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U-Test (Mann-Whitney) Example of a Mann-Whitney U-Test A physician is interested in the effect of an anaesthetic on reaction times. Two groups are compared, one with (A) and one without (B) taking the anaesthetic. Subjects had to react on a simple visual stimulus. Reaction times are not normally distributed in this experiment, so data is analysed with the Mann-Whitney U-Test for ordinal scaled measurements. The table below shows the rank-ordered data:

Mean RT Rank Group 131

1

B

135

2

A

138

3.5

B

138

3.5

B

139

5

A

141

6

B

142

8

B

142

8

A

142

8

B

143

10

B

144

11

A

145

12

B

156

13

B

158

14

A

165

15

A

167

16

B

171

17

A

178

18

A

191

19

B

230

20

B

244

21

A

245

22

A

256

23

A

267

24

A

268

25

A

289

26

A

Table showing Ranked Measures for each Group separately:

Group A Group B 2

1

5

3.5

8

3.5

11

6

14

8

15

8

17

10

18

12

21

13

22

16

23

19

24

20

25 26 Sum of Ranks

231

120

Average Ranks

16.5

10

The observed Z value is greater than the Z-value (5%, two-tailed). The anaesthetic group shows significantly slower reaction times than the non-anaesthetic group.

Chi-Square (2 x 2)

Example of a 2 x 2 Chi-Square Test Accidents were recorded and classified according to the severity of the accident (Heavy, Slight) and the blood alcohol level of the driver (No Alcohol, Alcohol). Heavy accidents are accidents with dead and injured persons. Slight accidents are accidents with property damage only. Drivers who had a blood alcohol level of 0 were assingned to the No Alcohol group. Drivers who had a blood alcohol level of greater than 0 were assigned to the Alcohol group. The observed frequencies are diplayed in the contingency table below:

Accident Heavy Slight Total

Alcohol 0 320 831 1151

>0 56 51 107

Total 376 882 1258

The expected frequencies are estimated from the sample and displayed in the following table:

Accident Heavy Slight Total

Alcohol 0 344.02 806.98 1151

>0 31.98 75.02 107

Total 376 882 1258

The residuals (fo - fe ) are diplayed in the following table:

Accident Heavy Slight

Alcohol 0 24.02 -24.02

For Chi-Square we get:

df = (2-1)(2-1) = 1

>0 -24.02 24.02

Critical Chi-Square Value 5 % = 3.84 The obtained Chi-Square is greater than the critical Chi-Square value. Hence, there must be some relationship between the two variables. Alcohol 0 Accident

Heavy Observed320 Expected344.02 Row %85.11 % Column %27.8 % Std. Res.-1.29 Slight Observed 831 Expected 806.98 Row % 94.22 % Column % 72.2 % Std. Res. 0.85 Total Observed1151 Expected1151 Row %91.49 % Column %100 %

>0

Total

56 31.98 14.89 % 52.34 % 4.25

376 376 100 % 29.89 %

51 75.02 5.78 % 47.66 % -2.77

822 882 100 % 70.11 %

107 107 8.51 % 100 %

1258 1258 100 % 100 %

We can either compare the column percentages of each cell with the total column percentages. For Heavy accidents we see, that total column percentage is 29.89 % and Slight acccidents it is 70.11 %. Further the percentages in the cells are 85.11 % (Heavy x no Alcohol), 94.22 % (Slight x no Alcohol) and 14.89 % (Heavy x Alcohol), 5.78 % (Slight Alcohol). For the no Alcohol cells the column percentages match the total column percentages quite well (27,8 % compared to 29.89 % and 72.2 % compared to 70.11 %), whereas the column percentages in the Alcohol cells deviate strongly from the total column percentages (52.34 % compared to 29.89 % and 47.66 % compared to 70.11 %). If we look closer to the Alcohol cells it is obvious, that there are more heavy accidents and less slight accident than expected. The standard residuals confirm this finding. In the “Heavy Accident x Alcohol” cell the standard residual is 4.25, indicating that in this cell there are far more observations than expected. In the “Slight Accident x Alcohol” cell the standard residual is –2.77, indicating that in this cell there are less observations than expected. In short: Alocohol leads to significant more heavy accidents.

