Using Statistics To Make Inferences 6 1

A novel treatment was compared against control values.

Treated Control

236.9 400.6 403.3 509.3 562.6 664.4 747.7 795.8 962.9 1146.4 1238.9 308.5 321.5 397.8 400.4 466.2 580.0 887.1

Perform a Mann-Whitney Test to assess the efficacy of the treatment. 1

Note that the original data is sorted to aid the calculations.

Data 236.90 308.50 321.50 397.80 400.40 400.60 403.30 466.20 509.30 562.60 580.00 664.40 747.70 795.80 887.10 962.90 1146.40 1238.90

Source Treated Control Control Control Control Treated Treated Control Treated Treated Control Treated Treated Treated Control Treated Treated Treated

Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Treated 1

Total

123

Control 2 3 4 5

6 7 8 9 10 11 12 13 14 15 16 17 18 48

nT nT  1 11 (11  1)  RT  11 7   123  20.0 (W = 123) 2 2 n n  1 7  (7  1)  nT  nC  C C  RC  11 7   48  57.0 2 2

U T  nT  nC  UC

(mid-point 1 nT  nC  38.5 so only need calculate one) 2 For nT  11 , nC  7 , the critical value from the tables for p=0.025 is 16 . The result is not significant at the 5% level, the two samples do not appear to differ. For p=0.05 it is 19 , the result is not significant at the 10% level. Close to the value U (see 0.1031 in the test reported).

z

1 nT nC  2 2  1.63 nT nC nT  nC  1 12 U

Which is greater than -1.96, there does not appear to be a significant difference.

Mike Cox

6.1

Version 1

MTB > Mann-Whitney 95.0 'Treated' 'Control'; SUBC> Alternative 0. Mann-Whitney Confidence Interval and Test Treated N = 11 Median = 664.4 Control N = 7 Median = 400.4 Point estimate for ETA1-ETA2 is 198.2 95.4 Percent CI for ETA1-ETA2 is (-70.8,496.7) W = 123.0 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.1031 Cannot reject at alpha = 0.05

2

A random sample of 15 male and 20 female students had their academic ability tested. The scores are displayed below. Men

10.00 15.20 20.00

11.00 16.00

12.00 17.00

12.40 18.50

12.50 19.00

12.50 19.25

14.00 19.50

Women 15.00 18.25 20.00

16.20 18.30 21.00

16.30 18.50 21.10

17.25 18.75 23.00

17.50 19.10 24.00

17.75 19.20 25.00

18.00 19.75

From the results would you conclude that there is no difference between the two groups (The samples are from populations with similar medians)? 2

A Mann-Whitney test is appropriate, since there is no reason to assume the scores are normally distributed. The data is already sorted. The only problem is ties. Rank 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

Mike Cox

True rank 1 2 3 4 5.5 5.5 7 8 9 10 11 12 13 14 15 16 17 18 19 20.5 20.5 22 23 24 25 26 27 28 29.5

6.2

Score 10.00 11.00 12.00 12.40 12.50 12.50 14.00 15.00 15.20 16.00 16.20 16.30 17.00 17.25 17.50 17.75 18.00 18.25 18.30 18.50 18.50 18.75 19.00 19.10 19.20 19.25 19.50 19.75 20.00

Men Men Men Men Men Men Men Women Men Men Women Women Men Women Women Women Women Women Women Men Women Women Men Women Women Men Men Women Men

True rank 1 Men 2 Men 3 Men 4 Men 5.5 Men 5.5 Men 7 Men 9 Men 10 Men 13 Men 20.5 Men 23 Men 26 Men 27 Men 29.5 Men 8 Women 11 Women 12 Women 14 Women 15 Women 16 Women 17 Women 18 Women 19 Women 20.5 Women 22 Women 24 Women 25 Women 28 Women

Version 1

Sum 186

30 31 32 33 34 35

29.5 31 32 33 34 35

20.00 21.00 21.10 23.00 24.00 25.00

Women Women Women Women Women Women

29.5 31 32 33 34 35

Women Women Women Women Women Women

Sum 444

Here nM = 15, nW = 20 with SM = 186 and SW = 444 UM = nM nW + ½ nM (nM + 1) - SM = 234 and UW = nM nW + ½ nW (nW + 1) - SW = 66

U = min(UM,UW) = 66 and z 

1 nM nW  2 2  2.78 nM nW nM  nW  1 U





12 Which is less than -1.96, there appears to be a significant difference. A check employing Minitab, plus a two sample t test, just in case. ROW

