Bootstrap likelihood ratio test for Weibull mixture models fitted to grouped data

Abstract

Weibull mixture models are widely used in a variety of fields for modeling phenomena caused by heterogeneous sources. We focus on circumstances in which original observations are not available, and instead the data comes in the form of a grouping of the original observations. We illustrate EM algorithm for fitting Weibull mixture models for grouped data and propose a bootstrap likelihood ratio test (LRT) for determining the number of subpopulations in a mixture model. The effectiveness of the LRT methods are investigated via simulation. We illustrate the utility of these methods by applying them to two grouped data applications.     

Missing data in clinical trials: from clinical assumptions to statistical analysis using pattern mixture models

The need to use rigorous, transparent, clearly interpretable, and scientifically justified methodology for preventing and dealing with missing data in clinical trials has been a focus of much attention from regulators, practitioners, and academicians over the past years. New guidelines and recommendations emphasize the importance of minimizing the amount of missing data and carefully selecting primary analysis methods on the basis of assumptions regarding the missingness mechanism suitable for the study at hand, as well as the need to stress‐test the results of the primary analysis under different sets of assumptions through a range of sensitivity analyses. Some methods that could be effectively used for dealing with missing data have not yet gained widespread usage, partly because of their underlying complexity and partly because of lack of relatively easy approaches to their implementation. In this paper, we explore several strategies for missing data on the basis of pattern mixture models that embody clear and realistic clinical assumptions. Pattern mixture models provide a statistically reasonable yet transparent framework for translating clinical assumptions into statistical analyses. Implementation details for some specific strategies are provided in an Appendix (available online as Supporting Information), whereas the general principles of the approach discussed in this paper can be used to implement various other analyses with different sets of assumptions regarding missing data. Copyright © 2013 John Wiley & Sons, Ltd.     

Bayesian variable selection for longitudinal substance abuse treatment data subject to informative censoring

Summary.  Measuring the process of care in substance abuse treatment requires analysing repeated client assessments at critical time points during treatment tenure. Assessments are frequently censored because of early departure from treatment. Most analyses accounting for informative censoring define the censoring time to be that of the last observed assessment. However, if missing assessments for those who remain in treatment are attributable to logistical reasons rather than to the underlying treatment process being measured, then the length of stay in treatment might better characterize censoring than would time of measurement. Bayesian variable selection is incorporated in the conditional linear model to assess whether time of measurement or length of stay better characterizes informative censoring. Marginal posterior distributions of the trajectory of treatment process scores are obtained that incorporate model uncertainty. The methodology is motivated by data from an on-going study of the quality of care in in-patient substance abuse treatment.     

Markov structure based logistic regression for repeated measures: An application to diabetes mellitus data

In longitudinal studies, as repeated observations are made on the same individual the response variables will usually be correlated. In analyzing such data, this dependence must be taken into account to avoid misleading inferences. The focus of this paper is to apply a logistic marginal model with Markovian dependence proposed by Azzalini [A. Azzalini, Logistic regression for autocorrelated data with application to repeated measures, Biometrika 81 (1994) 767–775] to the study of the influence of time-dependent covariates on the marginal distribution of the binary response in serially correlated binary data. We have shown how to construct the model so that the covariates relate only to the mean value of the process, independent of the association parameters. After formulating the proposed model for repeated measures data, the same approach is applied to missing data. An application is provided to the diabetes mellitus data of registered patients at the Bangladesh Institute of Research and Rehabilitation in Diabetes, Endocrine and Metabolic Disorders (BIRDEM) in 1984, using both time stationary and time varying covariates.     

Analysis of longitudinal data using a finite mixture model

This paper considers a finite mixture model for longitudinal data, which can be used to study the dependency of the shape of the respective follow-up curves on treatments or other influential factors and to classify these curves. An EM-algorithm to achieve the ml-estimate of the model is given. The potencies of the model are demonstrated using data of a clinical trial.     

Likelihood inference for correlated binary data without any information about the joint distributions

We propose a universal robust likelihood that is able to accommodate correlated binary data without any information about the underlying joint distributions. This likelihood function is asymptotically valid for the regression parameter for any underlying correlation configurations, including varying under- or over-dispersion situations, which undermines one of the regularity conditions ensuring the validity of crucial large sample theories. This robust likelihood procedure can be easily implemented by using any statistical software that provides naïve and sandwich covariance matrices for regression parameter estimates. Simulations and real data analyses are used to demonstrate the efficacy of this parametric robust method.     

Semiparametric analysis of mixture regression models with competing risks data

In the analysis of competing risks data, cumulative incidence function is a useful summary of the overall crude risk for a failure type of interest. Mixture regression modeling has served as a natural approach to performing covariate analysis based on this quantity. However, existing mixture regression methods with competing risks data either impose parametric assumptions on the conditional risks or require stringent censoring assumptions. In this article, we propose a new semiparametric regression approach for competing risks data under the usual conditional independent censoring mechanism. We establish the consistency and asymptotic normality of the resulting estimators. A simple resampling method is proposed to approximate the distribution of the estimated parameters and that of the predicted cumulative incidence functions. Simulation studies and an analysis of a breast cancer dataset demonstrate that our method performs well with realistic sample sizes and is appropriate for practical use.     

