Page 14 - Contributed Paper Session (CPS) - Volume 8

CPS2144 Laura Antonucci et al.
Dij = (Xij(post) −Xij(pre),Yij(post) −Yij(pre), Zij(post) −Zij(pre)),

i = 1,...,n and j ∈{apex, entry point}. Formalizing we are interested in testing
the following system of hypotheses
H0: Pj,post = Pj,pre ∀j
H1: Pj,post ≠ Pj,pre for at least one j
where Pj,post and Pj,pre are the multivariate distributions of responses post and
presurgery respectively, and j ∈ {apex, entry point}. If we assume in both
groups the multivariate errors of positioning to be normally distributed, an
unconditional solution is represented by the parametric paired Hotelling T 2
test. However, this distributional assumption may not be true, and departures
from this assumption can potentially lead to incorrect conclusions.
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Furthermore, the T test fails to provide an easily implemented one-sided
(directional) hypothesis test (Blair et al., 1994) and it does not allow to
investigate on partial aspects involved (marginal coordinates), giving only a
global result. It is also worth to underline that patients enrolled in this study
were not randomly selected, that is one of the assumptions regarding the
validity of Hotelling T test. For this reason, we used the permutation test and
2
in particular the nonparametric combination methodology described in
Pesarin and Salmaso 2010. Permutation tests are conditional inferential
procedures in which conditioning is with respect to the sub-space associated
with the set of sufficient statistics in the null hypothesis for all nuisance entities,
including the underlying known or unknown distribution. A sufficient
condition for properly applying permutation tests is that the null hypothesis
implies that observed data are exchangeable with respect to groups. When
exchangeability may be assumed in H0, the similarity and unbiasedness
property allow for a kind of weak extension of conditional to unconditional
inferences, irrespective of the underlying population distribution and the way
sampling data are collected. Therefore, this weak extension may be made for
any sampling data, even if they are not collected by welldesigned sampling
procedure (Pesarin, 2002). Permutation tests do not require assumptions
and/or approximations that may be difficult to meet in real data. In order to
solve the global hypothesis testing we have to face separately but
simultaneously - the two (multivariate) testing problems (one for each j {
apex, entry point}) and then combining them through the nonparametric
combination (NPC) methodology (Pesarin and Salmaso, 2010). A detailed
description of the algorithm used is described in (Antonucci et all 2019).

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