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CPS1972 Livio Corain et al.
brain region/layer and the interaction between population and region, and
are population-and-region varying scale coefficients which may depend,
2
through monotonic functions, on main treatment effects and brain
region/layer location effects . Basically, according to the so-called
multivariate generalized Behrens-Fisher problem (Yanagihara and Yuan, 2005),
the proposed data representation model is a quite general less-demanding
two-way linear effect nonparametric model where specific location and scale
effects are both allowed to differ across populations and brain regions.
Since the study’s main goal is to compare populations, both jointly across
all regions and separately within any given region, we are actually inferring on
the sum of location coefficients + () . By using the Roy’s Union-
Intersection testing approach (Roy, 1953; Pesarin and Salmaso, 2010), let us
formalize, separately for the location and scatter parameters, the comparison
between the j-th and the h-th population with the null and alternative
hypotheses as follow
where τ = τ + (τβ) ., and = 1, … , , = 1, . . , , are the reference
indexes for each brain region and univariate response, i.e. cell morphometric
indicator, respectively. It is worth noting that hypotheses (2) refers to a general
version of the so-called generalized Beherens-Fisher problem (Yanagihara and
Yuan, 2005). Under the null hypotheses of joint equality, either in location and
in scatter, actual data are exchangeable random components that can be
permuted between combinations of groups and strata in order to derive,
separately for the location and scatter problems, two multivariate directional
p-values. Once we removed the nuisance individual location effect + ()
by computing suitable individual-free residuals, as univariate location and
scatter test statistic we respectively used the difference of sample means and
squared deviations along with the Fisher’s combining function, as combination
strategy to derive the multivariate p-values (Pesarin and Salmaso, 2010). In this
view, when just one between 0(ℎ) and 0(ℎ) are true, the permutation
approach can be considered as an approximated nonparametric testing
solution. For a more in depth understanding of the testing procedure we
shortly sketched here we refer the readers to Pesarin and Salmaso (2010).
By exploiting the multivariate one-sided alternatives in expression (2), we
may derive two sorts of location and scatter rankings using the ranking
methodology proposed by Arboretti et al. (2014). In fact, by suitable
combining information from directional multivariate p-values, the possible
2
underline latent ordering among and parameters can be properly
estimated. In a nutshell, the rationale behind the ranking within a multivariate
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