Date: Fri, 2 May 2008 14:58:59 -0400
Reply-To: Susan Durham <sdurham@BIOLOGY.USU.EDU>
Sender: "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From: Susan Durham <sdurham@BIOLOGY.USU.EDU>
Subject: Re: Hierarchical Models--Centering predictors--WHY
Another advantage of centering when the covariate is involved in
interactions (whether with itself, as a quadratic term, or with another
explanatory variable) lies in the interpretation of the overall tests of
significance.
For example, in Steve's model
MODEL daily_yield=treatment|group covariate covariate*treatment;
the overall test of TREATMENT (from the MIXED procedure I know, and I think
for many other SAS procedures) compares treatment means estimated at
covariate=0. If a zero value for the covariate is far from the mean of the
covariate and if COVARIATE*TREATMENT is not significant, this test can be
very misleading because the comparison is extrapolated beyond the observed
range of the covariate and the regression lines for each level of treatment
are not exactly parallel. The test isn't wrong, it's just testing a
hypothesis that might be different than what many users (like *my*
biologists) expect.
(If COVARIATE*TREATMENT is significant, then we wouldn't be much interested
in the overall test of TREATMENT because treatment differences would vary
depending on the value of COVARIATE.)
This lack of scale invariance is clearly discussed in
LS Aiken & SG West, 1991, Multiple Regression: Testing and Interpreting
Interactions, Sage Publications.
I work with biologists, too, and although they must "suffer" through the
social science framework of this book, they still manage to glean the
information--plus we draw lots of pictures to illustrate the problem.
Cheers,
Susan
On Fri, 2 May 2008 11:33:34 -0700, Steve Denham <stevedrd@YAHOO.COM> wrote:
>Warren,
>
>Give it a try with a dataset. For group 1, let the covariate range from 55
to 85, and for group 2 from 80 to 120 (those are kind of typical pounds of
milk per day numbers in a pretreatment period). Now fit a model that looks
like (all random stuff left out):
>
>MODEL daily_yield=treatment|group covariate;
>
>and look at the lsmeans. All of the group 1 are adjusted upward, all of
group 2 downward, so that it would seem that adjusted means for heifers are
greater than those for mature cows.
>
>Now change the covariate to the deviation from the group mean (centering on
the group mean) and fit
>
>MODEL daily_yield=treatment|group deviate_covariate;
>
>and see what the results look like. Treatment differences remain constant,
but the adjusted means now do not seem paradoxical (mature cows>heifers).
>
>On to the last case, where the heterogeneous slopes model is fit--
>
>MODEL daily_yield=treatment|group covariate covariate*treatment;
>
>When you write the LSMEANS @<value> to explain heterogeneity, it's quickly
apparent that for a fixed pretreatment yield (say 82), groups will be
adjusted in opposite directions. By centering, we can easily look at values
that are 15 pounds below the contemporary group or 15 pounds above the
contemporary group.
>
>I never learned this stuff until I started working with animal geneticists,
who in the animal science field, seem to express everything as a deviation
from the contemporary group mean.
>
>There is a carry over advantage to expressing covariates as a deviation in
matrix operations as well. Think about inverting a big old sparse matrix
with ones and zeros everywhere except one column with values two or three
orders of magnitude larger. Compare the results to the matrix where that
column sums to zero (at least across factors of interest).
>
>Steve Denham
>Associate Director, Biostatistics
>MPI Research, Inc.
>
>----- Original Message ----
>From: Warren Schlechte <Warren.Schlechte@TPWD.STATE.TX.US>
>To: SAS-L@LISTSERV.UGA.EDU
>Sent: Friday, May 2, 2008 11:26:13 AM
>Subject: Re: Hierarchical Models--Centering predictors--WHY
>
>Steve,
>
>I don't follow, why would the LSMeans be higher without centering?
>
>Are you saying the treatment effect looks like it is doing more in first
>lactation cows because the cows are growing, and naturally producing
>more milk? If so, I don't see how centering helps.
>
>I am curious. I too work with biologists, and prefer not to be laughed
>out of the room.
>
>Warren Schlechte
>
>
> -----Original Message-----
> From: Steve Denham [mailto:stevedrd@YAHOO.COM]
> Sent: Thursday, May 01, 2008 12:36 PM
> Subject: Re: Hierarchical Models--Centering predictors--WHY
>
> Centering is good, standardizing not necessarily--as it changes
>the interpretation of any parameters. Peter hit the nail on the head
>regarding centering with the comment about quadratics. Here is another
>example:
>
> The best example I know of with centering is with regard to the
>analysis of milk production by dairy cows.
>
> First lactation animals are still growing and have a lower
>production than older cows.
>
> Now suppose you are running a trial and you have a pretreatment
>milk yield as a covariate. If you have lactation number (=1 for first
>lactation, =2 for all subsequent) in the model, and do not center the
>pretreatment yield on the mean for that lactation, you rapidly will find
>paradoxical (in the biological sense) effects:
>
> LSmeans for the first lactation animals will be greater than
>LSmeans for subsequent lactation animals.
>
> Treatment differences remain constant, provided you do not have
>treatment by covariate interaction included. If that particular
>interaction is included (heterogeneous slopes model) it is vitally
>important that you center the covariate. Failure to do so will almost
>certainly result in something that will cause the biologist you are
>consulting for to laugh you out of the room (does this sound like
>personal experience? Could be...)
>
> Steve Denham
> Associate Director, Biostatistics
> MPI Research, Inc.
>
>
> ----- Original Message ----
> From: Citam <citam.sasl@GMAIL.COM>
> To: SAS-L@LISTSERV.UGA.EDU
> Sent: Thursday, May 1, 2008 12:43:30 PM
> Subject: Re: Hierarchical Models--Centering predictors--WHY
>
> On 5/1/08, Tom White <tw2@mail.com> wrote:
> > Hi Peter and thanks for this response.
> >
> > (1)
> > So, then: If I don't care about easier interpretation or any
>interpretation at all,
> > I only care about how well the model fits (and
>predicts) my data,
> > then, I'll just don't mess with centering.
> > Am I right about this?
> > Because I really won't bother about centering my
>vars.
>
> Correct, adding a constant will not affect the variance.
>However, you may later want to interpret the results. As precedence,
>you may wish to make the extra effort, if it suits your purpose.
>
>
> > (2)
> > Do you know how centering is done?
>
> Consider the STANDARD procedure, available in BASE for one. You
>could also use the MEANS procedure and a datastep.
>
> "The STANDARD procedure standardizes variables in a SAS data set
>to a given mean and standard deviation, and it creates a new SAS data
>set containing the standardized values."
>
>
>
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>
>
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