Model diagnostics for marginal regression analysis of correlated binary data

We propose several diagnostic methods for checking the adequacy of marginal regression models for analyzing correlated binary data. We use a parametric marginal model based on latent variables and derive the projection (hat) matrix, Cook's distance, various residuals and Mahalanobis distance between the observed binary responses and the estimated probabilities for a cluster. Emphasized are several graphical methods including the simulated Q-Q plot, the half-normal probability plot with a simulated envelope, and the partial residual plot. The methods are illustrated with a real life example.     

Identification of average marginal effects under misspecification when covariates are normal

A previously known result in the econometrics literature is that when covariates of an underlying data generating process are jointly normally distributed, estimates from a nonlinear model that is misspecified as linear can be interpreted as average marginal effects. This has been shown for models with exogenous covariates and separability between covariates and errors. In this paper, we extend this identification result to a variety of more general cases, in particular for combinations of separable and nonseparable models under both exogeneity and endogeneity. So long as the underlying model belongs to one of these large classes of data generating processes, our results show that nothing else must be known about the true DGP—beyond normality of observable data, a testable assumption—in order for linear estimators to be interpretable as average marginal effects. We use simulation to explore the performance of these estimators using a misspecified linear model and show they perform well when the data are normal but can perform poorly when this is not the case.     

A Class of Multidimensional Latent Class IRT Models for Ordinal Polytomous Item Responses

We propose a class of multidimensional Item Response Theory models for polytomously-scored items with ordinal response categories. This class extends an existing class of multidimensional models for dichotomously-scored items in which the latent abilities are represented by a random vector assumed to have a discrete distribution, with support points corresponding to different latent classes in the population. In the proposed approach, we allow for different parameterizations for the conditional distribution of the response variables given the latent traits, which depend on the type of link function and the constraints imposed on the item parameters. Moreover, we suggest a strategy for model selection that is based on a series of steps consisting of selecting specific features, such as the dimension of the model (number of latent traits), the number of latent classes, and the specific parameterization. In order to illustrate the proposed approach, we analyze a dataset from a study on anxiety and depression on a sample of oncological patients.     

Exact or approximate inference in graphical models: why the choice is dictated by the treewidth,and how variable elimination can be exploited

Probabilistic graphical models offer a powerful framework to account for the dependence structure between variables, which is represented as a graph. However, the dependence between variables may render inference tasks intractable. In this paper, we review techniques exploiting the graph structure for exact inference, borrowed from optimisation and computer science. They are built on the principle of variable elimination whose complexity is dictated in an intricate way by the order in which variables are eliminated. The so‐called treewidth of the graph characterises this algorithmic complexity: low‐treewidth graphs can be processed efficiently. The first point that we illustrate is therefore the idea that for inference in graphical models, the number of variables is not the limiting factor, and it is worth checking the width of several tree decompositions of the graph before resorting to the approximate method. We show how algorithms providing an upper bound of the treewidth can be exploited to derive a ‘good' elimination order enabling to realise exact inference. The second point is that when the treewidth is too large, algorithms for approximate inference linked to the principle of variable elimination, such as loopy belief propagation and variational approaches, can lead to accurate results while being much less time consuming than Monte‐Carlo approaches. We illustrate the techniques reviewed in this article on benchmarks of inference problems in genetic linkage analysis and computer vision, as well as on hidden variables restoration in coupled Hidden Markov Models.     

Objective Bayes Covariate‐Adjusted Sparse Graphical Model Selection

We present an objective Bayes method for covariance selection in Gaussian multivariate regression models having a sparse regression and covariance structure, the latter being Markov with respect to a directed acyclic graph (DAG). Our procedure can be easily complemented with a variable selection step, so that variable and graphical model selection can be performed jointly. In this way, we offer a solution to a problem of growing importance especially in the area of genetical genomics (eQTL analysis). The input of our method is a single default prior, essentially involving no subjective elicitation, while its output is a closed form marginal likelihood for every covariate‐adjusted DAG model, which is constant over each class of Markov equivalent DAGs; our procedure thus naturally encompasses covariate‐adjusted decomposable graphical models. In realistic experimental studies, our method is highly competitive, especially when the number of responses is large relative to the sample size.     

Identifiability and estimation of recursive max‐linear models

We address the identifiability and estimation of recursive max‐linear structural equation models represented by an edge‐weighted directed acyclic graph (DAG). Such models are generally unidentifiable and we identify the whole class of DAG s and edge weights corresponding to a given observational distribution. For estimation, standard likelihood theory cannot be applied because the corresponding families of distributions are not dominated. Given the underlying DAG, we present an estimator for the class of edge weights and show that it can be considered a generalized maximum likelihood estimator. In addition, we develop a simple method for identifying the structure of the DAG. With probability tending to one at an exponential rate with the number of observations, this method correctly identifies the class of DAGs and, similarly, exactly identifies the possible edge weights.     

