Transcript ANCOVAx

How to use a covariate to get a more sensitive analysis
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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)
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Raw data:
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Suppose I analyze the data and only look at post
data (graph first)
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Now ANOVA
No apparent significant differences
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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.
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New analysis with ANCOVA model
This plot looks much more convincing
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ANCOVA table
Now it is clear that there is a treatment group effect, so go on to test group means.
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Means comparisons all groups with Tukey
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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
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Residuals vs. Predicted
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Normality
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Test of Normality
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Drugs vs. Placebo
This confirms that virtually all Treatment group variation is due to the
Placebo vs. Drugs.
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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.
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