regression with interaction of group variable

[Marnie LaNoue]

Hi All -

Any comments/advice appreciated!

I have a situation with 2 groups, and 3 continuous predictor variables. THe
research question is mostly regarding whether the predictors differ across
the groups. I created product vectors of (group x predictor) for all
crossings of group and predictors. Does it make sense to run a regression
with just the product vectors as predictors (ignoring the vectors of
predictor variables?)

Another note: I am using path analysis on this data as well, modeling the
relationships for the groups separately to adress certain predictions
regarding the relationships between predictors and outcomes in the groups,
but I want the regression as an omnibus test of differences in the
predictors. Am I correctly interpreting significant coefficients in the
regression of the product vectors as answering this question?

[Dale Glaser]

Marnie..the general (though I'm not sure how universally it is subscribed to) rule is in testing the categorical x continuous variable interaction (see Aiken and West, 1991) is to enter the individual variables (not sure why you call them 'vectors') a the first step and then the multiplicative term at the subsequent step of entry, and then examine the incremental statistics to assess if the interaction term added variance above and beyond the constituent variables (also, this assumes you have centered the continuous level predictor(s)).  This is akin to running a fixed-factor 2-way interaction where the interaction is examined over the main effects...........I recall a discussion on this listserv many years ago where someone provided a rationale for only entering the interaction term, but I don't recall what came of the justifcation.

  Dale

Marnie LaNoue <[hidden email]> wrote:
  Hi All -

Any comments/advice appreciated!

I have a situation with 2 groups, and 3 continuous predictor variables. THe
research question is mostly regarding whether the predictors differ across
the groups. I created product vectors of (group x predictor) for all
crossings of group and predictors. Does it make sense to run a regression
with just the product vectors as predictors (ignoring the vectors of
predictor variables?)

Another note: I am using path analysis on this data as well, modeling the
relationships for the groups separately to adress certain predictions
regarding the relationships between predictors and outcomes in the groups,
but I want the regression as an omnibus test of differences in the
predictors. Am I correctly interpreting significant coefficients in the
regression of the product vectors as answering this question?



Dale Glaser, Ph.D.
Principal--Glaser Consulting
Lecturer--SDSU/USD/CSUSM/AIU
3115 4th Avenue
San Diego, CA 92103
phone: [phone]
fax: [phone]
email: [hidden email]
website: www.glaserconsult.com

[JOHN ANTONAKIS]

Hi:

Comments below

> Hi All -
>
> Any comments/advice appreciated!
>
> I have a situation with 2 groups, and 3 continuous
> predictor variables. THe research question is mostly
> regarding whether the predictors differ across the groups.
> I created product vectors of (group x predictor) for all
> crossings of group and predictors. Does it make sense to
> run a regression with just the product vectors as
> predictors (ignoring the vectors of predictor variables?)
>

No. The cross-products are scaled-on the main effects and
the two-way effects. You MUST control for main and two-way
effects before you can use the three-way effects to predict
Y. See, for example,

Evans, M. G. 1991. The problem of analyzing multiplicative
composites. American Psychologist, 46: 6-15.

Any short of doing what I have suggested, will result in
"profoundly and fatally flawed" results (to quote from
Evans).

> Another note: I am using path analysis on this data as
> well, modeling the relationships for the groups separately
> to adress certain predictions regarding the relationships
> between predictors and outcomes in the groups, but I want
> the regression as an omnibus test of differences in the
> predictors. Am I correctly interpreting significant
> coefficients in the regression of the product vectors as
> answering this question?

It is not exactly the same thing, but it will give you
similar results.  In the multiple group situation you have
two equations (using one predictor here to keep things
simple):

y(G1)=b0 + b1x + e
y(G2)=b0 + b1x + e

If you estimate a model in which b1 is constrained to be
equal, you notice that the error terms are still
independent. If you pool the data and estimate the following
equation (where z is a grouping dummy variable) you have:

y=b0 + b1x + b2z + b3xz + e

The e term here is now pooled. Here, if b3 is significantly
different from zero it is the same as saying that b1 is
different in both groups. However, the t-statistic will not
be precisely the same because of the way in which the error
term is handled.

HTH,
John.