Men

Women

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

10.00 11.00 12.00 12.40 12.50 12.50 14.00 15.20 16.00 17.00 18.50 19.00 19.25 19.50 20.00

15.00 16.20 16.30 17.25 17.50 17.75 18.00 18.25 18.30 18.50 18.75 19.10 19.20 19.75 20.00 21.00 21.10 23.00 24.00 25.00

Mann-Whitney Confidence Interval and Test Men N = 15 Median = 15.200 Women N = 20 Median = 18.625 Point estimate for ETA1-ETA2 is -4.000 95.3 Percent C.I. for ETA1-ETA2 is (-6.250,-1.249) W = 186.0 Test of ETA1 = ETA2 vs. ETA1 = ETA2 is significant at 0.0054 The test is significant at 0.0054 (adjusted for ties) Men Women

N 15 20

MEAN 15.257 19.198

MEDIAN 15.200 18.625

TRMEAN 15.296 19.108

Men Women

MIN 10.000 15.000

MAX 20.000 25.000

Q1 12.400 17.562

Q3 19.000 20.750

STDEV 3.442 2.581

TWOSAMPLE T FOR Men VS Women

Mike Cox

6.3

Version 1

SEMEAN 0.889 0.577

Men Women

N 15 20

MEAN 15.26 19.20

STDEV 3.44 2.58

SE MEAN 0.89 0.58

95 PCT CI FOR MU Men - MU Women: ( -6.12,

-1.76)

TTEST MU Men = MU Women (VS NE): T= -3.72 P=0.0010 DF= 25

3

A company run a chain of chemists. A female pharmacist working for the chain sued alleging discrimination in the promotion from pharmacist to chief pharmacist. Given below are the data showing the time from initial employment to chief pharmacist in months. Female 229

453

Male 5 34

3

7 37

12 47

14 49

14 64

14 67

18 69

21 125

22 192

23 483

24

25

Does the evidence support the claim? The details of the Mann-Whitney calculation are shown below. Score 5 7 12 14 14 14 18 21 22 23 24 25 34 34 37 47 49 64 67 69 125 192 229 453 483

Rank Female Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Male Female Female Male Sum U mean var. z

Mike Cox

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 47 278 2 44 23.00 23.00 99.67 99.67 -2.104 2.10

6.4

Rank Female Male reversed 25 25 24 24 23 23 22 22 21 21 20 20 19 19 18 18 17 17 16 16 15 15 14 14 13 13 12 12 11 11 10 10 9 9 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 5 44 23.00 99.67 2.10

320 2 23.00 99.67 -2.104

Version 1

34

Here nM = 23, nF = 2 then SM = 278 and SF = 47 UM = nM nF + ½ nM (nM + 1) - SM = 44 and UF = nM nF + ½ nF (nF + 1) - SF = 2

1 nM nF  2 2  2.053 n M n F n M  n F  1 12 U

z

Note that ties need not be considered in detail since they occur in the same sample. There would appear to be a clear case of discrimination, when compared to the critical value of ±1.96 for a difference.

A Mann-Whitney test is expected, however just in case a two sample t test is attempted some basic statistics are also given.

Data Display Female 229

453

Data Display Male 5 34

7 37

12 47

14 49

14 64

14 67

18 69

21 125

22 192

23 483

24

Median 341 25.0

Q3 * 64.0

25

34

Descriptive Statistics: Female, Male Variable Female Male

N 2 23

N* 0 0

Mean 341 60.9

SE Mean 112 21.1

StDev 158 101.4

Minimum 229 5.0

Q1 * 14.0

Maximum 453 483.0

Two-Sample T-Test and CI: Female, Male Two-sample T for Female vs Male Female Male

N 2 23

Mean 341 61

StDev 158 101

SE Mean 112 21

Difference = mu (Female) - mu (Male) Estimate for difference: 280.130 95% CI for difference: (-1168.076, 1728.337) T-Test of difference = 0 (vs not =): T-Value = 2.46

P-Value = 0.246

DF = 1

Mann-Whitney Test and CI: Female, Male Female Male

N 2 23

Median 341.0 25.0

Point estimate for ETA1-ETA2 is 223.0 96.0 Percent CI for ETA1-ETA2 is (36.9,441.1)

Mike Cox

6.5

Version 1

W = 47.0 Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0400

The test is significant at 0.0398 (adjusted for ties)

Mike Cox

6.6

Version 1

Critical Values For Wilcoxon's Signed-Rank Test The body of the table contains the critical values for Wilcoxon's signed-rank test. Always enter the table with W+, the sum of the ranks of the positive deviations. If a critical value is missing, the hypothesis can not be rejected for this combination of n and α. One Tail Probability 0.05

