Biostatistics-Lecture 4 Analysis of Variance

1. Biostatistics-Lecture 4Analysis of Variance Ruibin Xi Peking University School of Mathematical Sciences

2. Analysis of Variance (ANOVA) • Consider the Iris data again • Want to see if the average sepal widths of the three species are the same • μ1 ,μ2, μ3 : the mean sepal width of Setosa, Versicolor, Virginica • Hypothesis: H0: μ1=μ2= μ3 H1: at least one mean is different

3. Analysis of Variance (ANOVA) • Used to compare ≥ 2 means • Definitions • Response variable (dependent)—the outcome of interest, must be continuous • Factors (independent)—variables by which the groups are formed and whose effect on response is of interest, must be categorical • Factor levels—possible values the factors can take

4. Sources of Variation in One-Way ANOVA • Partition the total variability of the outcome into components—source of variation • the sepal width of the jth plant from the ith species (group) Grand mean The ith group mean

5. Sources of Variation in One-Way ANOVA • SST: sum of squares total • SSB: sum of squares between • SSW (SSE): sum of squares within (error)

6. F-test in one-way ANOVA • The test statistic is called F-statistic Follows an F-distribution with (df1,df2) = (k-1,n-k) • For the Iris data • SSB=11.34, MSB = 5.67, SSE=16.96, MSE=0.12 • f= 49.16, df1=2,df2=147 • Critical value 3.06 at α=0.05, reject the null • Pvalue = P(F>f)=4.49e-17

7. One-way ANOVA • ANOVA table

8. One-way ANOVA • ANOVA table

9. ANOVA model • The statistical model Yij = μ + αi + eij error The effect of group j The ith response in the jth group grand mean

10. ANOVA assumptions • Normality • Homogeneity • Independence

11. Multiple Comparisons • After reject null hypothesis of ANOVA, we’d like to know which means differ from another • Use individual t-test to compare all pairs? • At significance level 0.05, 5% chance for a false positive • If there are n test, the chance of a false positive • 1-(1-α)n

12. Multiple Comparisons • Bonferroni method—conservative but simple • Divide the level of significance by the number of comparisons to be made Example: 3 comparisons 0.05/3=0.017 • Or adjusting your p-values • No need of ANOVA • Planned comparison

13. Multiple Comparisons • After ANOVA has resulted in a significant F-test • Tukey—can perform all pairwise comparisons • Based on studentizedrange distribution • Scheffe—more versatile, more conservative

14. Multiple Comparisons • After ANOVA has resulted in a significant F-test • Tukey—can perform all pairwise comparisons • Scheffe—more versatile, more conservative

15. Multiple Comparisons • Scheffe’s test • An arbitrary contrast is where • Estimate C by , for which the s.d. is • The 1-α confidence interval of Scheffe’s test is

16. Regression—an example • Cystic fibrosis (囊胞性纤维症) lung function data • PEmax (maximal static expiratory pressure) is the response variable • Potential explanatory variables • age, sex, height, weight, • BMP (body mass as a percentage of the age‐specific median) • FEV1 (forced expiratory volume in 1 second) • RV (residual volume) • FRC(funcAonalresidual capacity) • TLC (total lung capacity)

17. Regression—an example • Let’s first concentrate on the age variable • The model • Plot PEmaxvs age

18. Regression—an example • Let’s first concentrate on the age variable • The model • Plot PEmaxvs age

19. Simple Linear regression

20. Assumptions • Normality • Given x, the distribution of y is normal with mean α+βx with standard deviation σ • Homogeneity • σ does not depend on x • Independence

21. Residuals

22. Fitting the model

23. Fitting the model

24. Goodness of Fit

25. Inference about β

26. Inference about β

27. Inference about β: the CF data

28. Plotting the regression line

29. R2

30. Residual plot

31. Residual plot

32. Residual plot

33. Residual plot • The CF patients data

34. Linear Regression

35. Summary: simple linear regression

36. Multiple regression • See blackboard