Page 21 - Contributed Paper Session (CPS) - Volume 4

CPS2106 Julio M. Singer et al.
A longitudinal analysis of the behaviour of the response variable
corroborates its expected stable level before the onset of the symptom (a
decrease in the length of time during which the animal remains in the rotating
cylinder). Furthermore, individual differences in the moment where this occurs
as well as differences among the accelerations with which the intensity of the
symptom progresses are also visible. It also seems reasonable to expect a
change in the acceleration with which the intensity of the symptom progresses
after the disease onset.
Given that such conclusions are in line with the expected biological
behaviour, a random changepoint mixed polynomial segmented regression
model may be considered for the analysis.
Such models have an attractive practical appeal in many fields and have been
the object of statistical research for a long time as detailed in Muggeo et al.
(2014). These authors consider a frequentist approach as opposed to the
commonly Bayesian perspective usually employed in the statistical literature.
Keeping in mind the necessarily non-negative nature of the response, we
adopt a similar approach and consider an analysis of the ALS data based on
the model
2
yijk = αijI(tk < ψ2ij) + γij[tk − ψ1ij(λij)] I(ψ1ij ≤ tk < ψ2ij) + eijk (1)
(i = 1,...,6, j = 1,...,ni and k = 1,...,nij) where yijk denotes the response for the j-th
animal observed in the i-th group (defined by the combination of the levels of
treatment and sex) at the k-th evaluation instant, αij is the corresponding stable
level of the symptom prior to the first changepoint, γij is the coefficient of the
quadratic term for the curve that governs the response behaviour post
changepoint ψ1ij, with
ψ1ij(λij) = [L1 + L2 exp(λij)]/[1 + exp(λij)]
to restrict the value of ψ1ij to the interval (L1,L2) in which the observations are
obtained and ψ2ij denotes the instant where the response is null. We assume

>
2
that αij = αi+aij, γij = γi+cij, λij = λi + lij with bij = (aij,cij, lij) ∼ N(0,Gi) and eijk ∼ N(0,σi )
independent of bij.

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