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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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