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

From: Frank Harrell <f.harrell_at_vanderbilt.edu>
Date: Thu, 12 May 2011 04:01:32 -0700 (PDT)

I second David's first reply regarding the non-utility of individual coefficients, especially for low-order terms. Also, nonlinearity can be quite important. Properly modeling main effects through the use of flexible nonlinear functions can sometimes do away with the need for interaction terms.

Back to the original question, it is easy to get "total effects" for each predictor. The anova function in the rms package does this, by combining lower and higher-order effects (main effects + interactions). Frank

David Winsemius wrote:
>
> 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 &lt;Greg.Snow_at_imail.org&gt;
>> 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
>>>>> 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.
>>>>>
>>>>>
>>>>> 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
>>>>>
>>>>>     [[alternative HTML version deleted]]
>>>>>
>>>>> ______________________________________________
>>>>> R-help_at_r-project.org mailing list
>>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> guide.html
>>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>
>>>> David Winsemius, MD
>>>> West Hartford, CT
>>>>
>>>> ______________________________________________
>>>> R-help_at_r-project.org mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>> ______________________________________________
>>> R-help_at_r-project.org mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> http://www.R-project.org/posting-guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>
>>
>>
>> --
>> Matthew C Keller
>> Asst. Professor of Psychology
>> University of Colorado at Boulder
>> www.matthewckeller.com
```

>
> David Winsemius, MD
> West Hartford, CT
>
> ______________________________________________
> R-help@r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
>

Frank Harrell
Department of Biostatistics, Vanderbilt University
```--
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