����&."�����->��<%(0?���/�@(=�.�A- �� %=�A0?->��<%B�(�>� �'��(- C.>%=�A0?/�@(0?->��<% (->��<%EF=� C.>C#>%=�A0?/�@ G%�@ )
24/)12)(1(
4/)1(
+++−=nnn
nnTZ
9
48
3
24
)12)(1(
4/)1(
∑ ∑−−++
+−=ttnnn
nnTZ
���������
����� ��������� sysbp ���������������� �������!�� 40 $% bmi 20-25 �����+,�,��-�.� 120 mmHg 2�3 4��
-77-20120140
-4.54.5-15120135
-4.54.5-15120135
-1010-45120165
-99-39120159
-33-13120133
-11-10120130
-22-11120131
-88-31120151
-66-16120136
T+T-rankdi
yi
xi
10
T- = 55, T+ = 0
T=minimum(55, 0) = 0
-77-20120140
-4.54.5-15120135
-4.54.5-15120135
-1010-45120165
-99-39120159
-33-13120133
-11-10120130
-22-11120131
-88-31120151
-66-16120136
T+T-rankdi
yi
xi
48
3
24
)12)(1(
4/)1(
∑ ∑−−++
+−=
ttnnn
nnTZ
80.2
48
23
2
24
)1)10(2)(110(10
4/)110(100=
∑ ∑−−++
+−=Z
���������
11
��������������� ����: ��;�����<= 2 "?�Mann-Whitney Test'(��*��)%��&� 2 (E* ����<=+����"����+
@=A�������������>�[�+%��%��&� 2 ��E�����+ ����%���S�*�)���
'+#��"��E (�M��'���S�*�)��� (Rank) %��&�����+) $"����#�������+
−+
+
−+
+=
2T
2
1)2
(n2
n
2n
1n
1T
2
1)1
(n1
n
2n
1n
minu
Mann-Whitney Test����
n1 = aK$050-D5)B3$7C)743$(D71#-&@A+@aK$0500N)Bn2 = aK$050-D5)B3$7C)743$(D71#-&@A+@aK$050+$#T1 = O:%5+:K$'D*&@AM0#:,3+ n1T2 = O:%5+:K$'D*&@AM0#:,3+ n2
12
@���
��;��B���A������B��!@C�
2
1)n(nTu 1
+−=
1)/12n(nnn
2
nn-U
z2121
21
++=
�������� ������� �������� �������� cholesterol��!���"#$%�&�!'(� ������
%8'D* cholesterol %8953$71EF2$B 1EF9G.71EF2$B 1EF9G.7200 220230 225255 240
13
�������� (manual)1. �AF���?��G�B2. !@�=H��A��� ����<= $��B��������
=H��A��� ����AB!@�=H��A��� I=� �3. @�K=�����=H��A��� �AF 2 �=?��4. �H�B�;���������� Mann-Whitney
n1 = $H�B�B"?�����<="?��� 1n2 = $H�B�B"?�����<="?��� 2 �� � n1 < n2
5. �S�����@� p-value /��?�K=
Mann-Whitney U Test
���������)!���!* M1 "-.���/%0�����*���&$1�234 1
M2 "-.���/%0�����*���&$1�234 21. ������� �!"�#
Two Tailed Ho : M1 = M2
HA : M1 < M2
2. �%�&#��'���#���%�(�) 0.05
14
-M9N:K$'D*&@A /%5+O:%5+C)7 rank%8'D* cholesterol %8953$71EF2$B 1EF9G.7
1EF2$B 1EF9G.7200 (1) 220 (2)230 (4) 225 (3)255 (6) 240 (5)
%5+ 11 10
Mann-Whitney U Test(%�#�,(������-!�!
