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Analysis of Covariance Experimental Designs

Analysis of Covariance Experimental Designs. 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. 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.

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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).

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