Dear Tsung, The geomorph function, advanced.procD.lm, allows one to extract group slopes and model coefficients. In fact, procD.allometry is a specialized function that uses advanced.procD.lm to perform the HOS test and then uses procD.lm to produce an ANOVA table, depending on the results of the HOS test. It also uses the coefficients and fitted values from procD.lm to generate the various types of regression scores. In essence, procD.allometry is a function that carries out several analyses with geomorph base functions, procD.lm and advanced.procD.lm, in a specified way. By comparison, the output is more limited, but one can use the base functions to get much more output.
In advanced.procD.lm, if one specifies groups and a slope, one of the outputs is a matrix of slope vectors. Also, one can perform pairwise tests to compare either the correlation or angle between slope vectors. Regarding the operation of the HOS test, it is a permutational test that does the following: calculate the sum of squared residuals for a “full” model, shape ~ size + group + size:group and the same for a “reduced” model, shape ~ size + group. (The sum of squared residuals is the trace of the error SSCP matrix, which is the same of the sum of the summed squared residuals for every shape variable.) The difference between these two values is the sum of squares for the size:group effect. If significantly large (i.e., is found with low probability in many random permutations), one can conclude that the coefficients for this effect are collectively large enough to justify this effect should be retained, as the slope vectors are (at least in part) not parallel. If not significant, than the slope vectors are approximately parallel, and the effect can be removed from the model. A randomized residual permutation procedure is used, which randomizes the residual vectors of the reduced model in each random permutation to obtain random pseudo-values, repeating the sum of squares calculations each time. Regarding your final question, yes, you are correct. In a case like this, one might conclude that logCS is not a significant source of shape variation, and proceed with other analyses that do not include it as a covariate. In either case - whether is is retained as a covariate or excluded - advanced.procD.lm will allow one to perform pairwise comparison tests among groups. Cheers! Mike > On Dec 11, 2016, at 10:56 AM, Tsung Fei Khang <[email protected]> wrote: > > Dear Mike, > > Many thanks for the reply! > > When the procD.allometry function performs HOS test with multiple group > labels given, does it compute the regression vectors for each group, and then > tests whether the coefficients of these vectors were equal, using some > multivariate statistical test? If so, is there an option that outputs the > regression vectors? Given the high frequency of the latter being discussed in > the primary GM literature, it seems important to be able to extract this > result from the function. > > Finally, on the interpretation side - If group variation is significant, but > not logCS, then under the model shape~size+group, does this imply that shape > variation is mainly explained by variation in species, and allometry is > absent? > > Regards, > > T.F. > > On Thursday, December 8, 2016 at 6:08:17 PM UTC+8, Mike Collyer wrote: > 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 <[email protected] <>> 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 >> >> >> >> >> >> >> " PENAFIAN: E-mel ini dan apa-apa fail yang dikepilkan bersamanya ("Mesej") >> adalah ditujukan hanya untuk kegunaan penerima(-penerima) yang termaklum di >> atas dan mungkin mengandungi maklumat sulit. 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" >> >> -- >> MORPHMET may be accessed via its webpage at http://www.morphometrics.org >> <http://www.morphometrics.org/> >> --- >> You received this message because you are subscribed to the Google Groups >> "MORPHMET" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to morphmet+u...@ <>morphometrics.org <http://morphometrics.org/>. > > > " PENAFIAN: E-mel ini dan apa-apa fail yang dikepilkan bersamanya ("Mesej") > adalah ditujukan hanya untuk kegunaan penerima(-penerima) yang termaklum di > atas dan mungkin mengandungi maklumat sulit. Anda dengan ini dimaklumkan > bahawa mengambil apa jua tindakan bersandarkan kepada, membuat penilaian, > mengulang hantar, menghebah, mengedar, mencetak, atau menyalin Mesej ini atau > sebahagian daripadanya oleh sesiapa selain daripada penerima(-penerima) yang > termaklum di atas adalah dilarang. 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