Total effect of X on Y under presence of interaction effects

On May 11, 2011, at 6:26 PM, Matthew Keller wrote:

Not to rehash an old statistical argument, but I think David's reply
here is too strong ("In the presence of interactions there is little
point in attempting to assign meaning to individual coefficients.").
As David notes, the "simple effect" of your coefficients (e.g., a) has
an interpretation: it is the predicted effect of a when b, c, and d
are zero. If the zero-level of b, c, and d are meaningful (e.g., if
you have centered all your variables such that the mean of each one is
zero), then the coefficient of a is the predicted slope of a at the
mean level of all other predictors...

And there is internal evidence that such a procedure was not performed  
in this instance. I think my advice applies here.

-- 
David.

Matt



On Wed, May 11, 2011 at 2:40 PM, Greg Snow <Greg.Snow at imail.org>  
wrote:

Just to add to what David already said, you might want to look at  
the Predict.Plot and TkPredict functions in the TeachingDemos  
package for a simple interface for visualizing predicted values in  
regression models.

These plots are much more informative than a single number trying  
to capture total effect.

--
Gregory (Greg) L. Snow Ph.D.
Statistical Data Center
Intermountain Healthcare
greg.snow at imail.org
801.408.8111

-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-
project.org] On Behalf Of David Winsemius
Sent: Wednesday, May 11, 2011 7:48 AM
To: Michael Haenlein
Cc: r-help at r-project.org
Subject: Re: [R] Total effect of X on Y under presence of  
interaction
effects


On May 11, 2011, at 4:26 AM, Michael Haenlein wrote:

Dear all,

this is probably more a statistics question than an R question but
probably
there is somebody who can help me nevertheless.

I'm running a regression with four predictors (a, b, c, d) and all
their
interaction effects using lm. Based on theory I assume that a
influences y
positively. In my output (see below) I see, however, a negative
regression
coefficient for a. But several of the interaction effects of a with
b, c and
d have positive signs. I don't really understand this. Do I have to
add up
the coefficient for the main effect and the ones of all interaction
effects
to get a total effect of a on y? Or am I doing something wrong  
here?

In the presence of interactions there is little point in  
attempting to
assign meaning to individual coefficients. You need to use predict()
(possibly with graphical or tabular displays) and produce  
estimates of
one or two variable at relevant levels of  the other variables.

The other aspect about which your model is not informative, is the
possibility that some of these predictors have non-linear  
associations
with `y`.

(The coefficient for `a` examined in isolation might apply to a  
group
of subjects (or other units of analysis) in which the values of `b`,
`c`, and `d` were all held at zero. Is that even a situation that
would occur in your domain of investigation?)

--
David.

Thanks very much for your answer in advance,

Regards,

Michael


Michael Haenlein
Associate Professor of Marketing
ESCP Europe
Paris, France



Call:
lm(formula = y ~ a * b * c * d)

Residuals:
   Min      1Q  Median      3Q     Max
-44.919  -5.184   0.294   5.232 115.984

Coefficients:
           Estimate Std. Error t value Pr(>|t|)
(Intercept)  27.3067     0.8181  33.379  < 2e-16 ***
a           -11.0524     2.0602  -5.365 8.25e-08 ***
b            -2.5950     0.4287  -6.053 1.47e-09 ***
c           -22.0025     2.8833  -7.631 2.50e-14 ***
d            20.5037     0.3189  64.292  < 2e-16 ***
a:b          15.1411     1.1862  12.764  < 2e-16 ***
a:c          26.8415     7.2484   3.703 0.000214 ***
b:c           8.3127     1.5080   5.512 3.61e-08 ***
a:d           6.6221     0.8061   8.215 2.33e-16 ***
b:d          -2.0449     0.1629 -12.550  < 2e-16 ***
c:d          10.0454     1.1506   8.731  < 2e-16 ***
a:b:c         1.4137     4.1579   0.340 0.733862
a:b:d        -6.1547     0.4572 -13.463  < 2e-16 ***
a:c:d       -20.6848     2.8832  -7.174 7.69e-13 ***
b:c:d        -3.4864     0.6041  -5.772 8.05e-09 ***
a:b:c:d       5.6184     1.6539   3.397 0.000683 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 7.913 on 12272 degrees of freedom
Multiple R-squared: 0.8845,     Adjusted R-squared: 0.8844
F-statistic:  6267 on 15 and 12272 DF,  p-value: < 2.2e-16

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David Winsemius, MD
West Hartford, CT

-- 
Matthew C Keller
Asst. Professor of Psychology
University of Colorado at Boulder
www.matthewckeller.com

David Winsemius, MD
West Hartford, CT