1. The Analysis of Covariance ANACOVA

2. Multiple Regression 1.Dependent variable Y (continuous) 2.Continuous independent variables X 1, X 2, …, X p The continuous independent variables X 1, X 2, …, X p are quite often measured and observed (not set at specific values or levels)

3. Analysis of Variance 1.Dependent variable Y (continuous) 2.Categorical independent variables (Factors) A, B, C,… The categorical independent variables A, B, C,… are set at specific values or levels.

4. Analysis of Covariance 1.Dependent variable Y (continuous) 2.Categorical independent variables (Factors) A, B, C,… 3.Continuous independent variables (covariates) X 1, X 2, …, X p

5. Example 1.Dependent variable Y – weight gain 2.Categorical independent variables (Factors) i.A = level of protein in the diet (High, Low) ii.B = source of protein (Beef, Cereal, Pork) 3.Continuous independent variables (covariates) i.X 1 = initial wt. of animal.

6. Statistical Technique Independent variables continuouscategorical Multiple Regression× ANOVA× ANACOVA×× Dependent variable is continuous It is possible to treat categorical independent variables in Multiple Regression using Dummy variables.

7. The Multiple Regression Model

8. The ANOVA Model

9. The ANACOVA Model

10. ANOVA Tables

11. The Multiple Regression Model SourceS.S.d.f. RegressionSS Reg p ErrorSS Error n – p - 1 TotalSS Total n - 1

12. The ANOVA Model SourceS.S.d.f. Main Effects ASS A a - 1 BSS B b - 1 Interactions ABSS AB (a – 1)(b – 1)  ErrorSS Error n – p - 1 TotalSS Total n - 1

13. The ANACOVA Model SourceS.S.d.f. CovariatesSS Covaraites p Main Effects ASS A a - 1 BSS B b - 1 Interactions ABSS AB (a – 1)(b – 1)  ErrorSS Error n – p - 1 TotalSS Total n - 1

14. Example 1.Dependent variable Y – weight gain 2.Categorical independent variables (Factors) i.A = level of protein in the diet (High, Low) ii.B = source of protein (Beef, Cereal, Pork) 3.Continuous independent variables (covariates) X = initial wt. of animal.

15. The data

16. The ANOVA Table

17. Using SPSS to perform ANACOVA

18. The data file

19. Select Analyze  General Linear Model  Univariate

20. Choose the Dependent Variable, the Fixed Factor(s) and the Covaraites

21. The following ANOVA table appears

22. Covariate Dependent variable The Process of Analysis of Covariance

23. Covariate Adjusted Dependent variable The Process of Analysis of Covariance

24. The dependent variable (Y) is adjusted so that the covariate takes on its average value for each case The effect of the factors ( A, B, etc) are determined using the adjusted value of the dependent variable.

25. ANOVA and ANACOVA can be handled by Multiple Regression Package by the use of Dummy variables to handle the categorical independent variables. The results would be the same.