Transcript ANCOVA.pptx

How to use a covariate to get a more sensitive analysis
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)
Raw data:
Suppose I analyze the data and only look at post
data (graph first)
Now ANOVA
No apparent significant differences
Maybe we can improve the Model by including
another predictor variable, so ANCOVA
ANCOVA Model:
Yij=µ+τi+β*X+εij
so that i indexes the treatment group and X is now
included in the model as a regression variable with
slope β.
New analysis with ANCOVA model
This plot looks much more convincing
ANCOVA table
Now it is clear that there is a treatment group effect, so go on to test group means.
Means comparisons all groups with Tukey
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
Residuals vs. Predicted
Normality
Test of Normality
Drugs vs. Placebo
This confirms that virtually all Treatment group variation is due to the
Placebo vs. Drugs.
Why did all of this work?
Compare Mean Squares for both models:
 σ2 approximately 36.86 for the Model without
Covariate
 σ2 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.