Variable selection in Bayesian generalized linear‐mixed models: An illustration using candidate gene case‐control association studies

The problem of variable selection in the generalized linear‐mixed models (GLMMs) is pervasive in statistical practice. For the purpose of variable selection, many methodologies for determining the best subset of explanatory variables currently exist according to the model complexity and differences between applications. In this paper, we develop a “higher posterior probability model with bootstrap” (HPMB) approach to select explanatory variables without fitting all possible GLMMs involving a small or moderate number of explanatory variables. Furthermore, to save computational load, we propose an efficient approximation approach with Laplace's method and Taylor's expansion to approximate intractable integrals in GLMMs. Simulation studies and an application of HapMap data provide evidence that this selection approach is computationally feasible and reliable for exploring true candidate genes and gene–gene associations, after adjusting for complex structures among clusters.     

Testing homogeneity in Weibull-regression models

In survival studies with families or geographical units it may be of interest testing whether such groups are homogeneous for given explanatory variables. In this paper we consider score type tests for group homogeneity based on a mixing model in which the group effect is modelled as a random variable. As opposed to hazard-based frailty models, this model presents survival times that conditioned on the random effect, has an accelerated failure time representation. The test statistics requires only estimation of the conventional regression model without the random effect and does not require specifying the distribution of the random effect. The tests are derived for a Weibull regression model and in the uncensored situation, a closed form is obtained for the test statistic. A simulation study is used for comparing the power of the tests. The proposed tests are applied to real data sets with censored data.     

Semiparametric maximum likelihood for measurement error model regression

This paper presents an EM algorithm for semiparametric likelihood analysis of linear, generalized linear, and nonlinear regression models with measurement errors in explanatory variables. A structural model is used in which probability distributions are specified for (a) the response and (b) the measurement error. A distribution is also assumed for the true explanatory variable but is left unspecified and is estimated by nonparametric maximum likelihood. For various types of extra information about the measurement error distribution, the proposed algorithm makes use of available routines that would be appropriate for likelihood analysis of (a) and (b) if the true x were available. Simulations suggest that the semiparametric maximum likelihood estimator retains a high degree of efficiency relative to the structural maximum likelihood estimator based on correct distributional assumptions and can outperform maximum likelihood based on an incorrect distributional assumption. The approach is illustrated on three examples with a variety of structures and types of extra information about the measurement error distribution.     

Asymmetric regression models with limited responses with an application to antibody response to vaccine

We develop regression models for limited and censored data based on the mixture between the log‐power‐normal and Bernoulli‐type distributions. A likelihood‐based approach is implemented for parameter estimation and a small‐scale simulation study is conducted to evaluate parameter recovery, with emphasis on bias estimation. The main conclusion is that the approach is very much satisfactory for moderate and large sample sizes. A real data example, the safety and immunogenecity study of measles vaccine in Haiti, is presented to illustrate how different models can be used to fit this type of data. As shown, the asymmetric models considered seem to present the best fit for the data set under study, revealing significance of the explanatory variable sex, which is not found significant with the log‐normal model.     

Creep fracture in bones with different stiffnesses.

Creep fracture experiments were used to examine the differences in time to fracture of bones with very different Young's moduli (bovine bone and red deer antler) and the implications of these differences for the 'cumulative-damage' model of Caler and Carter [J. Biomechanics 22, 625-635 (1989)] for bone fracture. Using normalised stress as the explanatory variable, the slopes of the distributions agreed quite well with that of Caler and Carter for human bone. However, antler took far longer to fracture at any given normalised stress than did bovine bone. Using stress alone as the explanatory variable, the relationships within each bone type almost disappeared. Within any bone type strain is the important determinant of time to fracture, but less mineralised bone takes much longer to fracture at any given strain, or normalised stress, which seems not to be in accord with the cumulative-damage model. The rate of damage accumulation in lightly mineralised bone at high strains (greater than 1%) is much less than that occurring in more heavily mineralised bone.     

