Date:         Mon, 15 Dec 2008 15:51:11 -0800
Reply-To:     stringplayer_2@yahoo.com
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From:         Dale McLerran <stringplayer_2@YAHOO.COM>
Subject:      Re: Regression: do you always need main effects with interactions?
In-Reply-To:  <000001c95f07$a7bd2350$b3c9678e@soeh.ad>
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In general, I would concur that the main effects should always
be included in the model wherever there is an interaction effect
estimated.  There is at least one instance that I can think of
where this is not (in my opinion) necessary.  Suppose that one
has a multivariate response model.  The multivariate response
model with a vector of length q for each subject is often
estimated with code something like the following:

  proc mixed data=multvar_resp;
    class ID response response_indic ...;
    model y = response response*X1 ... response*X{k} / noint s;
    repeated response_indic / subject=ID type=un;
  run;

Here, each subject has a vector of responses of length up to m
RESPONSE_INDIC takes on values 1, 2, ..., m to indicate which
component of the vector is being modeled.  The variable
RESPONSE is assumed to be a character variable which has a 1-1
mapping with RESPONSE_INDIC.  RESPONSE identifies by a name
which variable is being modeled rather than by position in
the vector.

The effects on the model statement have the following
interpretations:

    RESPONSE     - the intercept for the j-th response variable,
                   j=1,2,...,m
    RESPONSE*X1 - the slope coefficient for predictor X1
                  regressed on the j-th response variable
    ...


We could rewrite the model statement as

    model y = response X1 response*X1 ... X{k} response*X{k} / s;

This revised model would produce exactly the same likelihood
statistics, but the parameters of the model would be interpreted
as

    Intercept   - The intercept for the last response variable
    RESPONSE    - Intercept offset for the j-th response
                  variable.  That is, intercept for response
                  variable j, j=1,2,...,m minus intercept for
                  response variable m.
    X1          - Slope for the last response variable for
                  predictor variable X1
    RESPONSE*X1 - Slope offset for the j-th response
                  variable.
    ...


The assumption in the multivariate response model is that there
are differences in slope and intercept terms across response
variables.  We have no interest in testing whether the difference
is significant.  It is assumed to exist, and we want to estimate
the parameters of the multivariate response model.

Although I would say that as a rule of thumb it is not acceptable
to eliminate main effects and only keep interaction effects in
the model, I am perfectly comfortable with the multivariate
response model which excludes the intercept and main effects of
the predictors X1, ..., X{k}.  There may be some other instances
where elimination of part of the main effects is acceptable.
Before passing judgment, I would want to know a little more
context.

Dale

---------------------------------------
Dale McLerran
Fred Hutchinson Cancer Research Center
mailto: dmclerra@NO_SPAMfhcrc.org
Ph:  (206) 667-2926
Fax: (206) 667-5977
---------------------------------------


--- On Mon, 12/15/08, Cornel Lencar <clencar@INTERCHANGE.UBC.CA> wrote:

> From: Cornel Lencar <clencar@INTERCHANGE.UBC.CA>
> Subject: Re: Regression: do you always need main effects with interactions?
> To: SAS-L@LISTSERV.UGA.EDU
> Date: Monday, December 15, 2008, 2:51 PM
> From an experimental design perspective, analyzing the
> interaction btw
> treatment A and B without any regards to the actual
> treatment effects,
> especially if they are non-significant, is a blasphemy...It
> has to do with
> the actual components of variance and I am not fit to go
> there without
> checking first a couple of refernces...
>
> Cornel
>
> -----Original Message-----
> From: SAS(r) Discussion [mailto:SAS-L@LISTSERV.UGA.EDU] On
> Behalf Of Mary
> Sent: Monday, December 15, 2008 12:59 PM
> To: SAS-L@LISTSERV.UGA.EDU
> Subject: Re: Regression: do you always need main effects
> with interactions?
>
> Hum,   I've never seen them left out.
>
> One thing of concern would be whether the interpretation of
> a
> significant weight on the interaction term would be the
> same mathematically
> if you left out the main terms.
>
> In Kleinbaum/Kupper, Applied Regression Analysis, I find
> this:
> "In general, then, one might say that "no
> interaction" is synonymous with
> "parallelism" in the sense that the response
> curves of Y versus T for
> various
> fixed of C are parallel; in other words, these response
> curves (which
> may be linear or nonlinear) have the same general
> shape..."
>
> It seems to me this implies that the main terms must be in
> the model.
>
> -Mary
>   ----- Original Message -----
>   From: Scott Czepiel
>   To: SAS-L@LISTSERV.UGA.EDU
>   Sent: Monday, December 15, 2008 2:43 PM
>   Subject: Regression: do you always need main effects with
> interactions?
>
>
>   I got into a statistical tussle with a colleague last
> week when I insisted
>   that any model containing interaction effects must also
> include the main
>   effects which compose the interactions.  In other words:
>
>   model y = a b a*b
>
>   I learned that if you have a model containing the a*b
> interaction, then
> you
>   must also keep both a and b in the model even if their
> individual
>   contributions are not significant.  My colleague
> disagreed and when I
> looked
>   it up in some of the usual references (Agresti, Draper
> & Smith, Scott
> Long,
>   Nelder & Wedderburn), I was actually shocked that I
> could not find any
>   corroborating evidence for this piece of dogma that has
> followed me ever
>   since my grad school days.
>
>   I was brought up to believe that not including those main
> effects would
>   render a model at best uninterpretable, and at worst,
> downright
>   mathematically unsound.  Without the main effects as
> anchor points, the
>   interaction between them is meaningless.  The intent of
> an interaction
>   effect is to explain the variance that cannot be
> explained with main
> effects
>   alone, in situations where the dependent variable varies
> differently at
>   different combinations of the main effects.  Still, you
> cannot remove the
>   main effects from such a model, because doing so would
> prevent you from
>   understanding the additive effect of the interaction over
> and above the
>   individual main effects.
>
>   Which is it?  Is there a real basis to my claim or was it
> just a
> hand-waving
>   simplification that an old professor threw at us to avoid
> making the
>   interpretation of the model more difficult to explain?
>
>   Mainly yours,
>   S.C.