BIOSTATISTICS

Statistical tests part IV: nonparametric tests

1. Mann-Whitney test

2. Wilcoxon test

3. Kruskal-Wallis test

• Applicability

• Definition

• Example

Copyright ©2012, Joanna Szyda

INTRODUCTION

SAMPLE STRUCTUREHYPOTHESES

TEST

Copyright ©2011, Joanna Szyda

INTRODUCTION

MANN-WHITNEY TEST

MANN-WHITNEY TEST

1. Comparison of means

2. Quantitative or ordered data (ranks)

3. No normal distribution required

4. Two independent samples

Copyright ©2012, Joanna Szyda

Copyright ©2012, Joanna Szyda

DATA SET

1. Shrimp length in different water salinities

3. Shrimp length [mm] at 4 weeks of age

MEDIUM HIGH5.5 6.06.0 7.05.0 7.57.0 6.05.5 7.56.0 8.07.0 11.06.0 9.08.0 8.07.0 11.06.0 8.07.0 8.06.0 7.08.0 7.06.0 7.07.0 9.0

02468

10

1 2 3 4 5 6 7 8 9

N

LENGTH

02468

1 2 3 4 5 6 7 8 9 10 11

N

LENGTH

MANN-WHITNEY TEST

Copyright ©20112 Joanna Szyda

1. Formulate hypotheses H0 and H1

H0: shrimp length does not depend on water salinity H1: shrimp length depends on water salinity H0: H = M H1: H ≠ M

2. Set the significance levelMAX = 0.05

3. Choose the statistical test and calculate test value

2 1

1 11

11212

2221 2

1,21min

n

i

n

iii rnnnnrnnnnU

Excel: exampleArkusz programu Microsoft Office Excel

MANN-WHITNEY TEST

Copyright ©2011, Joanna Szyda

3. Choose the statistical test and calculate test value

46210,46min

3462171616*16,182

2171616*16min

21,

21min

2 1

1 11

11212

2221

U

rnnnnrnnnnUn

i

n

iii

MANN-WHITNEY TEST

Copyright ©2011, Joanna Szyda

4. Determine distribution of the test• Nonparametric test – no known distribution• For n1n2 > 20 – approximated by a normal distribution:

1,0~

121

2

,~

2121

21

2

2

Nnnnn

nnUUz

NU

U

U

UU

no tables

tables

MANN-WHITNEY TEST

Copyright ©2011, Joanna Szyda

4. Determine distribution of the test

5. Determine t:

002.0t

1,0~09.3

123316*16216*1646

121

22121

21

Nnnnn

nnUz

Excel: exampleArkusz programu Microsoft Office Excel

or compare with a critical value:4618116,16,05.0 21

tnn UU

MANN-WHITNEY TEST

Copyright ©2012, Joanna Szyda

6. Decisiont < max Ut < U

H0 H1

shrimp length depends on water salinity

ATTENTION !!!

MANN-WHITNEY TEST

WILCOXON TEST

Copyright ©2011, Joanna Szyda

WILCOXON TEST

1. Nonparamteric test

2. Quantitative or ordered data (ranks)

3. No normal distribution required

4. Comparison of two paired samples

Copyright ©2011, Joanna Szyda

1. Feeding behaviour of sheep2. Data collected in1994-1996

in Canada, Rocky Mountains region

3. Differences in time of feeding with / without lamb

4. % time spent on feeding

DATA SETIID NO LAMB LAMB

1 72.00 55.502 62.35 43.803 55.77 66.804 59.98 68.005 51.60 57.886 61.48 61.907 52.57 45.408 52.50 56.679 56.43 73.30

10 60.13 77.5011 48.60 63.5312 42.90 54.5013 53.50 55.5814 70.43 91.1015 47.10 64.0516 50.08 71.40

WILCOXON TEST

0

1

2

3

-25 -20 -15 -10 -5 0 5 10 15 20

N

DIFFERENCE IN TIMECopyright ©2011, Joanna Szyda

DATA SETIID NO LAMB LAMB

1 72.00 55.502 62.35 43.803 55.77 66.804 59.98 68.005 51.60 57.886 61.48 61.907 52.57 45.408 52.50 56.679 56.43 73.30

10 60.13 77.5011 48.60 63.5312 42.90 54.5013 53.50 55.5814 70.43 91.1015 47.10 64.0516 50.08 71.40

WILCOXON TEST

Copyright ©2011, Joanna Szyda

1. Formulate hypotheses H0 and H1

H0: feeding time does not depend on a lamb H1: feeding time depends on a lamb

H0: J = B H1: J ≠ B

2. Set the significance levelMAX = 0.05

3. Choose the statistical test and calculate test value

1

1

1

,min1

nN

nii

n

ii rrW

Excel: exampleArkusz programu Microsoft Office Excel

WILCOXON TEST

Copyright ©2011, Joanna Szyda

3. Choose the statistical test and calculate test value

2929,107min,min1

1

1

1

nN

nii

n

ii rrW

WILCOXON TEST

Copyright ©2011, Joanna Szyda

4. Determine distribution of the test• Nonparametric test – no known distribution• For N > 15 – approximated by a normal distribution:

1,0~02.2

2433*17*16417*1629

1,0~

24121

41

1,0~

,~

2

2

Nz

NNNN

NNWz

NWz

NW

W

W

WW

WILCOXON TEST

Copyright ©2011, Joanna Szyda

5. Determine t:

0437.0t

or compare with a critical value :292916,05.0 tN WW

Excel: exampleArkusz programu Microsoft Office Excel

6. Decisiont < max Wt = W H0 H1

?

feeding time depends on a lamb ?

WILCOXON TEST

KRUSKAL-WALLIS TEST

KRUSKAL-WALLIS TEST

1. Comparing variability

2. Quantitative or ordered data (ranks)

3. No normal distribution required

4. Analysis of variance

Copyright ©2012, Joanna Szyda

Copyright ©2011, Joanna Szyda

DATA SETKRUSKAL-WALLIS TEST

1. Height of adult women in the USA2. Three age groups

20-29 30-39 40-49

161.925 164.465 173.990

173.355 171.450 175.260

158.115 173.355 167.640

170.815 175.260 166.370

179.705 164.465 168.910

Copyright ©2011, Joanna Szyda

1. Formulate hypotheses H0 and H1

H0: women's height is the same in each age interval H1: women's height differs across age intervals

H0: H1:

2. Set the significance level MAX = 0.05

3. Choose the statistical test and calculate test value

KRUSKAL-WALLIS TEST

22eage 22eage

21

2

1

~1

12

A

A

Ni

N

ii RRn

NNH

total no of observations

number of groups

mean ranking within group i

mean overall rankingR

R

N

N

i

A

Copyright ©2011, Joanna Szyda

KRUSKAL-WALLIS TEST

3. Choose the statistical test and calculate test value

4. Determine distribution of the test: 213~

5. Determine t: 0398.0t

6. Decision: t < max H0

H1

women's height differs across age intervals

45.688588582.7511515

121

12 2222

1

RRnNN

H i

N

ii

A

Excel: exampleArkusz programu Microsoft Office Excel

Copyright ©2012 Joanna Szyda

NONPARAMETRIC TESTS

xs