[R] Total effect of X on Y under presence of interaction effects

[David Winsemius]

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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>
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David Winsemius, MD
West Hartford, CT