___________________________________

Prof. John Antonakis
School of Management and Economics
University of Lausanne
Internef #527
CH-1015 Lausanne-Dorigny
Switzerland

Tel: ++41 (0)21 692-3438
Fax: ++41 (0)21 692-3305

http://www.hec.unil.ch/jantonakis
___________________________________

[JOHN ANTONAKIS]

Hi Dale:

You said: "someone provided a rationale for only entering
the interaction term, but I don't recall what came of the
justifcation."

It may have been me (and at that time I was mistaken--I have
now learned my lesson). What I did was to pool the main and
two-way effects into the error term. However, in the end,
you get the same results as if you estimated the main and
two-way effects first prior to entering the three-way
interaction.

Best,
John.


----- Original Message -----
Expéditeur: Dale Glaser <[hidden email]>
à: [hidden email]
Sujet: Re: regression with interaction of group variable
Date: Fri, 8 Sep 2006 11:32:02 -0700

> Marnie..the general (though I'm not sure how universally
> it is subscribed to) rule is in testing the categorical x
> continuous variable interaction (see Aiken and West, 1991)
> is to enter the individual variables (not sure why you
> call them 'vectors') a the first step and then the
> multiplicative term at the subsequent step of entry, and
> then examine the incremental statistics to assess if the
> interaction term added variance above and beyond the
> constituent variables (also, this assumes you have
> centered the continuous level predictor(s)).  This is akin
> to running a fixed-factor 2-way interaction where the
> interaction is examined over the main effects...........I
> recall a discussion on this listserv many years ago where
> someone provided a rationale for only entering the
> interaction term, but I don't recall what came of the
> justifcation.
>
>   Dale
>
> Marnie LaNoue <[hidden email]> wrote:
>   Hi All -
>
> Any comments/advice appreciated!
>
> I have a situation with 2 groups, and 3 continuous
> predictor variables. THe research question is mostly
> regarding whether the predictors differ across the groups.
> I created product vectors of (group x predictor) for all
> crossings of group and predictors. Does it make sense to
> run a regression with just the product vectors as
> predictors (ignoring the vectors of predictor variables?)
>
> Another note: I am using path analysis on this data as
> well, modeling the relationships for the groups separately
> to adress certain predictions regarding the relationships
> between predictors and outcomes in the groups, but I want
> the regression as an omnibus test of differences in the
> predictors. Am I correctly interpreting significant
> coefficients in the regression of the product vectors as
> answering this question?
>
>
>
> Dale Glaser, Ph.D.
> Principal--Glaser Consulting
> Lecturer--SDSU/USD/CSUSM/AIU
> 3115 4th Avenue
> San Diego, CA 92103
> phone: [phone]
> fax: [phone]
> email: [hidden email]
> website: www.glaserconsult.com

___________________________________

Prof. John Antonakis
School of Management and Economics
University of Lausanne
Internef #527
CH-1015 Lausanne-Dorigny
Switzerland

Tel: ++41 (0)21 692-3438
Fax: ++41 (0)21 692-3305

http://www.hec.unil.ch/jantonakis
___________________________________

[Dale Glaser]

ah yes, you just tapped my memory banks!!.....and John, I also recall aligned with that thread quite a few years ago I think someone commented that in SAS there is some type of function that permits (correcting the error term as you allude to?) testing such a model......to be honest, I still don't know how I would be able to justify it....Dale

John Antonakis <[hidden email]> wrote:  Hi Dale:

You said: "someone provided a rationale for only entering
the interaction term, but I don't recall what came of the
justifcation."

It may have been me (and at that time I was mistaken--I have
now learned my lesson). What I did was to pool the main and
two-way effects into the error term. However, in the end,
you get the same results as if you estimated the main and
two-way effects first prior to entering the three-way
interaction.

Best,
John.


----- Original Message -----
Expéditeur: Dale Glaser
à: [hidden email]
Sujet: Re: regression with interaction of group variable
Date: Fri, 8 Sep 2006 11:32:02 -0700