From distance sampling to spatial capture–recapture

Distance sampling and capture–recapture are the two most widely used wildlife abundance estimation methods. capture–recapture methods have only recently incorporated models for spatial distribution and there is an increasing tendency for distance sampling methods to incorporated spatial models rather than to rely on partly design-based spatial inference. In this overview we show how spatial models are central to modern distance sampling and that spatial capture–recapture models arise as an extension of distance sampling methods. Depending on the type of data recorded, they can be viewed as particular kinds of hierarchical binary regression, Poisson regression, survival or time-to-event models, with individuals’ locations as latent variables and a spatial model as the latent variable distribution. Incorporation of spatial models in these two methods provides new opportunities for drawing explicitly spatial inferences. Areas of likely future development include more sophisticated spatial and spatio-temporal modelling of individuals’ locations and movements, new methods for integrating spatial capture–recapture and other kinds of ecological survey data, and methods for dealing with the recapture uncertainty that often arise when “capture” consists of detection by a remote device like a camera trap or microphone.     

Embedding latent class regression and latent class distal outcome models into cluster-weighted latent class analysis: a detailed simulation experiment

Usually in latent class (LC) analysis, external predictors are taken to be cluster conditional probability predictors (LC models with external predictors), and/or score conditional probability predictors (LC regression models). In such cases, their distribution is not of interest. Class-specific distribution is of interest in the distal outcome model, when the distribution of the external variables is assumed to depend on LC membership. In this paper, we consider a more general formulation, that embeds both the LC regression and the distal outcome models, as is typically done in cluster-weighted modelling. This allows us to investigate (1) whether the distribution of the external variables differs across classes, (2) whether there are significant direct effects of the external variables on the indicators, by modelling jointly the relationship between the external and the latent variables. We show the advantages of the proposed modelling approach through a set of artificial examples, an extensive simulation study and an empirical application about psychological contracts among employees and employers in Belgium and the Netherlands.     

Generalized Linear Factor Models: A New Local EM Estimation Algorithm

In this article, a general approach to latent variable models based on an underlying generalized linear model (GLM) with factor analysis observation process is introduced. We call these models Generalized Linear Factor Models (GLFM). The observations are produced from a general model framework that involves observed and latent variables that are assumed to be distributed in the exponential family. More specifically, we concentrate on situations where the observed variables are both discretely measured (e.g., binomial, Poisson) and continuously distributed (e.g., gamma). The common latent factors are assumed to be independent with a standard multivariate normal distribution. Practical details of training such models with a new local expectation-maximization (EM) algorithm, which can be considered as a generalized EM-type algorithm, are also discussed. In conjunction with an approximated version of the Fisher score algorithm (FSA), we show how to calculate maximum likelihood estimates of the model parameters, and to yield inferences about the unobservable path of the common factors. The methodology is illustrated by an extensive Monte Carlo simulation study and the results show promising performance.     

Dynamic latent trait models with mixed hidden Markov structure for mixed longitudinal outcomes

We propose a general Bayesian joint modeling approach to model mixed longitudinal outcomes from the exponential family for taking into account any differential misclassification that may exist among categorical outcomes. Under this framework, outcomes observed without measurement error are related to latent trait variables through generalized linear mixed effect models. The misclassified outcomes are related to the latent class variables, which represent unobserved real states, using mixed hidden Markov models (MHMMs). In addition to enabling the estimation of parameters in prevalence, transition and misclassification probabilities, MHMMs capture cluster level heterogeneity. A transition modeling structure allows the latent trait and latent class variables to depend on observed predictors at the same time period and also on latent trait and latent class variables at previous time periods for each individual. Simulation studies are conducted to make comparisons with traditional models in order to illustrate the gains from the proposed approach. The new approach is applied to data from the Southern California Children Health Study to jointly model questionnaire-based asthma state and multiple lung function measurements in order to gain better insight about the underlying biological mechanism that governs the inter-relationship between asthma state and lung function development.     

Distributional aspects in latent variable models

For observable indicators with ordered categories one can assume underlying latent variables following certain marginal distributions. Transforming the latent variables changes its marginal distributions but not the observable qualitative indicators. The joint distribution of the latent variables can be constructed from the marginal distributions. There is a broad class of multivariate distributions for which the observable indicators are equivalent. By choosing the multivariate normal distribution from this class we can analyse a linear relationship between the transformed latent variables. This leads to latent structural equation models. Estimation of these latter models is therefore more general than the distributional assumption might initially suggest. Robustness of the estimation procedure is also discussed for deviations from this distribution family. Using ordinal business survey data of the German Ifo-institute we test the efficiency of firms' price expectations implied by the rational expectation hypothesis.     

