Analysis of Covariance Experimental Designs

Analysis of Covariance Experimental Designs

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Analysis of Covariance Experimental Designs

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1. Analysis of Covariance Experimental Designs

2. Design Notation R O X O R O O • Uses a pre-measure • Can be a pretest, but doesn’t have to be • Can have multiple covariates

3. 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 The Covariance Design Posttest Pretest

4. 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 The Covariance Design The range of y is about 70 points. Posttest Pretest

5. 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 How Does a Covariate Reduce Noise? We fit regression lines to describe the pretest-posttest relationship. Posttest Pretest

6. 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 How Does a Covariate Reduce Noise? Posttest We want to “adjust” the posttest scores for pretest variability. Pretest

7. 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 How Does a Covariate Reduce Noise? Posttest We do this by “subtracting out” the pretest -- by “subtracting out” the line. We want to “adjust” the posttest scores for pretest variability. Pretest

8. 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 How Does a Covariate Reduce Noise? Posttest We do this by “subtracting out” the pretest -- by “subtracting out” the line. Get the difference between the line and each point. Pretest

9. 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 How Does a Covariate Reduce Noise? When we subtract the line from each point, it’s as though we make each line parallel to an x-axis. Posttest Pretest

10. Posttest 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 Pretest 6 0 7 0 8 0 How Does a Covariate Reduce Noise? Or, it’s like rotating the entire distribution until the regression lines are horizontal.

11. Posttest 1 0 0 9 0 8 0 7 0 6 0 5 0 4 0 3 0 2 0 2 0 3 0 4 0 5 0 Pretest 6 0 7 0 8 0 How Does a Covariate Reduce Noise? Notice how much less variability there is on y after we have “removed” the relationship to the covariate.

12. 4 0 3 0 2 0 1 0 0 - 1 0 - 2 0 - 3 0 - 4 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 How Does a Covariate Reduce Noise? Here is the plot with the effect of the pretest removed. Notice the range on y is now only about 50 points instead of 70 (although the difference between the means remains the same). Pretest

13. Summary • The Analysis of Covariance adjusts posttest scores for variability on the covariate (pretest). • This is what we mean by adjusting for the effects of one variable on another.

14. Summary • You can use any continuous variable as the covariate, but the pretest is usually best. Why? • You can use multiple covariates, but if they are highly intercorrelated, you don’t improve the adjustment (and you pay a price for each covariate).