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Mean RT Rank Group 131

1

B

135

2

A

138

3.5

B

138

3.5

B

139

5

A

141

6

B

142

8

B

142

8

A

142

8

B

143

10

B

144

11

A

145

12

B

156

13

B

158

14

A

165

15

A

167

16

B

171

17

A

178

18

A

191

19

B

230

20

B

244

21

A

245

22

A

256

23

A

267

24

A

268

25

A

289

26

A

Table showing Ranked Measures for each Group separately:

Group A Group B 2

1

5

3.5

8

3.5

11

6

14

8

15

8

17

10

18

12

21

13

22

16

23

19

24

20

25 26 Sum of Ranks

231

120

Average Ranks

16.5

10

The observed Z value is greater than the Z-value (5%, two-tailed). The anaesthetic group shows significantly slower reaction times than the non-anaesthetic group.

Chi-Square (2 x 2)

Example of a 2 x 2 Chi-Square Test Accidents were recorded and classified according to the severity of the accident (Heavy, Slight) and the blood alcohol level of the driver (No Alcohol, Alcohol). Heavy accidents are accidents with dead and injured persons. Slight accidents are accidents with property damage only. Drivers who had a blood alcohol level of 0 were assingned to the No Alcohol group. Drivers who had a blood alcohol level of greater than 0 were assigned to the Alcohol group. The observed frequencies are diplayed in the contingency table below:

Accident Heavy Slight Total

Alcohol 0 320 831 1151

>0 56 51 107

Total 376 882 1258

The expected frequencies are estimated from the sample and displayed in the following table:

Accident Heavy Slight Total

Alcohol 0 344.02 806.98 1151

>0 31.98 75.02 107

Total 376 882 1258

The residuals (fo - fe ) are diplayed in the following table:

Accident Heavy Slight

Alcohol 0 24.02 -24.02

For Chi-Square we get:

df = (2-1)(2-1) = 1

>0 -24.02 24.02

Critical Chi-Square Value 5 % = 3.84 The obtained Chi-Square is greater than the critical Chi-Square value. Hence, there must be some relationship between the two variables. Alcohol 0 Accident

Heavy Observed320 Expected344.02 Row %85.11 % Column %27.8 % Std. Res.-1.29 Slight Observed 831 Expected 806.98 Row % 94.22 % Column % 72.2 % Std. Res. 0.85 Total Observed1151 Expected1151 Row %91.49 % Column %100 %

>0

Total

56 31.98 14.89 % 52.34 % 4.25

376 376 100 % 29.89 %

51 75.02 5.78 % 47.66 % -2.77

822 882 100 % 70.11 %

107 107 8.51 % 100 %

1258 1258 100 % 100 %

We can either compare the column percentages of each cell with the total column percentages. For Heavy accidents we see, that total column percentage is 29.89 % and Slight acccidents it is 70.11 %. Further the percentages in the cells are 85.11 % (Heavy x no Alcohol), 94.22 % (Slight x no Alcohol) and 14.89 % (Heavy x Alcohol), 5.78 % (Slight Alcohol). For the no Alcohol cells the column percentages match the total column percentages quite well (27,8 % compared to 29.89 % and 72.2 % compared to 70.11 %), whereas the column percentages in the Alcohol cells deviate strongly from the total column percentages (52.34 % compared to 29.89 % and 47.66 % compared to 70.11 %). If we look closer to the Alcohol cells it is obvious, that there are more heavy accidents and less slight accident than expected. The standard residuals confirm this finding. In the “Heavy Accident x Alcohol” cell the standard residual is 4.25, indicating that in this cell there are far more observations than expected. In the “Slight Accident x Alcohol” cell the standard residual is –2.77, indicating that in this cell there are less observations than expected. In short: Alocohol leads to significant more heavy accidents.