Small-sample confidence regions in exponential families

This article presents an algorithm for small-sample conditional confidence regions for two or more parameters for any discrete regression model in the generalized linear interactive model family. Regions are constructed by careful inversion of conditional hypothesis tests. This method presupposes the use of approximate or exact techniques for enumerating the sample space for some components of the vector of sufficient statistics conditional on other components. Such enumeration may be performed exactly or by exact or approximate Monte Carlo, including the algorithms of Kolassa and Tanner (1994, Journal of the American Statistical Association 89, 697-702; 1999, Biometrics 55, 246-251). This method also assumes that one can compute certain conditional probabilities for a fixed value of the parameter vector. Because of a property of exponential families, one can use this set of conditional probabilities to directly compute the conditional probabilities associated with any other value of the vector of the parameters of interest. This observation dramatically reduces the computational effort required to invert the hypothesis test to obtain the confidence region. To construct a region with confidence level 1 - alpha, the algorithm begins with a grid of values for the parameters of interest. For each parameter vector on the grid (corresponding to the current null hypothesis), one transforms the initial set of conditional probabilities using exponential tilting and then calculates the p value for this current null hypothesis. The confidence region is the set of parameter values for which the p value is at least alpha.     

A comparison of exact, mid-P, and score tests for matched case-control studies

A recently developed algorithm for generating the distribution of sufficient statistics for conditional logistic models can be put to a twofold use. First, it provides an avenue for performing inference for matched case-control studies that does not rely on the assumption of a large sample size. Second, joint distributions generated by this algorithm can be used to make comparisons of various inferential procedures that are free from Monte Carlo sampling errors. In this paper, these two features of the algorithm are utilized to compare small-sample properties of the exact, mid-P value, and score tests for a conditional logistic model with two unmatched binary covariates. Both uniparametric and multiparametric tests, performed at a nominal significance level of .05, were studied. It was found that the actual significance levels of the mid-P test tend to be closer to the nominal level when compared with those of the other two tests.     

Exact inference for matched case-control studies

In an epidemiological study with a small sample size or a sparse data structure, the use of an asymptotic method of analysis may not be appropriate. In this paper we present an alternative method of analyzing data for case-control studies with a matched design that does not rely on large-sample assumptions. A recursive algorithm to compute the exact distribution of the conditional sufficient statistics of the parameters of the logistic model for such a design is given. This distribution can be used to perform exact inference on model parameters, the methodology of which is outlined. To illustrate the exact method, and compare it with the conventional asymptotic method, analyses of data from two case-control studies are also presented.     

Monte Carlo EM for missing covariates in parametric regression models

We propose a method for estimating parameters for general parametric regression models with an arbitrary number of missing covariates. We allow any pattern of missing data and assume that the missing data mechanism is ignorable throughout. When the missing covariates are categorical, a useful technique for obtaining parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). We extend this method to continuous or mixed categorical and continuous covariates, and for arbitrary parametric regression models, by adapting a Monte Carlo version of the EM algorithm as discussed by Wei and Tanner (1990, Journal of the American Statistical Association 85, 699-704). In addition, we discuss the Gibbs sampler for sampling from the conditional distribution of the missing covariates given the observed data and show that the appropriate complete conditionals are log-concave. The log-concavity property of the conditional distributions will facilitate a straightforward implementation of the Gibbs sampler via the adaptive rejection algorithm of Gilks and Wild (1992, Applied Statistics 41, 337-348). We assume the model for the response given the covariates is an arbitrary parametric regression model, such as a generalized linear model, a parametric survival model, or a nonlinear model. We model the marginal distribution of the covariates as a product of one-dimensional conditional distributions. This allows us a great deal of flexibility in modeling the distribution of the covariates and reduces the number of nuisance parameters that are introduced in the E-step. We present examples involving both simulated and real data.     

Bayesian semiparametric copula estimation with application to psychiatric genetics

This paper proposes a semiparametric methodology for modeling multivariate and conditional distributions. We first build a multivariate distribution whose dependence structure is induced by a Gaussian copula and whose marginal distributions are estimated nonparametrically via mixtures of B‐spline densities. The conditional distribution of a given variable is obtained in closed form from this multivariate distribution. We take a Bayesian approach, using Markov chain Monte Carlo methods for inference. We study the frequentist properties of the proposed methodology via simulation and apply the method to estimation of conditional densities of summary statistics, used for computing conditional local false discovery rates, from genetic association studies of schizophrenia and cardiovascular disease risk factors.     

A probabilistic network for the diagnosis of acute cardiopulmonary diseases

In this paper, the development of a probabilistic network for the diagnosis of acute cardiopulmonary diseases is presented in detail. A panel of expert physicians collaborated to specify the qualitative part, which is a directed acyclic graph defining a factorization of the joint probability distribution of domain variables into univariate conditional distributions. The quantitative part, which is a set of parametric models defining these univariate conditional distributions, was estimated following the Bayesian paradigm. In particular, we exploited an original reparameterization of Beta and categorical logistic regression models to elicit the joint prior distribution of parameters from medical experts, and updated it by conditioning on a dataset of hospital records via Markov chain Monte Carlo simulation. Refinement was iteratively performed until the probabilistic network provided satisfactory concordance index values for several acute diseases and reasonable diagnosis for six fictitious patient cases. The probabilistic network can be employed to perform medical diagnosis on a total of 63 diseases (38 acute and 25 chronic) on the basis of up to 167 patient findings.     

Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models

Inference for Dirichlet process hierarchical models is typicallyperformed using Markov chain Monte Carlo methods, which canbe roughly categorized into marginal and conditional methods.The former integrate out analytically the infinite-dimensionalcomponent of the hierarchical model and sample from the marginaldistribution of the remaining variables using the Gibbs sampler.Conditional methods impute the Dirichlet process and updateit as a component of the Gibbs sampler. Since this requiresimputation of an infinite-dimensional process, implementationof the conditional method has relied on finite approximations.In this paper, we show how to avoid such approximations by designingtwo novel Markov chain Monte Carlo algorithms which sample fromthe exact posterior distribution of quantities of interest.The approximations are avoided by the new technique of retrospectivesampling. We also show how the algorithms can obtain samplesfrom functionals of the Dirichlet process. The marginal andthe conditional methods are compared and a careful simulationstudy is included, which involves a non-conjugate model, differentdatasets and prior specifications.     

An empirical assessment of saddlepoint approximations for testing a logistic regression parameter.

Saddlepoint methods provide quick and easy approximations to significance levels for conditional tests of logistic regression parameters. We evaluate the accuracies of saddlepoint approximations for three well-known conditional tests: Bartlett's test for no three-factor interaction in a 2 x 2 x 2 table, the test for trend in a series of probabilities, and the exact test of no association in stratified 2 x 2 tables with a common odds ratio. General recommendations are suggested regarding the use of saddlepoint approximations for exact conditional significance levels.     

Extension of the SAEM algorithm for nonlinear mixed models with 2 levels of random effects

This article focuses on parameter estimation of multilevel nonlinearmixed-effects models (MNLMEMs). These models are used to analyzedata presenting multiple hierarchical levels of grouping (clusterdata, clinical trials with several observation periods, ...).The variability of the individual parameters of the regressionfunction is thus decomposed as a between-subject variabilityand higher levels of variability (e.g. within-subject variability).We propose maximum likelihood estimates of parameters of thoseMNLMEMs with 2 levels of random effects, using an extensionof the stochastic approximation version of expectation–maximization(SAEM)–Monte Carlo Markov chain algorithm. The extendedSAEM algorithm is split into an explicit direct expectation–maximization(EM) algorithm and a stochastic EM part. Compared to the originalalgorithm, additional sufficient statistics have to be approximatedby relying on the conditional distribution of the second levelof random effects. This estimation method is evaluated on pharmacokineticcrossover simulated trials, mimicking theophylline concentrationdata. Results obtained on those data sets with either the SAEMalgorithm or the first-order conditional estimates (FOCE) algorithm(implemented in the nlme function of R software) are compared:biases and root mean square errors of almost all the SAEM estimatesare smaller than the FOCE ones. Finally, we apply the extendedSAEM algorithm to analyze the pharmacokinetic interaction oftenofovir on atazanavir, a novel protease inhibitor, from theAgence Nationale de Recherche sur le Sida 107-Puzzle 2 study.A significant decrease of the area under the curve of atazanaviris found in patients receiving both treatments.     

Simplified Bayesian Sensitivity Analysis for Mismeasured and Unobserved Confounders

Summary We examine situations where interest lies in the conditional association between outcome and exposure variables, given potential confounding variables. Concern arises that some potential confounders may not be measured accurately, whereas others may not be measured at all. Some form of sensitivity analysis might be employed, to assess how this limitation in available data impacts inference. A Bayesian approach to sensitivity analysis is straightforward in concept: a prior distribution is formed to encapsulate plausible relationships between unobserved and observed variables, and posterior inference about the conditional exposure–disease relationship then follows. In practice, though, it can be challenging to form such a prior distribution in both a realistic and simple manner. Moreover, it can be difficult to develop an attendant Markov chain Monte Carlo (MCMC) algorithm that will work effectively on a posterior distribution arising from a highly nonidentified model. In this article, a simple prior distribution for acknowledging both poorly measured and unmeasured confounding variables is developed. It requires that only a small number of hyperparameters be set by the user. Moreover, a particular computational approach for posterior inference is developed, because application of MCMC in a standard manner is seen to be ineffective in this problem.     

On the quasi-stationary distribution of the Ross malaria model.

Approximations are derived for the quasi-stationary distribution of the fully stochastic version of the classical Ross malaria model. The approximations are developed in two stages. In the first stage, the Ross process is approximated with a bivariate Markov chain without an absorbing state. The second stage of the approximation uses ideas from perturbation theory to derive explicit expressions that serve as approximations of the joint stationary distribution of the approximating process. Numerical comparisons are made between the approximations and the quasi-stationary distribution.     

