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Re: st: comparing anova and regression results

From   [email protected] (William Gould)
To   [email protected]
Subject   Re: st: comparing anova and regression results
Date   Wed, 17 Jul 2002 11:44:57 -0500

Linda Salisbury <[email protected]> notes, 

> I am getting somewhat different model results from an anova procedure and
> it's accompanying regression output.  When regressing Y on x1 and x2, the
> significance levels reported for x1 and x2 are very different for anova vs.
> regression.  My guess is that the procedures use somewhat different
> algorithms.  Why are the results different and how should I interpret them?

It is not that the two use somewhat different algorithms, it is that they
report statistics that are the answers to different questions.  I suspect,
because Linda is asking the question, what she wants are the results reported
by -regress- and not -anova-, or perhaps something else.

In the output Linda provided, she has a 2x2 design.  She estimates the model

        . anova y x1 x2 x1*x2

and x1 and x2 each take on two values.  The regression output for x1 reports
a test for b1=0 in 

        y = b0 + b1*x1 b2*x2 + b3*x1*x2

That test amounts to a test that x1 has no "direct" effect:  If x2==0,
changing x1 does not change y.  It can still be the case, however, that
changing x1 when x2 is not zero can have an effect.

The anova output for x1 reports whether x1 has a "main effect".  This is a 
test whether the means for y would be equal in a populations with x1=0 and 
x1=1, assuming the average x2=.5 in each!  If x2=.5, then 

       y = b0 + b1*x1 + b2*.5 + b3*x1*.5
         = b0 + b2*.5  +  (b1+b3/2)*x1

and the test ANOVA reports amounts to a test that b1+b3/2 == 0.  There is a
rather cute dicussion of "main effects" in the Technical Note in [R] anova on
on page 33 of the References A-G manual.

There is a third test reported by neither, and that may be the test 
Linda really wants:  Does x1 have any effect?  To test that, we would 
have to ask whether b1==b2==0.  -test- can do that test.

-- Bill
[email protected]
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