1. Biomeasurement Continued from last week: t- and Mann-Whitney U tests Background reading: Chapter 7 www.biomeasurement.net

2. Lecture Content Comparing t- & MWU tests t-test Mann Whitney U test

3. Mann-Whitney U Test Comparison to t-Test When to use Example data Four steps Example from literature

4. Difference Two samples Unrelated Similarities

5. Differences t-test Parametric Scale data only Mann-Whitney U test Nonparametric Scale or ordinal data

6. When to use Difference. Two samples. Unrelated data. Dependent variable: Scale level Do not use if... Comparing frequency distributions.

7. Example Data

9. Tests on counts/frequenciesTests of Relationship Tests of Difference 1 set of Categories2 sets of Categories For scale/ordinal data from two variables RegressionCorrelation Scatterplot One Pie Chart Two Pie Charts ParametricNonparametric For counts/frequencies in categories For scale/ordinal dependent variable data in categories distinguished by the independent variable ©Hawkins & Carter 2004 Pie Charts Errorplots or Boxplot ErrorplotsBoxplots Choosing Chart for Graphs Scatterplots

10. The Four Steps 1. Construct a Null Hypothesis (H o ). 2. Decide Critical Significance Level  ). 3. Calculate Statistic. 4. Reject or Accept the Null Hypothesis.

11. 1. Construct H o H o : There is no difference between the bone density of males and females over 50 years old. In general no difference between the samples.

12. 2. Decide   5% = 0.05.

13. 3. Calculate Statistic. U Sum of ranks of sample 2 Size of sample 1 U is the the lower value of U 1 or U 2. Check: U 1 + U 2 = n 1 n 2

14. R2R2 R1R1 More about R

15. U

16. U = 120.5 n 1 = 20 n 2 = 20

17. Using SPSS

19. Dependent Variable Independent Variable

21. 4. Reject or Accept. Using Critical Values. Reject if your t is bigger than t critical. Using P Values Reject if P is less than or equal to  Where do you get these from?

22. Using Critical Values. 120.5< 273  Reject Bone density between males & females over 50 is different From... Step 2:  = 0.05 Step 3: U=120.5, n 1 =20, n 2 =20 If U </= U critical  REJECT H 0  significant result.

23. Using P Values. If P </=   reject the null hypothesis. If P >   accept the null hypothesis. If P </=   reject the null hypothesis. If P >   accept the null hypothesis. 0.032< 0.05  Reject Bone density between males & females over 50 is different

24. Example from Literature Drews (1995), BEHAVIOUR 133 The pattern and context of injuries was studied in a troop of yellow baboons in Mikumi National Park (Tanzania).

25. TABLE 7. Median and range of healing times in days for baboon injuries. N MedianMin Max Total sample* 35 23.0 8120 Small injuries (<5cm) 15 14.0 8114 Large injuries (>5cm) 11 24.010120 * The sample is composed of 24 cuts, 4 punctures, 1 tear, 1 case of limping, 1 bruise, and 3 injuries of unspecified shape. Quote from results: The difference between small and large wounds (Table 7) was not statistically significant (Mann-Whitney test, Z= -1.461, N1=15, N2=11, p=0.14). TABLE 7. Median and range of healing times in days for baboon injuries. N MedianMin Max Total sample* 35 23.0 8120 Small injuries (<5cm) 15 14.0 8114 Large injuries (>5cm) 11 24.010120 * The sample is composed of 24 cuts, 4 punctures, 1 tear, 1 case of limping, 1 bruise, and 3 injuries of unspecified shape. Quote from results: The difference between small and large wounds (Table 7) was not statistically significant (Mann-Whitney test, Z= -1.461, N1=15, N2=11, p=0.14).

26. What is Z? - bone example revisted.

27. Back to baboons…. QUESTIONS: What were the two samples? Sample sizes? What might the raw data have looked like? Why might he have used a Mann- Whitney U test instead of a t-test?

28. Lecture Content Comparing t- & MWU tests t-test Mann Whitney U test