Markov regression models for time series: a quasi-likelihood approach

This paper discusses a quasi-likelihood (QL) approach to regression analysis with time series data. We consider a class of Markov models, referred to by Cox (1981, Scandinavian Journal of Statistics 8, 93-115) as "observation-driven" models in which the conditional means and variances given the past are explicit functions of past outcomes. The class includes autoregressive and Markov chain models for continuous and categorical observations as well as models for counts (e.g., Poisson) and continuous outcomes with constant coefficient of variation (e.g., gamma). We focus on Poisson and gamma data for illustration. Analogous to QL for independent observations, large-sample properties of the regression coefficients depend only on correct specification of the first conditional moment.     

Comparison of Bayesian and maximum-likelihood inference of population genetic parameters

Comparison of the performance and accuracy of different inference methods, such as maximum likelihood (ML) and Bayesian inference, is difficult because the inference methods are implemented in different programs, often written by different authors. Both methods were implemented in the program MIGRATE, that estimates population genetic parameters, such as population sizes and migration rates, using coalescence theory. Both inference methods use the same Markov chain Monte Carlo algorithm and differ from each other in only two aspects: parameter proposal distribution and maximization of the likelihood function. Using simulated datasets, the Bayesian method generally fares better than the ML approach in accuracy and coverage, although for some values the two approaches are equal in performance. MOTIVATION: The Markov chain Monte Carlo-based ML framework can fail on sparse data and can deliver non-conservative support intervals. A Bayesian framework with appropriate prior distribution is able to remedy some of these problems. RESULTS: The program MIGRATE was extended to allow not only for ML(-) maximum likelihood estimation of population genetics parameters but also for using a Bayesian framework. Comparisons between the Bayesian approach and the ML approach are facilitated because both modes estimate the same parameters under the same population model and assumptions.     

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.     

Exchangeability in multivariate Markov chain models.

Time-homogeneous Markov chain models with state space [0, 1]k are useful in analysis of binary follow-up data on k individuals that interact. The number of parameters increases exponentially with k so more restrictive models are imperative for statistical inference. The hypothesis that the matrix of transition probabilities is invariant under permutation of individuals is discussed. It is shown that if individuals are exchangeable, then the process counting the number of individuals occupying a given state is a Markov chain. This reduction of data is sufficient if either at most a single individual may change state between two consecutive time points or if a state is absorbing. Similar results are obtained for exchangeability within two subgroups. Inference in the multivariate process reduces to a univariate problem if individuals are independent given the group's previous response. It is shown how conditional independence could be tested assuming exchangeability. The different hypotheses re examined in an analysis of the occurrence of bacteria in milk samples of Danish dairy cattle.     

A Variational Bayes Approach to the Analysis of Occupancy Models

Detection-nondetection data are often used to investigate species range dynamics using Bayesian occupancy models which rely on the use of Markov chain Monte Carlo (MCMC) methods to sample from the posterior distribution of the parameters of the model. In this article we develop two Variational Bayes (VB) approximations to the posterior distribution of the parameters of a single-season site occupancy model which uses logistic link functions to model the probability of species occurrence at sites and of species detection probabilities. This task is accomplished through the development of iterative algorithms that do not use MCMC methods. Simulations and small practical examples demonstrate the effectiveness of the proposed technique. We specifically show that (under certain circumstances) the variational distributions can provide accurate approximations to the true posterior distributions of the parameters of the model when the number of visits per site (K) are as low as three and that the accuracy of the approximations improves as K increases. We also show that the methodology can be used to obtain the posterior distribution of the predictive distribution of the proportion of sites occupied (PAO).     

A comparison of strategies for Markov chain Monte Carlo computation in quantitative genetics

In quantitative genetics, Markov chain Monte Carlo (MCMC) methods are indispensable for statistical inference in non-standard models like generalized linear models with genetic random effects or models with genetically structured variance heterogeneity. A particular challenge for MCMC applications in quantitative genetics is to obtain efficient updates of the high-dimensional vectors of genetic random effects and the associated covariance parameters. We discuss various strategies to approach this problem including reparameterization, Langevin-Hastings updates, and updates based on normal approximations. The methods are compared in applications to Bayesian inference for three data sets using a model with genetically structured variance heterogeneity.     

The Highest Confidence Density Region and Its Usage for Joint Inferences about Constrained Parameters

Summary Suppose that we are interested in making joint inferences about a set of constrained parameters. Confidence regions for these parameters are often constructed via a normal approximation of the distribution of a consistent estimator for a transformation of the parameters. In this article, we utilize the confidence distribution, a frequentist counterpart to the posterior distribution in Bayesian statistics, to obtain optimal confidence regions for the parameters. Members of such a region can be generated efficiently via a standard Markov chain Monte Carlo algorithm. We then apply this technique to draw inferences about the temporal profile of the survival function with censored observations. We illustrate the new proposal with the survival data from the well‐known Mayo primary biliary cirrhosis study and show that the volume of the new 0.95 confidence region is only one thirty‐fourth of that of the conventional confidence band.