0.025

0.01

OneTail Probability

0.005

n

0.05

Two Tail Probability

Mike Cox

n

0.10

0.05

0.02

5

1

6

2

1

7

4

2

0

8

6

4

2

9

8

6

10

11

11

0.025

0.01

0.005

TwoTail Probability

0.01

n

0.10

0.05

0.02

0.01

28

130

117

102

92

29

141

127

111

100

30

152

137

120

109

0

31

163

148

130

118

3

2

32

175

159

141

128

8

5

3

33

188

171

151

138

14

11

7

5

34

201

183

162

149

12

17

14

10

7

35

214

195

174

160

13

21

17

13

10

36

228

208

186

171

14

26

21

16

13

37

242

222

198

183

15

30

25

20

16

38

256

235

211

195

16

36

30

24

19

39

271

250

224

208

17

41

35

28

23

40

287

264

238

221

18

47

40

33

28

41

303

279

252

234

19

54

46

38

32

42

319

295

267

248

20

60

52

43

37

43

336

311

281

262

21

68

59

49

43

44

353

327

297

277

22

75

66

56

49

45

371

344

313

292

23

83

73

62

55

46

389

361

329

307

24

92

81

69

61

47

408

379

345

323

25

101

90

77

68

48

427

397

362

339

26

110

98

85

76

49

446

415

380

356

27

120

107

93

84

50

466

434

398

373

6.7

Version 1

Critical Values Of U In The Mann-Whitney Test Critical Values of U at α = 0.025 with direction predicted or at α = 0.05 with direction not predicted.

n2

9

10

11

12

13

14

15

16

17

18

19

20

2

0

0

0

1

1

1

1

1

2

2

2

2

3

2

3

3

4

4

5

5

6

6

7

7

8

4

4

5

6

7

8

9

10

11

11

12

13

13

5

7

8

9

11

12

13

14

15

17

18

19

20

6

10

11

13

14

16

17

19

21

22

24

25

27

7

12

14

16

18

20

22

24

26

28

30

32

34

8

15

17

19

22

24

26

29

31

34

36

38

41

9

17

20

23

26

28

31

34

37

39

42

45

48

10

20

23

26

29

33

36

39

42

45

48

52

55

11

23

26

30

33

37

40

44

47

51

55

58

62

12

26

29

33

37

41

45

49

53

57

61

65

69

13

28

33

37

41

45

50

54

59

63

67

72

76

14

31

36

40

45

50

55

59

64

67

74

78

83

15

34

39

44

49

54

59

64

70

75

80

85

90

16

37

42

47

53

59

64

70

75

81

86

92

98

17

39

45

51

57

63

67

75

81

87

93

99

105

18

42

48

55

61

67

74

80

86

93

99

106

112

19

45

52

58

65

72

78

85

92

99

106

113

119

20

48

55

62

69

76

83

90

98

105

112

119

127

n1

Mike Cox

6.8

Version 1

Critical Values Of U In The Mann-Whitney Test Critical Values of U at α = 0.05 with direction predicted or at α = 0.10 with direction not predicted.

n2

9

10

11

12

13

14

15

16

17

18

19

20

2

1

1

1

2

2

2

3

3

3

4

4

4

3

3

4

5

5

6

7

7

8

9

9

10

11

4

6

7

8

9

10

11

12

14

15

16

17

18

5

9

11

12

13

15

16

18

19

20

22

23

25

6

12

14

16

17

19

21

23

25

26

28

30

32

7

15

17

19

21

24

26

28

30

33

35

37

39

8

18

20

23

26

28

31

33

36

39

41

44

47

9

21

24

27

30

33

36

39

42

45

48

51

54

10

24

27

31

34

37

41

44

48

51

55

58

62

11

27

31

34

38

42

46

50

54

57

61

65

69

12

30

34

38

42

47

51

55

60

64

68

72

77

13

33

37

42

47

51

56

61

65

70

75

80

84

14

36

41

46

51

56

61

66

71

77

82

87

92

15

39

44

50

55

61

66

72

77

83

88

94

100

16

42

48

54

60

65

71

77

83

89

95

101

107

17

45

51

57

64

70

77

83

89

96

102

109

115

18

48

55

61

68

75

82

88

95

102

109

116

123

19

51

58

65

72

80

87

94

101

109

116

123

130

20

54

62

69

77

84

92

100

107

115

123

130

138

n1

Mike Cox

6.9

Version 1