Date:         Fri, 24 Jun 2005 17:33:19 -0700
Reply-To:     cassell.david@EPAMAIL.EPA.GOV
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         "David L. Cassell" <cassell.david@EPAMAIL.EPA.GOV>
Subject:      Re: ANCOVA with unequal slope and quantitative treatments
In-Reply-To:  <LbEue.13$Fi1.31262@weber.videotron.net>
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Louis Blais <louis.blais@MRNF.GOUV.QC.CA> wrote back:
> The treatments are quantity of lime on trees and the levels are fixed,
but
> not equaly spaced. Accordingly to Montgomery's "Design and Analysis of
> Experiments, 6th edition" (2005), we can analyse this kind of
treatements
> with orthogonal polynomial contrasts (we can obtain the values of the
> contrast with the ORPOL function in SAS/IML product). Also, the
dependant
> variable is the growth of basal area and the covariable is the growth
of
> each tree before the treatements were applied. These two variables are
> highly correlated (more specificaly, I have an ANCOVAR because I have
to
> analyse a 10 years repeated measures of annual growth of basal area,
that's
> why I use PROC MIXED). So, the linear and the quadratic part of the
function
> is applied to the treatments levels, not on the covariable. This kind
of
> analysis respect the factorial design and the fact that trees are not
equal
> in term of growth of basal area before the application of the lime's
> treatments. But, my problem is that I have not the parallelism of the
slopes
> of the covariable and the dependent variable for each treatment. In
> qualitative treatments, we can compare treatments at some levels of
the
> covariable with the LSMEANS statement. I want to use the same kind of
> solution with the quantitative treatments with the ESTIMATE statement.

First off, I would analyze your data differently.  I think you
have a repeated measures problem, with the pre-treatment growth
of the tree as a point 0 in time.  Even if your time points are
not equally spaced, you can address this in PROC MIXED by using
one of the *spatial* covariance structures: SP(POW), SP(GAU), etc.
So you have some time variable (which for convenience and lack of
originality, I'll call TIME).  Create an exact duplicate (let's
call it TIME2) like this:

    time2 = time;

Declare TIME as one of your class variables, but not TIME2.
And then you can write a statement in PROC MIXED like this:

    repeated time / type=sp(pow)(time2) sub=tree r rcorr;

The trick is that TIME is treated as discrete, with the values
of TIME being the levels for each observation.  But TIME2 is *not*
in the CLASS statement, and so is treated as continuous, and can
therefore work in the SP(whatever) covariance structure.

I think that if you take this approach, you'll have fewer problems
with your model, and you hopefully won't have to struggle with the
interaction terms that were bugging you.

But you will need to check your diagnostic plots.  Since you're
working with basal area, you'll want to see if you do in fact get
normality.  Alternatively, you might find that the DBH is better
behaved, and might avoid some of the issues like quadratic terms.
There is a huge amount of lit on this in the silviculture journals,
and I imagine you're much more familiar with it than I am.

HTH,
David
--
David Cassell, CSC
Cassell.David@epa.gov
Senior computing specialist
mathematical statistician