Non-linear random effects models with continuous time autoregressive errors: a Bayesian approach

Classification of longitudinal profiles using semi-parametric nonlinear mixed models with P-Splines and the SAEM algorithm

In this work, we propose an extension of a semiparametric nonlinear mixed-effects model for longitudinal data that incorporates more flexibility with penalized splines (P-splines) as smooth terms. The novelty of the proposed approach consists of the formulation of the model within the stochastic approximation version of the EM algorithm for maximum likelihood, the so-called SAEM algorithm. The proposed approach takes advantage of the formulation of a P-spline as a mixed-effects model and the use of the computational advantages of the existing software for the SAEM algorithm for the estimation of the random effects and the variance components. Additionally, we developed a supervised classification method for these non-linear mixed models using an adaptive importance sampling scheme. To illustrate our proposal, we consider two studies on pregnant women where two biomarkers are used as indicators of changes during pregnancy. In both studies, information about the women's pregnancy outcomes is known. Our proposal provides a unified framework for the classification of longitudinal profiles that may have important implications for the early detection and monitoring of pregnancy-related changes and contribute to improved maternal and fetal health outcomes. We show that the proposed models improve the analysis of this type of data compared to previous studies. These improvements are reflected both in the fit of the models and in the classification of the groups.

CO-CLUSTERING OF SPATIALLY RESOLVED TRANSCRIPTOMIC DATA

Spatial transcriptomics is a groundbreaking technology that allows the measurement of the activity of thousands of genes in a tissue sample and maps where the activity occurs. This technology has enabled the study of the spatial variation of the genes across the tissue. Comprehending gene functions and interactions in different areas of the tissue is of great scientific interest, as it might lead to a deeper understanding of several key biological mechanisms, such as cell-cell communication or tumor-microenvironment interaction. To do so, one can group cells of the same type and genes that exhibit similar expression patterns. However, adequate statistical tools that exploit the previously unavailable spatial information to more coherently group cells and genes are still lacking. In this work, we introduce SpaRTaCo, a new statistical model that clusters the spatial expression profiles of the genes according to a partition of the tissue. This is accomplished by performing a co-clustering, i.e., inferring the latent block structure of the data and inducing two types of clustering: of the genes, using their expression across the tissue, and of the image areas, using the gene expression in the spots where the RNA is collected. Our proposed methodology is validated with a series of simulation experiments and its usefulness in responding to specific biological questions is illustrated with an application to a human brain tissue sample processed with the 10X-Visium protocol.

A Bayesian nonparametric model for classification of longitudinal profiles

Across several medical fields, developing an approach for disease classification is an important challenge. The usual procedure is to fit a model for the longitudinal response in the healthy population, a different model for the longitudinal response in the diseased population, and then apply Bayes' theorem to obtain disease probabilities given the responses. Unfortunately, when substantial heterogeneity exists within each population, this type of Bayes classification may perform poorly. In this article, we develop a new approach by fitting a Bayesian nonparametric model for the joint outcome of disease status and longitudinal response, and then we perform classification through the clustering induced by the Dirichlet process. This approach is highly flexible and allows for multiple subpopulations of healthy, diseased, and possibly mixed membership. In addition, we introduce an Markov chain Monte Carlo sampling scheme that facilitates the assessment of the inference and prediction capabilities of our model. Finally, we demonstrate the method by predicting pregnancy outcomes using longitudinal profiles on the human chorionic gonadotropin beta subunit hormone levels in a sample of Chilean women being treated with assisted reproductive therapy.

Joint Bayesian weight and height postnatal growth model to study the effects of maternal smoking during pregnancy

Growth models used for describing the dynamics of body weight and height generally consider each trait independently. We proposed modeling height and weight trajectories jointly with a nonlinear heteroscedastic mixed model based on the Jenss-Bayley growth function with correlated individual random effects and using Bayesian inference techniques. Simulations showed that our model provides good estimates of the growth parameters. We illustrated how it can be used to assess the associations between maternal smoking during pregnancy, an early-life factor potentially involved in prenatal programming of obesity, and children's growth from birth to 5 years of age. We used real data from the EDEN study, a large French mother-child cohort study with a high number of height and weight measurements (a total of approximately 30 000 measurements for each of the 2 traits across the 1666 children). Our results supported the existence of a relationship between maternal smoking during pregnancy and growth from birth to 5 years of age. Children from mothers who smoked throughout pregnancy were shown to display a higher body mass index from the first few months of life onwards compared to children from nonsmokers. At 5 years of age, their mean body mass index was 0.21 kg/m² higher than unexposed children. It was mainly explained by the fact that these children tended to be smaller at birth but rapidly exceeded the weight of children from nonsmokers postnatally.

Bayesian regression analysis of data with random effects covariates from nonlinear longitudinal measurements

Joint models for a wide class of response variables and longitudinal measurements consist on a mixed-effects model to fit longitudinal trajectories whose random effects enter as covariates in a generalized linear model for the primary response. They provide a useful way to assess association between these two kinds of data, which in clinical studies are often collected jointly on a series of individuals and may help understanding, for instance, the mechanisms of recovery of a certain disease or the efficacy of a given therapy. When a nonlinear mixed-effects model is used to fit the longitudinal trajectories, the existing estimation strategies based on likelihood approximations have been shown to exhibit some computational efficiency problems (De la Cruz et al., 2011). In this article we consider a Bayesian estimation procedure for the joint model with a nonlinear mixed-effects model for the longitudinal data and a generalized linear model for the primary response. The proposed prior structure allows for the implementation of an MCMC sampler. Moreover, we consider that the errors in the longitudinal model may be correlated. We apply our method to the analysis of hormone levels measured at the early stages of pregnancy that can be used to predict normal versus abnormal pregnancy outcomes. We also conduct a simulation study to assess the importance of modelling correlated errors and quantify the consequences of model misspecification.