Antimicrobial Resistance: The Major Contribution of Poor Governance and Corruption to This Growing Problem

Objectives

To determine how important governmental, social, and economic factors are in driving antibiotic resistance compared to the factors usually considered the main driving factors—antibiotic usage and levels of economic development.

Design

A retrospective multivariate analysis of the variation of antibiotic resistance in Europe in terms of human antibiotic usage, private health care expenditure, tertiary education, the level of economic advancement (per capita GDP), and quality of governance (corruption). The model was estimated using a panel data set involving 7 common human bloodstream isolates and covering 28 European countries for the period 1998–2010.

Results

Only 28% of the total variation in antibiotic resistance among countries is attributable to variation in antibiotic usage. If time effects are included the explanatory power increases to 33%. However when the control of corruption indicator is included as an additional variable, 63% of the total variation in antibiotic resistance is now explained by the regression. The complete multivariate regression only accomplishes an additional 7% in terms of goodness of fit, indicating that corruption is the main socioeconomic factor that explains antibiotic resistance. The income level of a country appeared to have no effect on resistance rates in the multivariate analysis. The estimated impact of corruption was statistically significant (p< 0.01). The coefficient indicates that an improvement of one unit in the corruption indicator is associated with a reduction in antibiotic resistance by approximately 0.7 units. The estimated coefficient of private health expenditure showed that one unit reduction is associated with a 0.2 unit decrease in antibiotic resistance.

Conclusions

These findings support the hypothesis that poor governance and corruption contributes to levels of antibiotic resistance and correlate better than antibiotic usage volumes with resistance rates. We conclude that addressing corruption and improving governance will lead to a reduction in antibiotic resistance.     

Dynamic simulation of an in vitro multi-enzyme system

Parameters often are tuned with metabolite concentration time series data to build a dynamic model of metabolism. However, such tuning may reduce the extrapolation ability (generalization capability) of the model. In this study, we determined detailed kinetic parameters of three purified Escherichia coli glycolytic enzymes using the initial velocity method for individual enzymes; i.e., the parameters were determined independently from metabolite concentration time series data. The metabolite concentration time series calculated by the model using the parameters matched the experimental data obtained in an actual multi-enzyme system consisting of the three purified E. coli glycolytic enzymes. Thus, the results indicate that kinetic parameters can be determined without using an undesirable tuning process.     

Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study

A stochastic discrete-time susceptible-exposed-infectious-recovered (SEIR) model for infectious diseases is developed with the aim of estimating parameters from daily incidence and mortality time series for an outbreak of Ebola in the Democratic Republic of Congo in 1995. The incidence time series exhibit many low integers as well as zero counts requiring an intrinsically stochastic modeling approach. In order to capture the stochastic nature of the transitions between the compartmental populations in such a model we specify appropriate conditional binomial distributions. In addition, a relatively simple temporally varying transmission rate function is introduced that allows for the effect of control interventions. We develop Markov chain Monte Carlo methods for inference that are used to explore the posterior distribution of the parameters. The algorithm is further extended to integrate numerically over state variables of the model, which are unobserved. This provides a realistic stochastic model that can be used by epidemiologists to study the dynamics of the disease and the effect of control interventions.     

Cross-cultural perspectives on physician and lay models of the common cold

We compare physicians and laypeople within and across cultures, focusing on similarities and differences across samples, to determine whether cultural differences or lay-professional differences have a greater effect on explanatory models of the common cold. Data on explanatory models for the common cold were collected from physicians and laypeople in South Texas and Guadalajara, Mexico. Structured interview materials were developed on the basis of open-ended interviews with samples of lay informants at each locale. A structured questionnaire was used to collect information from each sample on causes, symptoms, and treatments for the common cold. Consensus analysis was used to estimate the cultural beliefs for each sample. Instead of systematic differences between samples based on nationality or level of professional training, all four samples largely shared a single-explanatory model of the common cold, with some differences on subthemes, such as the role of hot and cold forces in the etiology of the common cold. An evaluation of our findings indicates that, although there has been conjecture about whether cultural or lay-professional differences are of greater importance in understanding variation in explanatory models of disease and illness, systematic data collected on community and professional beliefs indicate that such differences may be a function of the specific illness. Further generalizations about lay-professional differences need to be based on detailed data for a variety of illnesses, to discern patterns that may be present. Finally, a systematic approach indicates that agreement across individual explanatory models is sufficient to allow for a community-level explanatory model of the common cold.     

