Total effect of X on Y under presence of interaction effects

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Total effect of X on Y under presence of interaction effects

Michael Haenlein
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?

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

        [[alternative HTML version deleted]]


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Re: 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
>
> [[alternative HTML version deleted]]
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.

David Winsemius, MD
West Hartford, CT

______________________________________________
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Re: Total effect of X on Y under presence of interaction effects

Greg Snow-2
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
[hidden email]
801.408.8111


> -----Original Message-----
> From: [hidden email] [mailto:r-help-bounces@r-
> project.org] On Behalf Of David Winsemius
> Sent: Wednesday, May 11, 2011 7:48 AM
> To: Michael Haenlein
> Cc: [hidden email]
> 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
> >
> > [[alternative HTML version deleted]]
> >
> > ______________________________________________
> > [hidden email] mailing list
> > https://stat.ethz.ch/mailman/listinfo/r-help
> > PLEASE do read the posting guide http://www.R-project.org/posting-
> guide.html
> > and provide commented, minimal, self-contained, reproducible code.
>
> David Winsemius, MD
> West Hartford, CT
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-
> guide.html
> and provide commented, minimal, self-contained, reproducible code.

______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
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Re: Total effect of X on Y under presence of interaction effects

Matthew Keller
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...

Matt



On Wed, May 11, 2011 at 2:40 PM, Greg Snow <[hidden email]> 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
> [hidden email]
> 801.408.8111
>
>
>> -----Original Message-----
>> From: [hidden email] [mailto:r-help-bounces@r-
>> project.org] On Behalf Of David Winsemius
>> Sent: Wednesday, May 11, 2011 7:48 AM
>> To: Michael Haenlein
>> Cc: [hidden email]
>> 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
>> >
>> >     [[alternative HTML version deleted]]
>> >
>> > ______________________________________________
>> > [hidden email] mailing list
>> > https://stat.ethz.ch/mailman/listinfo/r-help
>> > PLEASE do read the posting guide http://www.R-project.org/posting-
>> guide.html
>> > and provide commented, minimal, self-contained, reproducible code.
>>
>> David Winsemius, MD
>> West Hartford, CT
>>
>> ______________________________________________
>> [hidden email] mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide http://www.R-project.org/posting-
>> guide.html
>> and provide commented, minimal, self-contained, reproducible code.
>
> ______________________________________________
> [hidden email] mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide 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

______________________________________________
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Re: Total effect of X on Y under presence of interaction effects

David Winsemius

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 <[hidden email]>  
> 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
>> [hidden email]
>> 801.408.8111
>>
>>
>>> -----Original Message-----
>>> From: [hidden email] [mailto:r-help-bounces@r-
>>> project.org] On Behalf Of David Winsemius
>>> Sent: Wednesday, May 11, 2011 7:48 AM
>>> To: Michael Haenlein
>>> Cc: [hidden email]
>>> 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
>>>>
>>>>     [[alternative HTML version deleted]]
>>>>
>>>> ______________________________________________
>>>> [hidden email] mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> PLEASE do read the posting guide http://www.R-project.org/posting-
>>> guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>> David Winsemius, MD
>>> West Hartford, CT
>>>
>>> ______________________________________________
>>> [hidden email] mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide http://www.R-project.org/posting-
>>> guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>
>> ______________________________________________
>> [hidden email] mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide 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

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Re: Total effect of X on Y under presence of interaction effects

vioravis
This is what I believe is referred to as "supression" in regression, where the correlation correlation between the independent and the dependent variable turns out to be of one sign whereas the regression coefficient turns out to be of the opposite sign.

Read here about supression:

http://www.uvm.edu/~dhowell/gradstat/psych341/lectures/MultipleRegression/multreg3.html

HTH

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Re: Total effect of X on Y under presence of interaction effects

Frank Harrell
In reply to this post by David Winsemius
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 <[hidden email]>  
> 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
>> [hidden email]
>> 801.408.8111
>>
>>
>>> -----Original Message-----
>>> From: [hidden email] [mailto:r-help-bounces@r-
>>> project.org] On Behalf Of David Winsemius
>>> Sent: Wednesday, May 11, 2011 7:48 AM
>>> To: Michael Haenlein
>>> Cc: [hidden email]
>>> 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
>>>>
>>>>     [[alternative HTML version deleted]]
>>>>
>>>> ______________________________________________
>>>> [hidden email] mailing list
>>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>>> PLEASE do read the posting guide http://www.R-project.org/posting-
>>> guide.html
>>>> and provide commented, minimal, self-contained, reproducible code.
>>>
>>> David Winsemius, MD
>>> West Hartford, CT
>>>
>>> ______________________________________________
>>> [hidden email] mailing list
>>> https://stat.ethz.ch/mailman/listinfo/r-help
>>> PLEASE do read the posting guide http://www.R-project.org/posting-
>>> guide.html
>>> and provide commented, minimal, self-contained, reproducible code.
>>
>> ______________________________________________
>> [hidden email] mailing list
>> https://stat.ethz.ch/mailman/listinfo/r-help
>> PLEASE do read the posting guide 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

______________________________________________
[hidden email] mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Frank Harrell
Department of Biostatistics, Vanderbilt University