> Marnie..the general (though I'm not sure how universally
> it is subscribed to) rule is in testing the categorical x
> continuous variable interaction (see Aiken and West, 1991)
> is to enter the individual variables (not sure why you
> call them 'vectors') a the first step and then the
> multiplicative term at the subsequent step of entry, and
> then examine the incremental statistics to assess if the
> interaction term added variance above and beyond the
> constituent variables (also, this assumes you have
> centered the continuous level predictor(s)). This is akin
> to running a fixed-factor 2-way interaction where the
> interaction is examined over the main effects...........I
> recall a discussion on this listserv many years ago where
> someone provided a rationale for only entering the
> interaction term, but I don't recall what came of the
> justifcation.
>
> Dale
>
> Marnie LaNoue wrote:
> Hi All -
>
> Any comments/advice appreciated!
>
> I have a situation with 2 groups, and 3 continuous
> predictor variables. THe research question is mostly
> regarding whether the predictors differ across the groups.
> I created product vectors of (group x predictor) for all
> crossings of group and predictors. Does it make sense to
> run a regression with just the product vectors as
> predictors (ignoring the vectors of predictor variables?)
>
> Another note: I am using path analysis on this data as
> well, modeling the relationships for the groups separately
> to adress certain predictions regarding the relationships
> between predictors and outcomes in the groups, but I want
> the regression as an omnibus test of differences in the
> predictors. Am I correctly interpreting significant
> coefficients in the regression of the product vectors as
> answering this question?
>
>
>
> Dale Glaser, Ph.D.
> Principal--Glaser Consulting
> Lecturer--SDSU/USD/CSUSM/AIU
> 3115 4th Avenue
> San Diego, CA 92103
> phone: [phone]
> fax: [phone]
> email: [hidden email]
> website: www.glaserconsult.com

___________________________________

Prof. John Antonakis
School of Management and Economics
University of Lausanne
Internef #527
CH-1015 Lausanne-Dorigny
Switzerland

Tel: ++41 (0)21 692-3438
Fax: ++41 (0)21 692-3305

http://www.hec.unil.ch/jantonakis
___________________________________



Dale Glaser, Ph.D.
Principal--Glaser Consulting
Lecturer--SDSU/USD/CSUSM/AIU
3115 4th Avenue
San Diego, CA 92103
phone: [phone]
fax: [phone]
email: [hidden email]
website: www.glaserconsult.com

[statisticsdoc]

Stephen Brand
www.statisticsdoc.com

Marnie,

To address the first question, the significance of the cross-product vectors must be assessed when they are entered after the main effects for the continuous and categorical variables.  By cross-product vectors, I mean the interaction terms for the vectors representing levels of the categorical variable times the continuous variables.

The cross-product vectors should be entered as a block - if the block of cross-product vectors is significant, then the significance of the beta for each of the cross-product vectors can be considered (in an equation containing all of the cross-product vectors and main effects).

To address the second question, a significant beta for a single cross-product term (under the conditions described above) indicates that the continuous variable predicts differentially in that group relative to how it predicts for the reference group (depending on how you set up the coding for the categorical variable).

HTH,

Stephen Brand


---- Marnie LaNoue <[hidden email]> wrote:

> Hi All -
>
> Any comments/advice appreciated!
>
> I have a situation with 2 groups, and 3 continuous predictor variables. THe
> research question is mostly regarding whether the predictors differ across
> the groups. I created product vectors of (group x predictor) for all
> crossings of group and predictors. Does it make sense to run a regression
> with just the product vectors as predictors (ignoring the vectors of
> predictor variables?)
>
> Another note: I am using path analysis on this data as well, modeling the
> relationships for the groups separately to adress certain predictions
> regarding the relationships between predictors and outcomes in the groups,
> but I want the regression as an omnibus test of differences in the
> predictors. Am I correctly interpreting significant coefficients in the
> regression of the product vectors as answering this question?

--
For personalized and experienced consulting in statistics and research design, visit www.statisticsdoc.com

[Richard Ristow]

A general comment; Dale Glaser's post is just a useful starting point.

At 02:32 PM 9/8/2006, Dale Glaser wrote:

>The general rule is in testing the categorical x continuous variable
>interaction (see Aiken and West, 1991) is to enter the individual
>variables at the first step and then the multiplicative term at the
>subsequent step of entry, and then examine the incremental statistics
>to assess if the interaction term added variance above and beyond the
>constituent variables (also, this assumes you have centered the
>continuous level predictor(s)).

This may go without saying, but it doesn't seem to have been said
explicitly: When you enter the category level x continuous variables,
also enter the category indicator variable, or variables; but do NOT
include the category indicator variables in the group test. That avoids
a test that's partly testing whether the two groups have different mean
values of the dependent.

That may be what "centering the continuous predictors" is meant to
avoid, as well. Unless I'm missing something badly, I'd think it
easier, and more reliable, to enter the level indicator variables (it
amounts to letting the 'constant' be adjusted between levels), and
don't worry about centering the continuous predictors.