Sequential imputation for models with latent variables assuming latent ignorability

Models that involve an outcome variable, covariates, and latent variables are frequently the target for estimation and inference. The presence of missing covariate or outcome data presents a challenge, particularly when missingness depends on the latent variables. This missingness mechanism is called latent ignorable or latent missing at random and is a generalisation of missing at random. Several authors have previously proposed approaches for handling latent ignorable missingness, but these methods rely on prior specification of the joint distribution for the complete data. In practice, specifying the joint distribution can be difficult and/or restrictive. We develop a novel sequential imputation procedure for imputing covariate and outcome data for models with latent variables under latent ignorable missingness. The proposed method does not require a joint model; rather, we use results under a joint model to inform imputation with less restrictive modelling assumptions. We discuss identifiability and convergence‐related issues, and simulation results are presented in several modelling settings. The method is motivated and illustrated by a study of head and neck cancer recurrence. Imputing missing data for models with latent variables under latent‐dependent missingness without specifying a full joint model.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

Multivariate modelling of spatial extremes based on copulas

To model extreme spatial events, a general approach is to use the generalized extreme value (GEV) distribution with spatially varying parameters such as spatial GEV models and latent variable models. In the literature, this approach is mostly used to capture spatial dependence for only one type of event. This limits the applications to air pollutants data as different pollutants may chemically interact with each other. A recent advancement in spatial extremes modelling for multiple variables is the multivariate max-stable processes. Similarly to univariate max-stable processes, the multivariate version also assumes standard distributions such as unit-Fréchet as margins. Additional modelling is required for applications such as spatial prediction. In this paper, we extend the marginal methods such as spatial GEV models and latent variable models into a multivariate setting based on copulas so that it is capable of handling both the spatial dependence and the dependence among multiple pollutants. We apply our proposed model to analyse weekly maxima of nitrogen dioxide, sulphur dioxide, respirable suspended particles, fine suspended particles, and ozone collected in Pearl River Delta in China.     

Modeling Dependence in High Dimensions With Factor Copulas

This article presents flexible new models for the dependence structure, or copula, of economic variables based on a latent factor structure. The proposed models are particularly attractive for relatively high-dimensional applications, involving 50 or more variables, and can be combined with semiparametric marginal distributions to obtain flexible multivariate distributions. Factor copulas generally lack a closed-form density, but we obtain analytical results for the implied tail dependence using extreme value theory, and we verify that simulation-based estimation using rank statistics is reliable even in high dimensions. We consider “scree” plots to aid the choice of the number of factors in the model. The model is applied to daily returns on all 100 constituents of the S&P 100 index, and we find significant evidence of tail dependence, heterogeneous dependence, and asymmetric dependence, with dependence being stronger in crashes than in booms. We also show that factor copula models provide superior estimates of some measures of systemic risk. Supplementary materials for this article are available online.     

Mixed Graphical Models with Missing Data and the Partial Imputation EM Algorithm

In this paper we discuss graphical models for mixed types of continuous and discrete variables with incomplete data. We use a set of hyperedges to represent an observed data pattern. A hyperedge is a set of variables observed for a group of individuals. In a mixed graph with two types of vertices and two types of edges, dots and circles represent discrete and continuous variables respectively. A normal graph represents a graphical model and a hypergraph represents an observed data pattern. In terms of the mixed graph, we discuss decomposition of mixed graphical models with incomplete data, and we present a partial imputation method which can be used in the EM algorithm and the Gibbs sampler to speed their convergence. For a given mixed graphical model and an observed data pattern, we try to decompose a large graph into several small ones so that the original likelihood can be factored into a product of likelihoods with distinct parameters for small graphs. For the case that a graph cannot be decomposed due to its observed data pattern, we can impute missing data partially so that the graph can be decomposed.     

Formulas for counting acyclic digraph Markov equivalence classes

Multivariate Markov dependencies between different variables often can be represented graphically using acyclic digraphs (ADGs). In certain cases, though, different ADGs represent the same statistical model, thus leading to a set of equivalence classes of ADGs that constitute the true universe of available graphical models. Building upon the previously known formulas for counting the number of acyclic digraphs and the number of equivalence classes of size 1, formulas are developed to count ADG equivalence classes of arbitrary size, based on the chordal graph configurations that produce a class of that size. Theorems to validate the formulas as well as to aid in determining the appropriate chordal graphs to use for a given class size are included.     

Generalized Linear Latent Variable Models with Flexible Distribution of Latent Variables

Abstract. We consider a semi‐nonparametric specification for the density of latent variables in Generalized Linear Latent Variable Models (GLLVM). This specification is flexible enough to allow for an asymmetric, multi‐modal, heavy or light tailed smooth density. The degree of flexibility required by many applications of GLLVM can be achieved through this semi‐nonparametric specification with a finite number of parameters estimated by maximum likelihood. Even with this additional flexibility, we obtain an explicit expression of the likelihood for conditionally normal manifest variables. We show by simulations that the estimated density of latent variables capture the true one with good degree of accuracy and is easy to visualize. By analysing two real data sets we show that a flexible distribution of latent variables is a useful tool for exploring the adequacy of the GLLVM in practice.     

On representing second-order uncertainty in multi-state systems via moments of mixtures of Dirichlet distributions†

Second-order probabilities have been proposed as representations of the uncertainty in the parameters of probabilistic models such as Bayesian belief networks. We investigate conditions under which second-order probabilities can be represented in terms of their marginal moments. We show that certain combinations of marginal means and variances do not correspond to any valid second-order joint distribution. By fitting a Dirichlet mixture to marginal mean and variance information, we derive sufficient conditions for a valid second-order joint distribution to exist.