Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.     

Exact Goodness‐of‐Fit Testing for the Ising Model

The Ising model is one of the simplest and most famous models of interacting systems. It was originally proposed to model ferromagnetic interactions in statistical physics and is now widely used to model spatial processes in many areas such as ecology, sociology, and genetics, usually without testing its goodness of fit. Here, we propose various test statistics and an exact goodness‐of‐fit test for the finite‐lattice Ising model. The theory of Markov bases has been developed in algebraic statistics for exact goodness‐of‐fit testing using a Monte Carlo approach. However, finding a Markov basis is often computationally intractable. Thus, we develop a Monte Carlo method for exact goodness‐of‐fit testing for the Ising model that avoids computing a Markov basis and also leads to a better connectivity of the Markov chain and hence to a faster convergence. We show how this method can be applied to analyze the spatial organization of receptors on the cell membrane.     

Centered parameterizations and dependence limitations in Markov random field models

Markov random field models incorporate terms representing local statistical dependence among variables in a discrete-index random field. Traditional parameterizations for models based on one-parameter exponential family conditional distributions contain components that would appear to reflect large-scale and small-scale model behaviors, and it is natural to attempt to match these structures with large-scale and small-scale patterns in a set of data. Traditional manners of parameterizing Markov random field models do not allow such correspondence, however. We propose an alternative centered parameterization that, while not leading to different models, allows a correspondence between model structures and data structures to be successfully accomplished. The ability to make these connections is important when incorporating covariate information into a model or if a sequence of models is fit over time to investigate and interpret possible changes in data structure. We demonstrate the improved interpretation that results from use of centered parameterizations. Centered parameterizations also lend themselves to computation of an interpretable decomposition of mean squared error, and this is demonstrated both analytically and through a simulated example. A breakdown in model behavior occurs even with centered parameterizations if dependence parameters in Markov random field models are allowed to become too large. This phenomenon is discussed and illustrated using an auto-logistic model.     

Testing goodness of fit via nonparametric function estimation techniques

An overview is given of methodology for testing goodness of fit of parametric models using nonparametric function estimation techniques. The ideas are illustrated in two settings: the classical one-sample goodness-of-fit scenario and testing the goodness of fit of a polynomial regression model.     

Goodness of fit and homogeneity tests on the basis of N-distances

In this paper we propose an application of N-distance theory [Klebanov, L.B., 2005. N-distances and their applications. Karolinum, Prague] for testing simple hypotheses of goodness of fit and homogeneity. The asymptotic null distribution of test statistics is established and coincides with the distribution of infinite quadratic form of independent standard normal random variables. A construction of multivariate free-of-distribution homogeneity test is considered. The power of proposed criteria is compared with classical tests using Monte-Carlo simulations.     

Evaluation of Bayesian multiple stage estimation under spatial CAR model variants

In this study, an evaluation of Bayesian hierarchical models is made based on simulation scenarios to compare single-stage and multi-stage Bayesian estimations. Simulated datasets of lung cancer disease counts for men aged 65 and older across 44 wards in the London Health Authority were analysed using a range of spatially structured random effect components. The goals of this study are to determine which of these single-stage models perform best given a certain simulating model, how estimation methods (single- vs. multi-stage) compare in yielding posterior estimates of fixed effects in the presence of spatially structured random effects, and finally which of two spatial prior models – the Leroux or ICAR model, perform best in a multi-stage context under different assumptions concerning spatial correlation. Among the fitted single-stage models without covariates, we found that when there is low amount of variability in the distribution of disease counts, the BYM model is relatively robust to misspecification in terms of DIC, while the Leroux model is the least robust to misspecification. When these models were fit to data generated from models with covariates, we found that when there was one set of covariates – either spatially correlated or non-spatially correlated, changing the values of the fixed coefficients affected the ability of either the Leroux or ICAR model to fit the data well in terms of DIC. When there were multiple sets of spatially correlated covariates in the simulating model, however, we could not distinguish the goodness of fit to the data between these single-stage models. We found that the multi-stage modelling process via the Leroux and ICAR models generally reduced the variance of the posterior estimated fixed effects for data generated from models with covariates and a UH term compared to analogous single-stage models. Finally, we found the multi-stage Leroux model compares favourably to the multi-stage ICAR model in terms of DIC. We conclude that the mutli-stage Leroux model should be seriously considered in applications of Bayesian disease mapping when an investigator desires to fit a model with both fixed effects and spatially structured random effects to Poisson count data.     