Foliar interception of radionuclides in dry conditions: a meta-analysis using a Bayesian modeling approach

Uncertainty on the parameters that describe the transfer of radioactive materials into the (terrestrial) environment may be characterized thanks to datasets such as those compiled within International Atomic Energy Agency (IAEA) documents. Nevertheless, the information included in these documents is too poor to derive a relevant and informative uncertainty distribution regarding dry interception of radionuclides by the pasture grass and the leaves of vegetables. In this paper, 145 sets of dry interception measurements by the aboveground biomass of specific plants were collected from published scientific papers. A Bayesian meta-analysis was performed to derive the posterior probability distributions of the parameters that reflect their uncertainty given the collected data. Four competing models were compared in terms of both fitting performances and predictive abilities to reproduce plausible dry interception data. The asymptotic interception factor, applicable whatever the species and radionuclide to the highest aboveground biomass values (e.g. mature leafy vegetables), was estimated with the best model, to be 0.87 with a 95% credible interval (0.85, 0.89).

Parameter Estimation of Population Pharmacokinetic Models with Stochastic Differential Equations: Implementation of an Estimation Algorithm

Population pharmacokinetic (PPK) models play a pivotal role in quantitative pharmacology study, which are classically analyzed by nonlinear mixed-effects models based on ordinary differential equations. This paper describes the implementation of SDEs in population pharmacokinetic models, where parameters are estimated by a novel approximation of likelihood function. This approximation is constructed by combining the MCMC method used in nonlinear mixed-effects modeling with the extended Kalman filter used in SDE models. The analysis and simulation results show that the performance of the approximation of likelihood function for mixed-effects SDEs model and analysis of population pharmacokinetic data is reliable. The results suggest that the proposed method is feasible for the analysis of population pharmacokinetic data.

The linear mixed model and the hierarchical Ornstein-Uhlenbeck model: some equivalences and differences

We focus on comparing different modelling approaches for intensive longitudinal designs. Two methods are scrutinized, namely the widely used linear mixed model (LMM) and the relatively unexplored Ornstein-Uhlenbeck (OU) process based state-space model. On the one hand, we show that given certain conditions they result in equivalent outcomes. On the other hand, we consider it important to emphasize that their perspectives are different and that one framework might better address certain types of research questions than the other. We show that, compared to a LMM, an OU process based approach can cope with modelling inter-individual differences in aspects that are more substantively interesting. However, the estimation of the LMM is faster and the model is more straightforward to implement. The models are illustrated through an experience sampling study.

Bayesian analysis of growth curves using mixed models defined by stochastic differential equations

Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.

A Bayesian regression model for multivariate functional data

In this paper we present a model for the analysis of multivariate functional data with unequally spaced observation times that may differ among subjects. Our method is formulated as a Bayesian mixed-effects model in which the fixed part corresponds to the mean functions, and the random part corresponds to individual deviations from these mean functions. Covariates can be incorporated into both the fixed and the random effects. The random error term of the model is assumed to follow a multivariate Ornstein-Uhlenbeck process. For each of the response variables, both the mean and the subject-specific deviations are estimated via low-rank cubic splines using radial basis functions. Inference is performed via Markov chain Monte Carlo methods.

A mathematical model on the effect of M. denticulata weed on different winter crops

Many weeds interfere with the productivity of wheat and other winter annual crops; however, one common weed Medicago denticulata, appears to have a beneficial effect. An experiment was conducted at Agricultural Experimental Farm of the Indian Statistical Institute, Giridih, Jharkhand to observe the effect of M. denticulata (a legume) on different crops yield of barley (Hordeum vulgare Sensu lato), linseed (Linum usitatissimum L.), Indian mustard [Brassica juncea (L.) Czernj. & Cosson] and wheat (Triticum aestivum L.). The commensalisms interaction was studied, in which one species benefits, while the other is unaffected, between the crops, M. denticulata and other weeds, to observe the role of medicago as biological control agent of other weeds for enhancement of crop productivity. Further, a statistical test was performed to establish the effect of M. denticulata and weed infestations based on one-step Markov structure of the covariance matrix of the biomass. This permitted development of a mathematical model of a crop-M. denticulata-other weeds system and their interactions through direct and indirect competition for resources. Crops yield decreased considerably due to the presence of other weeds, while it increased due to introduction of medicago and hence medicago acted as a controlling agent in a crop-M. denticulata-other weeds system. Our mathematical analysis supported the experimental findings. The model parameters are estimated through a nonlinear least square and stochastic differential equations. Moreover, some thresholds for the growth of medicago population came out from our analysis that may be used as a biological control threshold for the enhancement of crop productivity.

Bayesian statistics in medicine: a 25 year review

This review examines the state of Bayesian thinking as Statistics in Medicine was launched in 1982, reflecting particularly on its applicability and uses in medical research. It then looks at each subsequent five-year epoch, with a focus on papers appearing in Statistics in Medicine, putting these in the context of major developments in Bayesian thinking and computation with reference to important books, landmark meetings and seminal papers. It charts the growth of Bayesian statistics as it is applied to medicine and makes predictions for the future. From sparse beginnings, where Bayesian statistics was barely mentioned, Bayesian statistics has now permeated all the major areas of medical statistics, including clinical trials, epidemiology, meta-analyses and evidence synthesis, spatial modelling, longitudinal modelling, survival modelling, molecular genetics and decision-making in respect of new technologies.