Evolve III: A discrete events model of an evolutionary ecosystem

Evolve III is a discrete events model of an evolutionary ecosystem. The model includes three levels of organization: population, organism and genetic structure. Each of these components was modeled independently, so that selective replacement of subsystems can be used to create families of models capable of testing alternative hypotheses about the real system. To demonstrate the use of the model we describe an experiment on the relationship between adaptability of populations and the variability of the environment. Populations cultured in a constant environment usually dominated those cultured in a variable environment when both were placed in a variable environment at an early stage of development, whereas the opposite is the case at later stages of development. This agrees with experiments on laboratory microcosms and lends credence to the potential predictive value of the model.     

Nonlinear biological oscillators: Action-angle variable approach with an application to Cowan's model of neuroelectric activity

We obtain within the action-angle variable approach new expressions, involving the Dirac delta function, for time periods and time averages of dynamical variables which are useful for nonlinear biological oscillator problems. We combine these with Laplace transformation techniques for evaluating the required perturbation expansions. The radii of convergence of these series are determined through a complex variable approach. The method is powerful enough to yield explicit results for such systems as the two species Volterra model, Goodwin's model of protein synthesis etc. and as an illustration, is applied here to Cowan's model of neuroelectric activity. We also point out the usefulness of the action integral in the case where parameters occurring in dynamics have slow time variations.     

The cox proportional hazards model with a continuous latent variable measured by multiple binary indicators

This article is motivated by the Women's Health and Aging Study, where information about physical functioning was recorded along with death information in a group of elderly women. The focus is on determining whether having difficulties in daily living tasks is accompanied by a higher mortality rate. To this end, a two-parameter logistic regression model is used for the modeling of binary questionnaire data assuming an underlying continuous latent variable, difficulty in daily living. The Cox model is used for the survival information, and the continuous latent variable is included as an explanatory variable along with other observed variables. Parameters are estimated by maximizing the likelihood for the joint distribution of the items and the time-to-event information. In addition to presenting a new statistical model, this article also illustrates the use of the model in a real data setting and addresses the more practical issues of model building, diagnostics, and parameter interpretation.     

A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

We formulate a deterministic epidemic model for the spread of Hepatitis C containing an acute, chronic and isolation class and analyse the effects of the isolation class on the transmission dynamics of the disease. We calculate the basic reproduction number R₀ and show that for R₀≤1, the disease-free equilibrium is globally asymptotically stable. In addition, it is shown that for a special case when R₀>1, the endemic equilibrium is locally asymptotically stable. Furthermore, an analogous stochastic epidemic model for Hepatitis C is formulated using a continuous time Markov chain. Numerical simulations are used to estimate the mean, variance and probability distributions of the discrete random variables and these are compared to the steady-state solutions of the deterministic model. Finally, the expected time to disease extinction is estimated for the stochastic model and the impact of isolation on the time to extinction is explored.     

A multi-type branching model with varying environment for bacterial dynamics with postantibiotic effect

A multi-type branching process with varying environment was used to construct a pharmacokinetic/pharmacodynamic (PK/PD) model that captures the postantibiotic effect (PAE) seen in bacterial populations after exposure of antibiotics. This phenomenon of continued inhibition of bacterial growth even after removal of the antibiotic from the growth medium is of high relevance in the context of optimizing dosing regimens. The clinical implication of long PAEs lies in the interesting possibility of increasing the intervals between drug administrations.The model structure is generalizable to most types of antibiotics and is useful both as a theoretical framework for understanding the time properties of PAE and to explore optimal antibiotic dosing regimens. Data from an in vitro study with Escherichia coli exposed to different dosing regimens of cefotaxime were used to evaluate the model.     

