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The Analysis of Covariance

The Analysis of Covariance. ANACOVA. Multiple Regression. Dependent variable Y (continuous) 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).

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The Analysis of Covariance

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  1. The Analysis of Covariance ANACOVA

  2. Multiple Regression • Dependent variable Y (continuous) • Continuous independent variables X1, X2, …, Xp The continuous independent variables X1, X2, …, Xp are quite often measured and observed (not set at specific values or levels)

  3. Analysis of Variance • Dependent variable Y (continuous) • 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 • Dependent variable Y (continuous) • Categorical independent variables (Factors) A, B, C,… • Continuous independent variables (covariates) X1, X2, …, Xp

  5. Example • Dependent variable Y – weight gain • Categorical independent variables (Factors) • A= level of protein in the diet (High, Low) • B = source of protein (Beef, Cereal, Pork) • Continuous independent variables (covariates) • X1= initial wt. of animal.

  6. 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

  12. The ANOVA Model

  13. The ANACOVA Model

  14. Example • Dependent variable Y – weight gain • Categorical independent variables (Factors) • A= level of protein in the diet (High, Low) • B = source of protein (Beef, Cereal, Pork) • 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. The Process of Analysis of Covariance Dependent variable Covariate

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

  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.

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