A Case Study for Modelling Cancer Incidence Using Bayesian Spatio‐Temporal Models

Researchers familiar with spatial models are aware of the challenge of choosing the level of spatial aggregation. Few studies have been published on the investigation of temporal aggregation and its impact on inferences regarding disease outcome in space–time analyses. We perform a case study for modelling individual disease outcomes using several Bayesian hierarchical spatio‐temporal models, while taking into account the possible impact of spatial and temporal aggregation. Using longitudinal breast cancer data from South East Queensland, Australia, we consider both parametric and non‐parametric formulations for temporal effects at various levels of aggregation. Two temporal smoothness priors are considered separately; each is modelled with fixed effects for the covariates and an intrinsic conditional autoregressive prior for the spatial random effects. Our case study reveals that different model formulations produce considerably different model performances. For this particular dataset, a classical parametric formulation that assumes a linear time trend produces the best fit among the five models considered. Different aggregation levels of temporal random effects were found to have little impact on model goodness‐of‐fit and estimation of fixed effects.     

A generalized-moments specification test for the logistic link

The authors consider the problem of testing the validity of the logistic regression model using a random sample. Given the values of the response variable, they observe that the sample actually consists of two independent subsets of observations whose density ratio has a known parametric form when the model is true. They are thus led to propose a generalized-moments specification test in detail. In addition, they show that this test can be derived using Neyman's smooth tests for goodness of fit. They present simulation results and apply the methodology to the analysis of two real data sets.     

Rank-Based Classification Using Robust Discriminant Functions

The Pareto distribution model assumption in the peaks over threshold method, will be tested by making using of the Kolmogorov–Smirnov goodness of fit method. Pareto distributed variables can be transformed to exponential, and the test will be for exponentiality. It was found that the statistic can be used as an indication of where to choose the threshold and to check the Pareto model assumption.     

On random grouping in goodness of fit tests of discrete distributions

We consider the problem of random grouping of data from discrete distributions to form χ² goodness of fit tests. In general the random partitions need not converge to the partition one gets using the true distribution function and the same partitioning scheme. We present a method of guaranteeing the convergence of the random partition and yielding the usual χ² asymptotics.     

A Goodness of Fit Test for the Multilevel Logistic Model

It is crucial to test the goodness of fit of a model before it is used to make statistical inferences. However, no satisfactory goodness of fit test is available for the case of categorical multilevel data which occur when categorical data are clustered or hierarchical in nature. Hence the aim of this paper is to develop a new goodness of fit test for multilevel binary data based on Hosmer and Lemeshow and Lipsitz et.al. In order to identify the properties of the developed test, simulation studies were carried out to assess the Type I error and the power.     

Factors influencing Soay sheep survival

We present a survival analysis of Soay sheep mark recapture and recovery data. Unlike previous conditional analyses, it is not necessary to assume equality of recovery and recapture probabilities; instead these are estimated by maximum likelihood. Male and female sheep are treated separately, with the higher numbers and survival probabilities of the females resulting in a more complex model than that used for the males. In both cases, however, age and time aspects need to be included and there is a strong indication of a reduction in survival for sheep aged 7 years or more. Time variation in survival is related to the size of the population and selected weather variables, by using logistic regression. The size of the population significantly affects the survival probabilities of male and female lambs, and of female sheep aged 7 or more years. March rainfall and a measure of the North Atlantic oscillation are found to influence survival significantly for all age groups considered, for both males and females. Either of these weather variables can be used in a model. Several phenotypic and genotypic individual covariates are also fitted. The only covariate which is found to influence survival significantly is the type of horn of first-year female sheep. There is a substantial variation in the recovery probabilities over time, reflecting in part the increased effort when a population crash was expected. The goodness of fit of the model is checked by using graphical procedures.     