Prediction of plant species distribution in lowland river valleys in Belgium: modelling species response to site conditions

In ecological modelling, limitations in data and their applicability for predictive modelling are more rule than exception. Often modelling has to be performed on sub-optimal data, as explicit and controlled collection of (more) appropriate data would not be feasible. An example of predictive ecological modelling is given with application of generalized additive and generalized linear models fitted to presence–absence records of plant species and site condition data from four nutrient-poor Flemish lowland valleys. Standard regression procedures are used for modelling, although explanatory and response data do not meet all the assumptions implicit in these procedures. Data were non-randomly collected and are spatially autocorrelated; model residuals retain part of that correlation. The scale of most site-condition records does not match the scale of the response variable (species distribution). Hence, interpolated and up-scaled explanatory variables are used. Data are aggregated from distinct phytogeographical regions to allow for generalized models, applicable to a wider population of river valleys in the same region. Nevertheless, ecologically sound models are obtained, which predict well the distribution of most plant species for the Flemish river valleys considered.     

Moving toward an unstable equilibrium: saddle nodes in population systems

1. We identify an unstable equilibrium with a two-dimensional stable manifold and a one-dimensional unstable manifold in a three-state variable (larva, pupa, adult) insect population growth model.
2. The saddle node forecasts that the time series of some initial numbers of larvae, pupae and adults are drawn closely to the unstable equilibrium before approaching the asymptotic stable attractor (a two-cycle), while the time series of other initial points are not.
3. Using two quantitative indices, we examine time series from a Tribolium experiment for evidence of the predicted saddle node. We conclude that a saddle node accounts for the transient dynamics in these data and for the differences between the transient behaviour of different replicates of the same experiment.     

Variable selection – A review and recommendations for the practicing statistician

Statistical models support medical research by facilitating individualized outcome prognostication conditional on independent variables or by estimating effects of risk factors adjusted for covariates. Theory of statistical models is well‐established if the set of independent variables to consider is fixed and small. Hence, we can assume that effect estimates are unbiased and the usual methods for confidence interval estimation are valid. In routine work, however, it is not known a priori which covariates should be included in a model, and often we are confronted with the number of candidate variables in the range 10–30. This number is often too large to be considered in a statistical model. We provide an overview of various available variable selection methods that are based on significance or information criteria, penalized likelihood, the change‐in‐estimate criterion, background knowledge, or combinations thereof. These methods were usually developed in the context of a linear regression model and then transferred to more generalized linear models or models for censored survival data. Variable selection, in particular if used in explanatory modeling where effect estimates are of central interest, can compromise stability of a final model, unbiasedness of regression coefficients, and validity of p‐values or confidence intervals. Therefore, we give pragmatic recommendations for the practicing statistician on application of variable selection methods in general (low‐dimensional) modeling problems and on performing stability investigations and inference. We also propose some quantities based on resampling the entire variable selection process to be routinely reported by software packages offering automated variable selection algorithms.     

A model for the transmission dynamics of Orientia tsutsugamushi among its natural reservoirs

The bacteria Orientia tsutsugamushi is the causative agent of scrub typhus, a prevalent disease in Asian countries that can affect humans and which shows an alarming increase of cases during the last years, especially in rural areas. Unfortunately, there is no vaccine for scrub typhus, and antibiotic treatments successfully used in the past appear to be inefficient to treat some strains of O. tsutsugamushi. We introduce a mathematical model that approximates the dynamics of the bacteria among its natural reservoirs. After computing the basic reproductive number from the proposed model, we explore its sensitivity to the parameter values that may be affected by application of control measures. This theoretical model may be of interest to pest managers as well as health authorities interested in gaining insight into the public management of the disease, through a better understanding of its qualitative dynamics.