AN ´ ALISE ESTAT´ ISTICA

Uma an´alise descritiva do comportamento longitudinal da vari´avel rotarod sugere que o n´ıvel da resposta se mant´em est´avel at´e um certo instante e que come¸ca a decrescer a partir da´ı, indicando o in´ıcio do aparecimento desse sintoma da doen¸ca. Al´em disso, pode-se conjecturar que o comportamento n˜ao ´e uniforme para os seis grupos correspondentes `as combina¸c˜oes de tratamento e sexo. Tendo em vista que essas conclus˜oes s˜ao compat´ıveis com a explica¸c˜ao biol´ogica do desenvolvimento da doen¸ca, propusemos uma estrat´egia de an´alise baseada no ajuste do seguinte modelo misto

yijk = αij+{γij[tk− ψij(λij)]2}I(tk > ψij) + eijk (1)

em que yijk ´e a resposta do j-´esimo animal observado sob o i-´esimo grupo no k-´esimo ins-

tante de avalia¸c˜ao, αij ´e o correspondente coeficiente linear da curva que representa o com-

portamento da resposta antes do ponto de mudan¸ca, γij ´e coeficiente do termo quadr´atico

associado `a curva que governa a resposta p´os ponto de mudan¸ca, ψij, com ψij(λij) =

[a1 + a2exp(λij)]/[1 + exp(λij)] para restringir o valor de ψij ao intervalo (a1, a2) em que

as respostas s˜ao observadas, αij = αi + aij, γij = γi+ cij, λij = λi+ `ij sob a suposi¸c˜ao

bij = (aij, cij, `ij)> ∼ N(0, G), G denota uma matriz de covariˆancias (n˜ao estruturada) e

eijk ∼ N(0, σ2) independente de bij.

O ajuste ´e adaptado de Muggeo (2014) e Fasola et al. (2018) e ´e baseado na expans˜ao de Taylor de f [tk, ψij(λij)] = γij[tk− ψij(λij)]2}I[tk > ψij(λij)] Explicitamente, f [tk, ψij(λij)]≈ f[tk, ψij(bλij)] + (λij− bλij) ∂f [tk, ψij] ∂ψij ∂ψij(λij) λij   λij=bλij com ∂f [tk, ψij] ∂ψij = hij(λij) =−2γij[tk − ψij(λij)]I[tk > ψij(λij)] e ∂ψij(λij) λij = gij(λij) = (a2− a1) exp(λij) [1 + exp(λij]2

Consequentemente, pode-se aproximar o modelo (1) como

yijk ≈ αij+ f [tk, ψij(bλij)]− bλijhij(bλij)gij(bλij) + λijhij(bλij)gij(bλij) + eijk.

Considerando pseudo observa¸c˜oes definidas por y_ijk∗ = yijk + bλijhij(bλij)gij(bλij), o modelo

y∗

ijk = αij+ λijbλijhij(bλij)gij(bλij) + eijk sugere o seguinte algoritmo para o ajuste de (1)

1) Fixar ψij = ψ(0) e y_ijk(0) = yijk.

2) Ajustar o modelo y(0)_ijk = αij+ γij(tk− ψ(0))2I(tk > ψ(0)) + eijk para obter α(0)ij , γ (0)

ij e

λ(0)_ij = log[(ψ(0)_{− a}1)/(a2− ψ(0))] .

3) Fixar r = 1.

4) Calcular y_ijk(r) = y_ijk(r−1)+ λ(r_ij−1)hij(λ(rij−1))gij(λ(rij−1)).

5) Ajustar o modelo y(r)_ijk = αij+γij(tk−ψ(r−1))2I(tk > ψ(r−1))+λijhij(λ(rij−1))gij(λ(rij−1))+

e(r_ijk−1) para obter α(r)_ij , γ_ij(r), λ(r)_ij e ψ(r)_{= [a}

1+ a2exp(λ(r)ij )]/[1 + exp(λ (r) ij )].

6) Parar se algum crit´erio de convergˆencia estiver satisfeito; em caso contr´ario, fazer r = r + 1 e repetir os passos 4-6.