−+
+
−+
+=
2T
2
1)2
(n2
n
2n
1n
1T
2
1)1
(n1
n
2n
1n
minu
=
−+
+
−+
+=
5
4
102
1)3(33(3)
112
1)3(33(3)
minu
2
1)n(nTu 1
+−=
410 =+
−=2
1)3(3u
15
./0������ k .�23� n1=n2=3 ; p-value >.10 [non significant]
.ranksum chol, by(sex)
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
+DJB/$0%8'D* cholesterol %8953$71EF2$B6:81EF9G.7H+36-#-3$7#D0 (p=.8273)
16
cholmale cholfemale
200 1 300 2
210 1 310 2
220 1 320 2
350 1 330 2
230 1 340 2
�������� ������� �������� �������� cholesterol��!���"#$%�&�!'(� ������
Confidence interval for difference between median (independent)
300 310 320 330 340
200 -100 -110 -120 -130 -140
210 -90 -100 -110 -120 -130
220 -80 -90 -100 -110 -120
350 50 40 30 20 10
230 -70 -80 -90 -100 -110
Confidence interval for difference between median (independent)
-4K$05045$+6-#-3$7CN)+_: 2 2,'-9$-K$69037235712eA)+DA0
( )12
)1(
2
21212/1
21 ++−= −
nnnnz
nnk α
17
( ) 311.312
)155)(5(596.1
2
)5(5≅=
++−=k
-9$-K$69037235712eA)+DA0&@A 95% +@43$&@A-K$69037 (k,n+1-k)3 6:8 (25+1)-3 = 23 1&3$#D* -130 i7 30
-140 -120 -100 -90 10
-130 -110 -100 -90 20
-130 -110 -100 -80 30
-120 -110 -100 -80 40
-120 -110 -90 -70 50
1%@B7:K$'D*CN)+_:45$+6-#-3$7 a$#0N)BH\+$#
#$%4K$050 95% CI Mann-Whitney�� �� � STATA: cid variable,by(group var) unpaired median
. cid chol,by(gr) unpaired median
Rank-based confidence interval for difference in medians by gr
Variable | Obs Estimate K [95% Conf. Interval]
---------+-------------------------------------------------------------
chol | 10 -100 3 -130 30
cholgr cholgr
200 1 300 2
210 1 310 2
220 1 320 2
350 1 330 2
230 1 340 2
*default group1-group2
18
#$%4K$050 95% CI Hodges-Lehmann�./�� �� � STATA: npshift variable,by(group var)
. npshift chol,by(gr)
Hodges-Lehmann Estimates of Shift Parameters
-------------------------------------------------------------
Point Estimate of Shift : Theta = Pop_2 - Pop_1 = 100
95% Confidence Interval for Theta: [-30 , 130]
-------------------------------------------------------------
cholgr cholgr
200 1 300 2
210 1 310 2
220 1 320 2
350 1 330 2
230 1 340 2
*default group2-group1
Wilcoxon Match paired sign rank test/A��?����0+ I��-�->��<% 2 #,A/�@���"���0�0�+�(0�-Pretest-Post test (repeated measure) -Twins, litter mates-match pair
19
Wilcoxon Matched-pair Signed rank Test���� T = minimum(T+ or T- )
T+ = <��������=>?�@��@�����A�������B���T- = <��������=>?�@��@�����A�������B��
1)/241)(2nn(n
4
1)n(n-T
z++
+
=
48
3
24
)12)(1(
4/)1(
∑ ∑−−++
+−=
ttnnn
nnTZ
���������
�2 342567 Wilcoxon sum rank test 34256589:58.;/6�<5
20
Wilcoxon Matched-pair Signed rank Test-D5)B3$7 #�$:;#<��$=�',#�$2�>%&?((>2$����4�+�'��� FEV1 *+#+��,#�(#�'���D#&?(2$='E�� ?:;#<���4? &?((>2$��GH��)� 10 �� )�%��� FEV1 2 �$�K, )��"����+$)G&�$$LM�N '% *�.M��2��,G�#��+$)G�$�K,*$# 6 �%='�
