1. Analysis of Covariance Goals: 1)Reduce error variance. 2)Remove sources of bias from experiment. 3)Obtain adjusted estimates of population means.

2. Analysis of Covariance Y XX.1 X.2 X.. Y X X.1 X.2 X.. Random assignment conditional distribution is less variable Not randomly assigned Need to block

3. Analysis of Covariance Selection of covariates: 1)Experiment contains one or more nuisance, uncontrollable, variables that we know. 2)Control of these variables is either not possible or not feasible. 3)It is possible to obtain a measure of the nuisance variable that doesn’t include effects attributable to the factors: a) the covariate is obtained prior to presentation of factor levels, b) the covariates are obtained after the levels are set but before they affect, c) it can be assumed that they are not affected by the factor levels. (These are not always necessary).

4. Analysis of Covariance Model:

5. Analysis of Covariance Y ij Response of the j th experimental unit in the i th factor. X ij Covariate of subject j in factor level i. Mean of covariate  The grand mean of all factor. ii Factor effect for population i, and should obey the condition:  Slope of the regression line.  ij The error effect associated with Y ij and is equal to:

6. Analysis of Covariance Some more assumptions for the above model: 1)The slope  does not equal zero. 2)The relationship between Y ij and X ij is linear. 3)The regression coefficients for each factor level are equal. 4)The covariate variable X ij is not affected by the levels of the factor.

7. Hypotheses: Analysis of Covariance

8. 1)Simplest, most reduced model: The fitted model: Total residual:

9. 2.Want to explain more, add covariate. The linear model becomes: Analysis of Covariance The fitted model: Residual of reduced model:

10. Analysis of Covariance 3)Adding the main effect after accounting for the covariate: The fitted model: Residual of full model:

11. Analysis of Covariance How to perform the analysis of covariance: 3)Wanted to explain more, added the factor. The linear model became: The fitted model:

12. Analysis of Covariance 4.Compare the errors: Part of residual not accounted for after regression along with its df. Part of residual not accounted for after adjusting using the covariate and the factor Along with its df.

13. Analysis of Covariance 4)Compare the errors (Cont.): The difference is what the factor accounts for.