Gr´aficos com os valores observados e previstos para as respostas de cada animal de um dos grupos analisados est˜ao dispostos na Figura 2.

Figura 2: Gr´aficos de valores observados e previstos para as respostas

Week Time 0 50 100 150 200 30 8 10 12 14 16 18 20 31 32 8 10 12 14 16 18 20 33 8 10 12 14 16 18 20 34 35 8 10 12 14 16 18 20 0 50 100 150 200 36 PA SS

Compara¸c˜oes entre os parˆametros correspondentes `as curvas esperadas associadas `as com- bina¸c˜oes dos n´ıveis dos diferentes tratamentos e sexos podem ser realizadas por meio de testes de Wald com base nos resultados do ajuste do modelo final. Os resultados s˜ao com- parados com aqueles de modelos em que as curvas associadas aos per´ıodos pr´e e p´os ponto de mudan¸ca s˜ao retas, obtidos por meio das t´ecnicas propostas por Muggeo et al. (2014) e Fasola et al. (2018).

A constru¸c˜ao de algoritmos para o ajuste de modelos lineares mistos com mais do que um ponto de mudan¸ca ´e um tema interessante para futuras pesquisas.

AGRADECIMENTOS

Este trabalho recebeu apoio financeiro do Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq, processo 3304126/2015-2) e Funda¸c˜ao de Amparo `a Pesquisa do Estado de S˜ao Paulo (FAPESP, processo 2013/21728-2), Brasil.

Referˆencias

[1] Coatti, G.C. (2015). Avalia¸c˜ao do potencial terapˆeutico de pericitos e de c´elulas mesenquimais no camundongo SOD1, modelo animal para esclerose lateral amiotr´ofica. Tese de doutorado, Departamento de Biocˆencias, Universidade de S˜ao Paulo.

http://www.teses.usp.br/teses/disponiveis/41/41131/tde-14012016-143346/pt-br.php

[2] Fasola, S., Muggeo, V.M.R. and K¨uchenhoff, H. (2018). A heuristic, iterative algorithm for change-point detection in abrupt change models. Compuational Statistics 33, 997-1015.

[3] Muggeo, V.M.R., Atkins, D.C., Gallop, R.J. and Dimidjian, S. (2014). Segmented mixed models with random changepoints: a maximum likelihood approach with application to treatment for depression study. Statistical Modelling 14,: 293-313.

III Luso-Galician Meeting on Biometry Aveiro, 28 - 30 June 2018

POPULATION DYNAMICS EQUILIBRIUM AND EXTREME GROWTH M. F´atima Brilhante1, M. Ivette Gomes2 e Dinis Pestana3

1_{Faculdade de Ciˆencias e Tecnologia da Universidade dos A¸cores and Centro de Estat´ıstica e}

Aplica¸c˜oes da Universidade de Lisboa

2_{Centro de Estat´ıstica e Aplica¸c˜oes da Universidade de Lisboa and Instituto de Investiga¸c˜ao}

Cient´ıfica Bento da Rocha Cabral

3_{Centro de Estat´ıstica e Aplica¸c˜oes da Universidade de Lisboa and Instituto de Investiga¸c˜ao}

Cient´ıfica Bento da Rocha Cabral

ABSTRACT

Over the years the Verhulst model for population dynamics has been the building block for other population growth models. Since the solution of the Verhulst model and of some generalized versions are connected to max-geometric stable distribu- tions or to generalized extreme value distributions, we prove that the exponent linked to the retroaction factor of some generalized models, whose solutions are not these distributions, determines on its own which limit law is appropriate for modeling extreme population growth.

Keywords and key sentences: Population Dynamics, Verhulst Model, Generalized Ver- hulst Models, Max-Geometric Stable Distributions, Generalized Extreme Value Distributions, Extreme Population Growth.

