A note on equivalent mathematical models

A mathematical model of a process contains parameters supposedly characterizing the system which manifests the process. If the parameters are statistically distributed in a population of such systems, the process manifested by the entire population will in general be described by a different mathematical model. Thus a choice is always at hand between two or more mathematical models, depending on which parameters (if any) are assumed to be distributed and, if so, how. Examples of such alternative interpretations are given for mathematical models of some behavioral processes.     

A note on craniofacial sutural growth

The paper presents and discusses some features of sutural structures. Fibre orientation in sutures seems to be quite variable and associated with minute local growth phenomena. Trabecular patterns in bones, reflecting growth, appear to support the idea of the outside determination of growth. The bevelled and interdigitated structures seen in many sutures may be interpreted as an expedient solution to the problem of fast growth.     

Solubility prediction in octanol: A technical note

Summary  The purpose of this work was to derive an equation for the rapid estimation of octanol solubilities of organic compounds. Solubilities ranging over 4 orders of, magnitude were predicted with an average absolute error of 0.39 logarithmic units using melting point alone. The greatest error in prediction occurred for strongly bonded compounds. Published: March 24, 2006     

A note on stability of discrete population models

P. Cull (1981) and G. Rosencranz (1983) studied a discrete population model described by the first-order difference equation xt+1 = g(xt) and obtained an important result on the global stability of the equilibrium point means when g(x) has only one extreme point (a maximum) in (0, means). Motivated by work of M. Kot and W. M. Schaffer (1984), a more general case is considered in which g(x) can have more then one maximum point in (0, means), and results on global stability are obtained. These results are applied to develop tests for global stability of the equilibrium point that imply other results in the literature on global stability.     

A note on multidimensional models of neutral mutation

A generalisation of the Ohta-Kimura charge-state model for neutral mutation, in which alleles are represented by points in d-dimensional space, seems appropriate in the light of recent experimental developments. It is noted that existing methods of analysis extend almost trivially to this more general model, and the point is illustrated by calculating the effective number of alleles.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet-Kolmogorov L(1)-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

A note on generation times in epidemic models

The time between the infection of a primary case and one of its secondary cases is called a generation time. The distribution (and mean) of the generation times is derived for a rather general class of epidemic models. The relation to assumptions on distributions of latency times and infectious times or more generally on random time varying infectiousness, is investigated. Serial times, defined as the times between occurrence of observable events in the progress of an infectious disease (e.g., the onset of clinical symptoms), are also considered.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet–Kolmogorov L ¹-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

A note on conditional AIC for linear mixed-effects models

The conventional model selection criterion, the Akaike informationcriterion, AIC, has been applied to choose candidate modelsin mixed-effects models by the consideration of marginal likelihood.Vaida & Blanchard (2005) demonstrated that such a marginalAIC and its small sample correction are inappropriate when theresearch focus is on clusters. Correspondingly, these authorssuggested the use of conditional AIC. Their conditional AICis derived under the assumption that the variance-covariancematrix or scaled variance-covariance matrix of random effectsis known. This note provides a general conditional AIC but withoutthese strong assumptions. Simulation studies show that the proposedmethod is promising.     

A note on simplifying likelihoods for site occupancy models

We show how a simple reparameterization can reduce the number of parameters that need to be estimated by numerical maximum likelihood in site occupancy models. Three examples are provided.     

A note on the interpretation of tree-based regression models

Tree-based models are a popular tool for predicting a response given a set of explanatory variables when the regression function is characterized by a certain degree of complexity. Sometimes, they are also used to identify important variables and for variable selection. We show that if the generating model contains chains of direct and indirect effects, then the typical variable importance measures suggest selecting as important mainly the background variables, which have a strong indirect effect, disregarding the variables that directly influence the response. This is attributable mainly to the variable choice in the first steps of the algorithm selecting the splitting variable and to the greedy nature of such search. This pitfall could be relevant when using tree-based algorithms for understanding the underlying generating process, for population segmentation and for causal inference.