Matching by fixing and sampling: a local model based on internality

Undermatching and overmatching in concurrent schedules of reinforcement have been traditionally described as changes in the slope of the Generalized Matching Law function. More recently, it has been suggested that deviations from strict matching may be better described as following a policy of mostly fixing on the preferred schedule, and occasionally sampling the alternative schedule. So far, no model of local performance predicts the global outcome of this policy. We describe one such model; it assumes immediate and long-term effects of reinforcement on local performance. The model assumes long-term effects as changes in the internal state of the organism. Formally, the model is analogous to the Axiom of Repeated Choice [Lefebvre, V.A., 2004. Bipolarity, choice, and entro-field. In: Proceedings of the Eighth World Multi-Conference on Systemics, Cybernetics and Informatics, vol. IV, pp. 95-99].     

Robust two-stage estimation in hierarchical nonlinear models

Hierarchical models encompass two sources of variation, namely within and among individuals in the population; thus, it is important to identify outliers that may arise at each sampling level. A two-stage approach to analyzing nonlinear repeated measurements naturally allows parametric modeling of the respective variance structure for the intraindividual random errors and interindividual random effects. We propose a robust two-stage procedure based on Huber's (1981, Robust Statistics) theory of M-estimation to accommodate separately aberrant responses within an experimental unit and subjects deviating from the study population when the usual assumptions of normality are violated. A toxicology study of chronic ozone exposure in rats illustrates the impact of outliers on the population inference and hence the advantage of adopting the robust methodology. The robust weights generated by the two-stage M-estimation process also serve as diagnostics for gauging the relative influence of outliers at each level of the hierarchical model. A practical appeal of our proposal is the computational simplicity since the estimation algorithm may be implemented using standard statistical software with a nonlinear least squares routine and iterative capability.     

Latent variable models for longitudinal data with multiple continuous outcomes

Multiple outcomes are often used to properly characterize an effect of interest. This paper proposes a latent variable model for the situation where repeated measures over time are obtained on each outcome. These outcomes are assumed to measure an underlying quantity of main interest from different perspectives. We relate the observed outcomes using regression models to a latent variable, which is then modeled as a function of covariates by a separate regression model. Random effects are used to model the correlation due to repeated measures of the observed outcomes and the latent variable. An EM algorithm is developed to obtain maximum likelihood estimates of model parameters. Unit-specific predictions of the latent variables are also calculated. This method is illustrated using data from a national panel study on changes in methadone treatment practices.     

Statistical models for analyzing repeated quality measurements of horticultural products. Model evaluations and practical example

In the field of postharvest quality assessment of horticultural products, research on the development of non-destructive quality sensors, replacing destructive and often time consuming sensors, has spurred in the last decennium offering the possibility of taking repeated quality measures on the same product. Repeated measures analysis is gaining importance during recent years and several software packages offer a broad class of routines. A dataset dealing with the postharvest quality evolution of different tomato cultivars serves as practical example for the comparison and discussion of four different statistical model types. Starting from an analysis at each time point and an ordinary least squares regression model as standard and widely used methods, this contribution aims at comparing these two methods to a repeated measures analysis and a longitudinal mixed model. It is shown that the flexibility of such a mixed model, both towards the repeated measures design of the experiments as towards the large product variability inherent to these horticultural products, is an important advantage over classical techniques. This research shows that different conclusions could be drawn depending on which technique is used due to the basic assumptions of each model and which are not always fulfilled. The results further demonstrate the flexibility of the mixed model concept. Using a mixed model for repeated measures, the different sources of variability, being inter-tomato variability, intra-tomato variability and measurement error were characterized being of great benefit to the researcher.     

Semiparametric regression analysis of longitudinal data with informative drop-outs

Informative drop-out arises in longitudinal studies when the subject's follow-up time depends on the unobserved values of the response variable. We specify a semiparametric linear regression model for the repeatedly measured response variable and an accelerated failure time model for the time to informative drop-out. The error terms from the two models are assumed to have a common, but completely arbitrary joint distribution. Using a rank-based estimator for the accelerated failure time model and an artificial censoring device, we construct an asymptotically unbiased estimating function for the linear regression model. The resultant estimator is shown to be consistent and asymptotically normal. A resampling scheme is developed to estimate the limiting covariance matrix. Extensive simulation studies demonstrate that the proposed methods are suitable for practical use. Illustrations with data taken from two AIDS clinical trials are provided.     