Degree-based goodness-of-fit tests for heterogeneous random graph models: Independent and exchangeable cases

The degrees are a classical and relevant way to study the topology of a network. They can be used to assess the goodness of fit for a given random graph model. In this paper, we introduce goodness-of-fit tests for two classes of models. First, we consider the case of independent graph models such as the heterogeneous Erdös-Rényi model in which the edges have different connection probabilities. Second, we consider a generic model for exchangeable random graphs called the W-graph. The stochastic block model and the expected degree distribution model fall within this framework. We prove the asymptotic normality of the degree mean square under these independent and exchangeable models and derive formal tests. We study the power of the proposed tests and we prove the asymptotic normality under specific sparsity regimes. The tests are illustrated on real networks from social sciences and ecology, and their performances are assessed via a simulation study.     

The 'demon' problem of Youden: Exponential case

We consider the problem of finding the probability of a sample mean falling above the (n - k)th-order statistic in a random sample of size n. Explicit expressions are obtained for the exponential distribution. Some applications that pertain to testing for outliers and goodness of fit are given.     

A General formula for the upper tail significance levels of empirical distribution function test statistics

In this paper ve obtain an asymptotic expression for the upper tail area of the distribution of an infinite weighted sum of chi-square random variables and show how this can be applied to distributions of various goodness of fit test statistics. Results obtained by this general approach are comparable with those reported previously in the literature. In the case of the Cramer-von Mises statistic an empirical adjustment is given vhich significantly improves on previous approximations. For the Kuiper statistic the corresponding empirical adjustment leads to an existing highly accurate approximation.     

A goodness of fit test for the multilevel proportional odds model

It is essential to test the goodness of fit of the model before making inferences based on it. Multilevel modeling of ordinal categorical responses is not as developed as for continuous responses. Assessing model adequacy in terms of the goodness of fit with ordinal categorical responses is still being developed and no satisfactory tests are available so far. As a consequence of that, this study concentrates on developing such a goodness of fit test for Multilevel Proportional Odds models and to study the properties of the test.     

Statistical Analysis for the Inverse Trinomial Distribution

This article considers parameter estimation, goodness of fit, likelihood ratio and score tests, and model selection by Akaike information criterion for the inverse trinomial (IT) distribution, a classical one-dimensional random walk distribution. The IT distribution has a cubic variance function of the mean and is a generalization of the negative binomial distribution. Basic distributional properties and expressions for the probability mass function, recurrence formula, moments, and score functions are also presented.     

Elliptically contoured model and factorization of wllks1 A

In a series of papers, Kshirsagar (1964, 1971) and McHenry and Kshirsagar (1977), factorize Wilks' A into a number of factors and find the independent null multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of independent or dependent variables. Essentially the factors of Wilks' A are the factors of certain multivariate beta distributed matrix or its determinant. The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distribution of these factors. The present paper generalizes Kshirsagar's (1971) normal theory to the elliptically contoured model, and shows that his results are null robust for the elliptically contoured model.     

A compound Poisson model for word occurrences in DNA sequences

Summary. We present a compound Poisson model describing the occurrence process of a set of words in a random sequence of letters. The model takes into account the frequency of the words and their overlapping structure. The model is compared with a Markov chain model in terms of fit and parsimony. Special attention is given to the detection of poor or rich regions. Several applications of the model are presented and a combination of the Markov and compound Poisson models is proposed.     

Moments of nonparametric probability density functions of entropy estimators applied to testing the inverse Gaussian distribution

The inverse Gaussian (IG) distribution is widely used to model data and then it is important to develop efficient goodness of fit tests for this distribution. In this article, we introduce some new test statistics for examining the IG goodness of fit based on correcting moments of nonparametric probability density functions of entropy estimators. These tests are consistent against all alternatives. Critical points and power of the tests are explored by simulation. We show that the proposed tests are more powerful than competitor tests. Finally, the proposed tests are illustrated by real data examples.