Jackknife-based test for equality of two regressions between distances. Given two groups of objects, this tests whether the regression involving the distances within one of the groups is compatible with the regression involving the same within-group distances together with the between group distances.
regdistbetweenone(dmx,dmy,grouping,groups=levels(as.factor(grouping))[1:2],rgroup)list of class "regdistbetween" with components
p-value.
difference between regression fits (within-group
together with between-groups distances
minus within-group distances only) at xcenterbetween, see
below.
condition numbers of regressions, see kappa.
list. Output objects of lm within the two groups.
output object of jackknife for difference
between regression fitted values at xcenterbetween.
mean of within-group distances for group species
of explanatory variable, used for centering.
mean of between-groups distances of explanatory
variable (after centering by xcenter); at this point
regression fitted values are computed.
t-statistic.
degrees of freedom of t-statistic according to Welch-Sattertwaithe approximation.
jackknife-estimator of difference between regression
fitted values at xcenterbetween.
jackknife-standard error for
jackest.
vector of jacknife pseudovalues on which the test is based.
see above.
see above.
title to be printed out when using
print.regdistbetween.
dissimilarity matrix or object of class
dist. Explanatory dissimilarities (often these will be proper
distances, but more general dissimilarities that do not
necessarily fulfill the triangle inequality can be used, same for dmy).
dissimilarity matrix or object of class
dist. Response dissimilarities.
something that can be coerced into a factor,
defining the grouping of
objects represented by the dissimilarities dmx and dmy
(i.e., if grouping has length n, dmx and dmy
must be dissimilarities between n objects).
vector of two levels. The two groups defining the
regressions to be compared in the test. These can be
factor levels, integer numbers, or strings, depending on the entries
of grouping.
one of the levels in groups, denoting the group
of which within-group dissimilarities are considered.
Christian Hennig christian.hennig@unibo.it https://www.unibo.it/sitoweb/christian.hennig/en
The null hypothesis that the regressions based on the distances
within group species and based on these distances together with
the between-groups distances are
equal is tested using jackknife pseudovalues. The test statistic is
the difference between fitted
values with x (explanatory variable) fixed at the center of the
between-group distances. The test is run one-sided, i.e., the null
hypothesis is only rejected if the between-group distances are larger
than expected under the null hypothesis, see below. For the jackknife,
observations from both groups are left out one at a time. However, the
roles of the two groups are different (observations from group
species are used in both regressions whereas observations from
the other group are only used in one of them), and therefore the
corresponding jackknife pseudovalues can have different variances. To
take this into account, variances are pooled, and the degrees of
freedom of the t-test are computed by the Welch-Sattertwaithe
approximation for aggregation of different variances.
The test cannot be run and many components will be NA in case that
within-group regressions or jackknifed within-group regressions are
ill-conditioned.
This was implemented having in mind an application in which the
explanatory distances represent geographical distances, the response
distances are genetic distances, and groups represent species or
species-candidates. In this application, for testing whether the
regression patterns are compatble with the two groups behaving like a
single species, one would first use regeqdist to test whether a
joint regression for the within-group distances of both groups makes
sense. If this is not rejected, regdistbetween is run to see
whether the between-group distances are compatible with the
within-group distances.
If a joint regression on
within-group distances is rejected by regeqdist,
regdistbetweenone can be
used to test whether the between-group distances are at least
compatible with the within-group distances of one of the groups, which
can still be the case within a single species, see Hausdorf and Hennig
(2019). This
is only rejected if the between-group
distances are larger than expected under equality of regressions,
because if they are smaller, this is not an indication against the
groups belonging together genetically. To this end,
regdistbetweenone needs to be run twice using both groups as
species. This will produce two p-values. The null hypothesis
that the regressions are compatible for at least one group can be
rejected if the maximum of the two p-values is smaller than the chosen
significance level.
Hausdorf, B. and Hennig, C. (2019) Species delimitation and geography. Submitted.
regeqdist, regdistbetweenone
options(digits=4)
data(veronica)
ver.geo <- coord2dist(coordmatrix=veronica.coord[173:207,],file.format="decimal2")
vei <- prabinit(prabmatrix=veronica[173:207,],distance="jaccard")
species <-c(rep(1,13),rep(2,22))
loggeo <- log(ver.geo+quantile(as.vector(as.dist(ver.geo)),0.25))
rtest3 <-
regdistbetweenone(dmx=loggeo,dmy=vei$distmat,grouping=species,groups=c(1,2),rgroup=1)
print(rtest3)
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