1. How to use a covariate to get a more sensitive analysis

2. Placebo and two treatments Suppose I have a drug treatment study: a-brand name drug (eg. Synthroid) d-generic version of first drug a (L-thyroxin) Placebo- inert substance (sham treatment) X- initial value of physiological parameter which may correlate with Response (related to thyroid function) Response- post-treatment value of physiological parameter (T4 which directly measures thyroid function) Either the brand name or generic should increase the physiological parameter (i.e. increase thyroid function)

3. Raw data:

4. Suppose I analyze the data and only look at post data (graph first)

5. Now ANOVA No apparent significant differences

6. Maybe we can improve the Model by including another predictor variable, so ANCOVA ANCOVA Model: Y ij =µ+τ i +β*X+ε ij so that i indexes the treatment group and X is now included in the model as a regression variable with slope β.

7. New analysis with ANCOVA model This plot looks much more convincing

8. ANCOVA table Now it is clear that there is a treatment group effect, so go on to test group means.

9. Means comparisons all groups with Tukey

10. Could have done contrasts (not really needed in this case because of the results of the Range test) This contrast tests Drug a vs. Drug d

11. Residuals vs. Predicted

12. Normality

13. Test of Normality

14. Drugs vs. Placebo This confirms that virtually all Treatment group variation is due to the Placebo vs. Drugs.

15. Why did all of this work? Compare Mean Squares for both models: σ 2 approximately 36.86 for the Model without Covariate σ 2 approximately 16.046 for the Model with Covariate It worked because including a meaningful explanatory variable, i.e. the covariate, reduced our estimate of experimental error.