24
1 Nonparametric Statistics . Web: http://home.kku.ac.th/nikom Email: [email protected] : - - -Interval/Ratio Scale Normal

nonpa nk 2551 - home.kku.ac.th16 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

  • Upload
    others

  • View
    2

  • Download
    0

Embed Size (px)

Citation preview

Page 1: nonpa nk 2551 - home.kku.ac.th16 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

1

��������������� ����Nonparametric Statistics

��. ���� ���� �� !��"�#�#�"�����$%&'�&#�(�)�����

�*&��+��*�,-)����� �.�"�/ �%0 -��$(��Web: http://home.kku.ac.th/nikom

Email: [email protected]

��������������� �����������

��������� ���������: ������-������������������� !"���#$�#$�%��&�-��������'(���)%��&����������*�"*�) +�������+*�)����-Interval/Ratio Scale %��&�� Normal

Page 2: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 3: nonpa nk 2551 - home.kku.ac.th16 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

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�+�+���

Page 4: nonpa nk 2551 - home.kku.ac.th16 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

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

���$��) %��&� )�� ���#<� Cholesterol <��� #� hei ����#$�#$��<���

Page 5: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 6: nonpa nk 2551 - home.kku.ac.th16 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

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\

Page 7: nonpa nk 2551 - home.kku.ac.th16 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

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\

Page 8: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 9: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 10: nonpa nk 2551 - home.kku.ac.th16 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

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

���������

Page 11: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 12: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 13: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 14: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 15: nonpa nk 2551 - home.kku.ac.th16 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

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)

Page 16: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 17: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 18: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 19: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 20: nonpa nk 2551 - home.kku.ac.th16 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

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=

Page 21: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 22: nonpa nk 2551 - home.kku.ac.th16 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

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

+=

Page 23: nonpa nk 2551 - home.kku.ac.th16 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

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

Page 24: nonpa nk 2551 - home.kku.ac.th16 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

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

.