Dear Tsung,
The procD.allometry function performs two basic processes when groups are
provided. First, it does a homogeneity of slopes (HOS) test. This test
ascertains whether two or more groups have parallel or unique slopes (the
latter meaning at least one groups’s slope is different than the others). The
HOS test constructs two linear models: shape ~ size + group and shape ~ size +
group + size:group, and performs an analysis of variance to determine if the
size:group interaction significantly reduces the residual error produced.
(Note: log(size) is a possible and default choice in this analysis.)
After this test, procD.allometry then provides an analysis of variance on each
term in the resulting model from the HOS test.
Regarding your question, if the HOS test reveals there is significant
heterogeneity in slopes, the coefficients returned allow one to find the unique
linear equations, by group, which would be found from separate runs on
procD.allometry, one group at a time. If the HOS test reveals that there is
not significant heterogeneity in slopes, the coefficients constrain the slopes
for different groups to be the same (parallel).
Finally, and I think more to your point, the projected regression scores are
found by using for a (in the Xa calculation you note) the coefficients that
represent a common or individual slope from the linear model produced. The
matrix of coefficients, B, is arranged as first row = intercept, second row =
common slope, next rows (if applicable) are coefficients for the group factor
(essentially change the intercept, by group), and finally, the last rows are
the coefficients for the size:group interaction (if applicable), which change
the common slope to match each group’s unique slope. Irrespective of the
complexity of this B matrix, a is found as the second row. If you run
procD.allometry group by group, it is the same as (1) asserting that group
slopes are unique and (2) changing a to match not the common slope, but the
summation of the common slope and the group-specific slope adjustment. One
could do that, but would lose the ability to compare the groups in the same
plot, as each group would be projected on a different axis.
Hope that helps.
Mike
> On Dec 8, 2016, at 3:37 AM, Tsung Fei Khang <tfkh...@um.edu.my> wrote:
>
> Hi all,
>
> I would like to use procD.allometry to study allometry in two species.
>
> I understand that the function returns the regression score for each specimen
> as Reg.proj, and that the calculation is obtained as:
> s = Xa, where X is the nxp matrix of Procrustes shape variables, and a is the
> px1 vector of regression coefficients normalized to 1. I am able to verify
> this computation from first principles when all samples are presumed to come
> from the same species.
>
> However, what happens when we are interested in more than 1 species (say 2)?
> I could run procD.allometry by including the species labels via f2=~gps,
> where gps gives the species labels. Is there just 1 regression vector (which
> feels weird, since this should be species-specific), or 2? If so, how can I
> recover both vectors? What is the difference of including f2=~gps using all
> data, compared to if we make two separate runs of procD.allometry, one for
> samples from species 1, and another for samples from species 2?
>
> Thanks for any help.
>
> Rgds,
>
> TF
>
>
>
>
>
>
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