1. 2nd Half Review ANOVA (Ch. 11) Non-Parametric (7.11, 9.5) Regression (Ch. 12) ANCOVA Categorical (Ch. 10) Correlation (Ch. 12)

2. The Exam Thursday, April 27, 9:00am –TC 348  Abdi to Middleton –TC 348a  Minto to Shetty –TC 348b  Siddiqui to Zdravic 50 Questions Not Cumulative 3hr Bring a calculator No formula Sheets

3. Variance Partitions –Total = Among + Within Grand Mean, Group Mean and associated Deviations When do we reject based on variance ratio??? ANOVA

4. ANOVA Table SourcedfSSMSF Among Treatmentsk-1SS among df among MS among MS within Within Treatmentsn-kSS within df within Totaln-1

5. ANOVA When do we use?? Model I vs Model II vs Model III?? Multi-Factors?? Main Effects vs Interactions??

6. Example From Text Question #11.40, p. 518 10 women in an aerobic exercise class, 10 women in a modern dance class, and a control group of 9 women were studied. One measurement made on each women was change in fat-free mass over the course of the 16-week training period. Summary statistics are given in the table. The ANOVA SS(between) is 2.465 and the SS(within) is 50.133. AerobicsDanceControl Mean00.440.71 SD1.311.171.68 n10 9 a)State the null hypothesis b)Construct the ANOVA table and test the null hypothesis (α = 0.05)

7. Example From Text Question #11.57, p. 522 A new investigational drug was given to 4 male and 4 female dogs, at doses 8 mg/kg and 25 mg/kg. The variable recorded was alkaline phosphatase level (U/Li). Dose (mg/kg)MaleFemale 8171150 154127 104152 143105 Avg143133.5 2580101 149113 138161 131197 Avg124.5143 (SS(sex) = 81, SS(dose) = 81, SS(interaction) = 784, and SS(within) = 12604 a)Construct the ANOVA Table b)Carry out an F test for interactions: use (α = 0.05). c)Test the null hypothesis that does has no effect on alkaline phosphatase level. (α = 0.05)

8. Extra Questions from the Text 11.4-11.6 11.9-11.11 11.17, 11.19 11.42, 11.43, 11.50, 11.54

9. Non-Parametric When to use?? –Normality –Homogenous of Variance –Independent Observations What about Power?? What do they use to compare data??

10. Mann-Whitney Test Compares two samples Replaces two-sample t-test If either U or U’ is greater than the critical value of U, then you should reject the H o

11. Critical U If either U or U’ is greater than the critical value of U, then you should reject the H o if n 1 < n 2 if n 1 > n 2

12. One-tailed Mann-Whitney U test Use U or U’ depending on whether you expect sample 1 or sample 2 to be bigger

13. Wilcoxon paired sample test Compares two paired samples Replaces Paired t-test

14. Critical T Sum the positive ranks - T + Sum the negative ranks - T - If either T + or T - is less than or equal to T 0.05, (2),n then reject H o Can also do these one tailed: H o : Measurement 1  Measurement 2 H A : Measurement 1 > Measurement 2 --> reject H o if T -  T 0.05, (1),n H o : Measurement 1  Measurement 2 H A : Measurement 1 < Measurement 2 --> reject H o if T +  T 0.05, (1),n

15. Kruskal Wallis Test Test for three or more groups Replaces ANOVA Rank each individual sample across all groups Sum ranks within each group = R Critical value -

16. Correction factor for tied ranks

17. Critical Value

18. Practice Questions Mann-Whitney –7.79, 7.80, 7.82-7.84 Wilcoxon –9.30 – 9.33 Kruskal-Wallis –11.54, 11.57  use Kruskal-Wallis instead of ANOVA

19. Regression Two or more continuous variables Linear relationship between Intercept Slope

20. Least Squares Line with smallest residual sum of squares YY i ˆ  Residual as

21. ANOVA Table Source of Variation SS DFMSF Regression Residual Total n-2 1 n-1

22. Coefficient of Determination - proportion of variation explained

23. Slope Intercept

24. Confidence Interval for Slope

25. Practice Questions 12.45 12.49-12.54

26. ANCOVA Continuous Dependent Continuous and Discrete Independents Compares relationship of two variables across two groups

28. Categorical Data Discrete Response variable Interested in Frequencies

29. Chi-Squared Test Observed vs Expected Frequencies f expected is hypothesized ratio e.g. 50:50 males to females

30. Contingency Table Test for independence among variables 2x2, 2x3, 3x3 etc. 2 variables with 2 levels, 2 variables with 3 levels, 3 variables with 3 levels etc.

31. R1 = sum of observed in Row 1 R2 = sum of observed in Row 2 C1 = sum of observed in Column 1 C2 = sum of observed in Column 2 C3 = sum of observed in Column 3 Total = sum of all observed

32. Expected calculation /

33. Practice Questions 10.73-10.78 10.84-10.88

34. Logistic Regression Discrete dependent – usually dichotomous Continuous independent