In general (you don't give enough information for a more detailed
answer), when the regression slopes are the same (or, when one is
willing to believe in homogeneity of regression), it doesn't really
matter where (that is, for what particular value of the covariate) one
asks what the differences are: the lines (or hyperplanes) are parallel,
the differences are the same everywhere. Consequently one chooses (or
perhaps more realistically it is usually chosen for one!) to assess
those differences where one's precision is greatest, that is at the mean
of the covariate (or at the center of gravity of the several
covariates).
When one DOESN'T believe in homogeneity of regression (that is,
when the regression lines (or hyperplanes) are NOT parallel, the
differences between groups are NOT the same everywhere. If the lines
actually cross (in the domain of interest: since they're not parallel,
they have to cross SOMEwhere, but the crossing(s) may be outside the
region one can observe), then for some values of the covariate group A
(say) has higher scores than group B, for other values group A has lower
scores than group B, and for intermediate values the groups are not
detectably different on average scores. With three groups, there are a
distinctly larger number of possibilities.
This doesn't prevent one from carrying out an analysis (unless
one is trying to use a program that will not permit heterogeneous slopes
-- in that case, use a multiple regression program instead of an "ANCOV"
program): it just makes the description of results more complicated,
both in terms of your own perception and understanding of what appears
to be going on in your data and in terms of conveying that understanding
to your readers.
If in fact the regression lines do not cross in the region of
interest, and if the groups differ enough that they appear separated
throughout the region, one might be willing to oversimplify the model by
fitting an ANCOV with a common slope (in order to get some improvement
in precision), while acknowledging in discussion that the model appears
to be simpler than the reality. (Of course, your ANOVA model that you
started with is even simpler.:-) You could even display two small
graphs for illustration: one showing parallel lines and one showing the
nonparallel lines.
I hope this will have been helpful. -- DFB.
On Wed, 26 Feb 2003, Clark Dickin [aka DCD] wrote (edited):
> [In] some data [on] older adults [I] found a significant difference
> (p < .05) between the three groups in terms of their activity level.
> Subsequent[ly] I am contrasting the three groups on a balance measure
> and deciding whether or not to use activity level as a covariate ...
>
> The [potential problem] is ... that the main effect for activity level
> violated the assumptions of the homogeneity of regression slopes when
> compared with one of the DV used in the study.
It is a little unclear precisely how this relates to the analysis you
are contemplating. You seem to be saying that the slope of one (only
one?) of the DVs (on, presumably, some predictor(s) or other) was not
homogeneous when "activity level", your proposed covariate, was one of
the predictors. In your proposed analysis, the DV appears to be "a
balance measure"; is that the DV on which you found heterogeneity of
regression on "activity level"? If so, my comments at the beginning of
this post are relevant. If not, why are you worried?
> I do have a small sample size
This of course has its own problems. You may not have enough power to
detect what would otherwise be useful differences, both of slopes and
between groups. But you presumably know that.
> and consequently did not use the covariate analysis but wanted to
> see if I was justified based on the violation listed above.
"Justified"? Are you seeking justification by faith, or justification
by works? Neither seems altogether germane to a statistical analysis.
What you should really be asking (and you can only ask it of your data,
not of us folks!) is whether a more complicated model for the data
<> helps you to understand the behavior of the data
<> permits you to see things in the data that a simpler analysis
doesn't turn up.
If the answer to either of these is "yes", then the more complicated
model is, to some degree, justified. (Notice that it's the MODEL, not
YOU, that might be thus justified!) If not, then the additional
complexity has not bought you anything worthwhile and you might well be
moved to work with the simpler model.
-----------------------------------------------------------------------
Donald F. Burrill [EMAIL PROTECTED]
56 Sebbins Pond Drive, Bedford, NH 03110 (603) 626-0816
.
.
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