idno fev1 fev1p1 83 802 76 763 80 774 76 745 75 736 78 707 77 728 85 799 80 7510 77 71
�������� (manual)1. �AF���?��G�B2. @�K=���������<= 2 "?� (di)2. !@�=H��A��� (di) $��B��������
=H��A��� ����AB!@�=H��A��� I=� � 3. @�K=�����=H��A��� (T+ or T-)4. �H�B�;���������� 5. �S�����@� p-value /��?�K=
21
idno fev1 fev1p di T- T+1 83 80 3 - 3.52 76 76 0 - ?<.@ABC 3 80 77 3 - 3.54 76 74 2 - 1.55 75 73 2 - 1.56 78 70 8 - 97 77 72 5 - 5.58 85 79 6 - 7.59 80 75 5 - 5.510 77 71 6 - 7.5 6� 0 45
di= fev1-fev1p1. ���,��I��H0 : Md = 0 HA : Md 0
2. (=�.�A�&A0?�0 �=��0Z 0.05
3. �<��T = min(T- or T+)= min(0, 45)= 0
≠
4. P-Value 1\k'-$%$7 T=0,n = 9 ; p-value<.0001.signrank fev1= fev1pWilcoxon signed-rank test
sign | obs sum ranks expected
---------+---------------------------------
positive | 9 54 27
negative | 0 0 27
zero | 1 1 1
---------+---------------------------------
all | 10 55 55
unadjusted variance 96.25
adjustment for ties -0.50
adjustment for zeros -0.25
----------
adjusted variance 95.50
Ho: fev1 = fev1p
z = 2.763
Prob > |z| = 0.0057
22
�0+ I���&A0? FEV1 (���$�(���(0?.%0. ,A�<??,.��@ � �����0 �=��0Z/������ (Wilcoxon matched-pair sign rank testp-value = 0.0057)
Confidence interval for difference between median (paired)-4K$05045$+6-#-3$7CN)+_: 2 2,',0K$+$9$43$1?:@AB -9$-K$69037235712eA)+DA0
( )24
)12)(1(
4
)1(2/1
++−
+= −
nnnz
nnk α
d 3 0 3 2 2 8 5 6 5 6
3 3 1.5 3 2.5 2.5 5.5 4 4.5 4 4.5
0 0 1.5 1 1 4 2.5 3 2.5 3
3 3 2.5 2.5 5.5 4 4.5 4 4.5
2 2 2 5 3.5 4 3.5 4
2 2 5 3.5 4 3.5 4
8 8 6.5 7 6.5 7
5 5 5.5 5 5.5
6 6 5.5 6
5 5 5.5
6 6
2
jiij
ddu
+=
23
( ) 82710115.824
)1)10(2)(110(1096.1
4
)110(10≅=
++−
+=k
-9$-K$69037235712eA)+DA0&@A 95% +@43$&@A-K$69037 (c,n+1-c)8 6:8 (55+1-8) = 48 1&3$#D* 2 i7 6
0 2.5 3.5 4 5 6.5
1 2.5 3.5 4 5.5 6.5
1 2.5 3.5 4.5 5.5 7
1.5 2.5 4 4.5 5.5 7
1.5 3 4 4.5 5.5 8
2 3 4 4.5 5.5
2 3 4 5 5.5
2 3 4 5 6
2.5 3 4 5 6
2.5 3.5 4 5 6
1%@B7:K$'D*CN)+_:45$+6-#-3$7a$#0N)BH\+$#
#$%4K$050 95% CI n'B5.J@ Wilcoxon'N5Bn\%6#%+ STATAcid var1 var2 ,median
. cid fev1 fev1p,median[paired data assumed]
Rank-based confidence interval for difference in paired medians
Variable | Obs Estimate K [95% Conf. Interval]
---------+-------------------------------------------------------------
fev1 | 10 4 8 2 6
24
#$%4K$050 95% CI n'B5.J@ Percentile'N5Bn\%6#%+ STATAcentile <difference>(���������� ���������� � http://home.kku.ac.th/nikom)
. gen diff = fev1 - fev1p
. centile diff
-- Binom. Interp. --
Variable | Obs Percentile Centile [95% Conf. Interval]
-------------+----------------------------------------------------------
diff | 10 50 4 2 6
.