Joint modelling of longitudinal measurements and event time data

This paper formulates a class of models for the joint behaviour of a sequence of longitudinal measurements and an associated sequence of event times, including single-event survival data. This class includes and extends a number of specific models which have been proposed recently, and, in the absence of association, reduces to separate models for the measurements and events based, respectively, on a normal linear model with correlated errors and a semi-parametric proportional hazards or intensity model with frailty. Special cases of the model class are discussed in detail and an estimation procedure which allows the two components to be linked through a latent stochastic process is described. Methods are illustrated using results from a clinical trial into the treatment of schizophrenia.     

Analysis of left-censored longitudinal data with application to viral load in HIV infection

The classical model for the analysis of progression of markers in HIV-infected patients is the mixed effects linear model. However, longitudinal studies of viral load are complicated by left censoring of the measures due to a lower quantification limit. We propose a full likelihood approach to estimate parameters from the linear mixed effects model for left-censored Gaussian data. For each subject, the contribution to the likelihood is the product of the density for the vector of the completely observed outcome and of the conditional distribution function of the vector of the censored outcome, given the observed outcomes. Values of the distribution function were computed by numerical integration. The maximization is performed by a combination of the Simplex algorithm and the Marquardt algorithm. Subject-specific deviations and random effects are estimated by modified empirical Bayes replacing censored measures by their conditional expectations given the data. A simulation study showed that the proposed estimators are less biased than those obtained by imputing the quantification limit to censored data. Moreover, for models with complex covariance structures, they are less biased than Monte Carlo expectation maximization (MCEM) estimators developed by Hughes (1999) Mixed effects models with censored data with application to HIV RNA Levels. Biometrics 55, 625-629. The method was then applied to the data of the ALBI-ANRS 070 clinical trial for which HIV-1 RNA levels were measured with an ultrasensitive assay (quantification limit 50 copies/ml). Using the proposed method, estimates obtained with data artificially censored at 500 copies/ml were close to those obtained with the real data set.     

Goodness-of-fit methods for generalized linear mixed models

We develop graphical and numerical methods for checking the adequacy of generalized linear mixed models (GLMMs). These methods are based on the cumulative sums of residuals over covariates or predicted values of the response variable. Under the assumed model, the asymptotic distributions of these stochastic processes can be approximated by certain zero-mean Gaussian processes, whose realizations can be generated through Monte Carlo simulation. Each observed process can then be compared, both visually and analytically, to a number of realizations simulated from the null distribution. These comparisons enable one to assess objectively whether the observed residual patterns reflect model misspecification or random variation. The proposed methods are particularly useful for checking the functional form of a covariate or the link function. Extensive simulation studies show that the proposed goodness-of-fit tests have proper sizes and are sensitive to model misspecification. Applications to two medical studies lead to improved models.     

Effects of range position,inter-annual variation and density on demographic transition rates of <Emphasis Type="Italic">Hornungia petraea</Emphasis> populations

The central-marginal model assumes unfavourable and more variable environmental conditions at the periphery of a species’ distribution range to negatively affect demographic transition rates, finally resulting in reduced population sizes and densities. Previous studies on density-dependence as a crucial factor regulating plant population growth have mainly focussed on fecundity and survival. Our objective is to analyse density-dependence in combination with the effect of inter-annual variation and range position on all life stages of an annual plant species, Hornungia petraea, including germination and seed incorporation into the seed bank. As previous studies on H. petraea had revealed a pattern opposite to existing theory with lower population densities at the distribution centre in Italy than at the periphery in Germany, we hypothesised that (1) demographic transition rates are lower, (2) the inter-annual variation in demographic transition rates is higher and (3) the intensity of density-dependence is weaker in Italy than in Germany. To analyse demographic transition rates, we used an autoregressive covariance strategy for repeated measures including density and inter-annual variation. All the three hypotheses were confirmed, but the impact of range position, density-dependence and inter-annual variation differed among the transition steps. All transition rates except fecundity were higher in the German populations than in the Italian populations. Germination rate and incorporation rate into the seed bank were strongly density-dependent. Central populations showed a larger inter-annual variation in fecundity and winter survival rate. Winter survival rate was the only transition step with a stronger density-dependence in peripheral populations. In most cases, these differences between distribution centre and periphery would not have emerged without taking density-dependence and inter-annual variation into account. We conclude that including range position, inter-annual variation and density-dependence in one single statistical model is an important tool for the interpretation of demographic patterns regarding the central-marginal model. Electronic Supplementary Material Supplementary material is available for this article at