Analysis of logistic growth models

A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most successful predictive models are shown to be based on extended forms of the classical Verhulst logistic growth equation. We further review and compare several such models and calculate and investigate properties of interest for these. We also identify and detail several previously unreported associated limitations and restrictions. A generalized form of the logistic growth curve is introduced which is shown incorporate these models as special cases. The reported limitations of the generic growth model are shown to be addressed by this new model and similarities between this and the extended growth curves are identified. Several of its properties are also presented. We furthermore show that additional growth characteristics are accommodated by this new model, enabling previously unsupported, untypical population dynamics to be modelled by judicious choice of model parameter values alone

Identifiability of multivariate logistic mixture models

Mixture models have been widely used in modeling of continuous observations. For the possibility to estimate the parameters of a mixture model consistently on the basis of observations from the mixture, identifiability is a necessary condition. In this study, we give some results on the identifiability of multivariate logistic mixture models

Grid multi-category response logistic models.

BackgroundMulti-category response models are very important complements to binary logistic models in medical decision-making. Decomposing model construction by aggregating computation developed at different sites is necessary when data cannot be moved outside institutions due to privacy or other concerns. Such decomposition makes it possible to conduct grid computing to protect the privacy of individual observations.MethodsThis paper proposes two grid multi-category response models for ordinal and multinomial logistic regressions. Grid computation to test model assumptions is also developed for these two types of models. In addition, we present grid methods for goodness-of-fit assessment and for classification performance evaluation.ResultsSimulation results show that the grid models produce the same results as those obtained from corresponding centralized models, demonstrating that it is possible to build models using multi-center data without losing accuracy or transmitting observation-level data. Two real data sets are used to evaluate the performance of our proposed grid models.ConclusionsThe grid fitting method offers a practical solution for resolving privacy and other issues caused by pooling all data in a central site. The proposed method is applicable for various likelihood estimation problems, including other generalized linear models

Conditionally conjugate mean-field variational Bayes for logistic models

Variational Bayes (VB) is a common strategy for approximate Bayesian inference, but simple methods are only available for specific classes of models including, in particular, representations having conditionally conjugate constructions within an exponential family. Models with logit components are an apparently notable exception to this class, due to the absence of conjugacy between the logistic likelihood and the Gaussian priors for the coefficients in the linear predictor. To facilitate approximate inference within this widely used class of models, Jaakkola and Jordan (2000) proposed a simple variational approach which relies on a family of tangent quadratic lower bounds of logistic log-likelihoods, thus restoring conjugacy between these approximate bounds and the Gaussian priors. This strategy is still implemented successfully, but less attempts have been made to formally understand the reasons underlying its excellent performance. To cover this key gap, we provide a formal connection between the above bound and a recent P\'olya-gamma data augmentation for logistic regression. Such a result places the computational methods associated with the aforementioned bounds within the framework of variational inference for conditionally conjugate exponential family models, thereby allowing recent advances for this class to be inherited also by the methods relying on Jaakkola and Jordan (2000)

Logistic mixed models to investigate implicit and explicit belief tracking

We investigated the proposition of a two-systems Theory of Mind in adults’ belief tracking. A sample of N = 45 participants predicted the choice of one of two opponent players after observing several rounds in an animated card game. Three matches of this card game were played and initial gaze direction on target and subsequent choice predictions were recorded for each belief task and participant. We conducted logistic regressions with mixed effects on the binary data and developed Bayesian logistic mixed models to infer implicit and explicit mentalizing in true belief and false belief tasks. Although logistic regressions with mixed effects predicted the data well a Bayesian logistic mixed model with latent task- and subject-specific parameters gave a better account of the data. As expected explicit choice predictions suggested a clear understanding of true and false beliefs (TB/FB). Surprisingly, however, model parameters for initial gaze direction also indicated belief tracking. We discuss why task-specific parameters for initial gaze directions are different from choice predictions yet reflect second-order perspective taking