1. INTRODUCTION

Let N (t) be the size of a population at time t. Under certain regularity conditions, Verhulst (1838) proposed the logistic differential equation

d dtN (t) = rN (t)  1₋N (t) K  (1) to model population dynamics, where r > 0 is the intrinsic growth rate and K > 0 is the carrying capacity, i.e. the limiting size the population may reach without disruptive effects on the availability of resources. In the right hand of equation (1), N (t) is considered the growth factor and 1₋ N (t)_K the retroaction factor, which is responsible for curbing down population growth. The solution of (1) is N (t) = KN0

N0+(K−N0)e−rt, a member of the family of

logistic functions, hence the name logistic (N0 is the initial population size).

In spite of its limitations, the Verhulst model is still quite popular. One limitation of the model is being only suitable for modeling sustainable growth, or modeling stable populations. Therefore, over the years the Verhulst model has been used as basis for building several other

growth models, stating, for instance, that either _dtdN (t) or _dtd ln N (t) is a decreasing function of the population density N (t)_K , such as in model (1). For example, the family of models

d dtln N (t) = r 1₋N (t)_K ν ν ⇔      d dtN (t) = rN (t)  1−  N (t) K ν ν , ν > 0 d dtN (t) = rN (t)  − lnN (t)K  , ν = 0 , (2)

which is based on the Box-Cox family of transformations, contains the Verhulst model (ν = 1). The subfamily for ν > 0 was considered in Richards (1959), and the solution for ν = 0 is N (t) = Kexp ln N0

K

e−rt
, which is known in population dynamics as the Gompertz growth model. This model is closely connected to the Gumbel distribution and has been used for modeling the growth of cancer tumors.

On the other hand, Blumberg (1968) extended the Verhulst equation by considering the hiperlogistic equation d dtN (t) = r N (t)
α 1₋ N (t) K β , α, β > 0 . (3)

However, equation (3) does not contain a closed form analytical solution, except for some special values of α and β. Generalizations of equation (3) and of family (2) for ν = 0 was considered in Brilhante et al. (2011, 2012), in connection to the BetaBoop familiy of densities. We would like to point out that all extended versions of the Verhulst model are intended to be more flexible, allowing in some cases the possibility of modeling different types of unrestricted population growth. For more information on other growth models cf. Tsoularis (2001). If we rewrite Verhulst’s logistic equation solely as a function of the population density δ(t) = N (t)_K , i.e. _dtdδ(t) = rδ(t) 1_{− δ(t)}
, the normalized solution δ(t) = _1+e1−rt belongs to

the logistic family of distributions. Additionally, Brilhante et al. (2011) showed that the solution of the differential equation

d

dtN (t) = rN (t) (− ln N(t))

1+ξ

, ξ_{∈ IR ,} (4)

which generalizes family (2) when ν = 0 (and K = 1), belongs to the family of generalized extreme value (GEV) distributions for maxima. In particular, if ξ > 0, we have the Fr´echet distribution, if ξ = 0, the Gumbel distribution and if ξ < 0, the Weibull distribution for max- ima. But, in Statistics, more precisely, in Extreme Value Theory, the logistic distribution is a max-geometric stable distribution, i.e. a stable distribution for random maxima of sequences of independent and identically distributed (iid) random variables, with a geometric subordi- nator (cf. Rachev and Resnick, 1991), and GEV distributions are max-stable distributions of sequences of maxima of iid random variables.

As a note, max-stable distributions are necessarily of the generalized extreme value type Gξ(x) = exp



− (1 + ξx)−1/ξ , 1 + ξx > 0, ξ_{∈ IR ,} whilst max-geometric stable distributions are of the type

Hξ(x) =

1 1_{− ln G}ξ(x)

= 1

1 + (1 + ξx)−1/ξ , 1 + ξx > 0, ξ∈ IR .

The three types of max-geometric stable distributions are the log-logistic distribution (ξ > 0), the logistic distribution (ξ = 0) and the backward log-logistic distribution (ξ < 0).

Therefore, there seems to be a connection between population dynamics equilibrium and extreme growth for some extended Verhulst models. In the next section we shall investigate the type of connection that does occurs.

2. GENERALIZED VERHULST MODELS AND EXTREME GROWTH