A multiple imputation method for incomplete correlated ordinal data using multivariate probit models

The multiple imputation technique has proven to be a useful tool in missing data analysis. We propose a Markov chain Monte Carlo method to conduct multiple imputation for incomplete correlated ordinal data using the multivariate probit model. We conduct a thorough simulation study to compare the performance of our proposed method with two available imputation methods – multivariate normal-based and chain equation methods for various missing data scenarios. For illustration, we present an application using the data from the smoking cessation treatment study for low-income community corrections smokers.     

Prediction of transplant-free survival in idiopathic pulmonary fibrosis patients using joint models for event times and mixed multivariate longitudinal data

We implement a joint model for mixed multivariate longitudinal measurements, applied to the prediction of time until lung transplant or death in idiopathic pulmonary fibrosis. Specifically, we formulate a unified Bayesian joint model for the mixed longitudinal responses and time-to-event outcomes. For the longitudinal model of continuous and binary responses, we investigate multivariate generalized linear mixed models using shared random effects. Longitudinal and time-to-event data are assumed to be independent conditional on available covariates and shared parameters. A Markov chain Monte Carlo algorithm, implemented in OpenBUGS, is used for parameter estimation. To illustrate practical considerations in choosing a final model, we fit 37 different candidate models using all possible combinations of random effects and employ a deviance information criterion to select a best-fitting model. We demonstrate the prediction of future event probabilities within a fixed time interval for patients utilizing baseline data, post-baseline longitudinal responses, and the time-to-event outcome. The performance of our joint model is also evaluated in simulation studies.     

A class of weighted dependence measures for bivariate failure time data

This paper considers a class of summary measures of the dependence between a pair of failure time variables over a finite follow-up region. The class consists of measures that are weighted averages of local dependence measures, and includes the cross-ratio-measure and finite region version of Kendall's τ; recently proposed by the authors. Two new special cases are identified that can avoid the need to estimate the bivariate survivor function and that admit explicit variance estimators. Nonparametric estimators of such dependence measures are proposed and are shown to be consistent and asymptotically normal with variances that can be consistently estimated. Properties of selected estimators are evaluated in a simulation study, and the method is illustrated through an analysis of Australian Twin Study data.     

Comparative GMM and GQL logistic regression models on hierarchical data

We often rely on the likelihood to obtain estimates of regression parameters but it is not readily available for generalized linear mixed models (GLMMs). Inferences for the regression coefficients and the covariance parameters are key in these models. We presented alternative approaches for analyzing binary data from a hierarchical structure that do not rely on any distributional assumptions: a generalized quasi-likelihood (GQL) approach and a generalized method of moments (GMM) approach. These are alternative approaches to the typical maximum-likelihood approximation approach in Statistical Analysis System (SAS) such as Laplace approximation (LAP). We examined and compared the performance of GQL and GMM approaches with multiple random effects to the LAP approach as used in PROC GLIMMIX, SAS. The GQL approach tends to produce unbiased estimates, whereas the LAP approach can lead to highly biased estimates for certain scenarios. The GQL approach produces more accurate estimates on both the regression coefficients and the covariance parameters with smaller standard errors as compared to the GMM approach. We found that both GQL and GMM approaches are less likely to result in non-convergence as opposed to the LAP approach. A simulation study was conducted and a numerical example was presented for illustrative purposes.     

A longitudinal study of children's aggressive behaviours based on multivariate mixed models with incomplete data

The authors consider children's behavioural and emotional problems and their relationships with possible predictors. They propose a multivariate transitional mixed‐effects model for a longitudinal study and simultaneously address non‐ignorable missing data in responses and covariates, measurement errors in covariates, and multivariate modelling of the responses and covariate processes. A real dataset is analysed in details using the proposed method with some interesting results. The Canadian Journal of Statistics 37: 435–452; 2009 © 2009 Statistical Society of Canada     

The estimation of multidimensional fixed effects panel data models

This article introduces the appropriate within estimators for the most frequently used three-dimensional fixed effects panel data models. It analyzes the behavior of these estimators in the cases of no self-flow data, unbalanced data, and dynamic autoregressive models. The main results are then generalized for higher dimensional panel data sets as well.     

Testing equality of correlation coefficients for paired binary data from multiple groups

Paired binary data arise naturally when paired body parts are investigated in clinical trials. One of the widely used models for dealing with this kind of data is the equal correlation coefficients model. Before using this model, it is necessary to test whether the correlation coefficients in each group are actually equal. In this paper, three test statistics (likelihood ratio test, Wald-type test, and Score test) are derived for this purpose. The simulation results show that the Score test statistic maintains type I error rate and has satisfactory power, and therefore is recommended among the three methods. The likelihood ratio test is over conservative in most cases, and the Wald-type statistic is not robust with respect to empirical type I error. Three real examples, including a multi-centre Phase II double-blind placebo randomized controlled trial, are given to illustrate the three proposed test statistics.     

Outlier detection in contingency tables using decomposable graphical models

For high-dimensional data, it is a tedious task to determine anomalies such as outliers. We present a novel outlier detection method for high-dimensional contingency tables. We use the class of decomposable graphical models to model the relationship among the variables of interest, which can be depicted by an undirected graph called the interaction graph. Given an interaction graph, we derive a closed-form expression of the likelihood ratio test (LRT) statistic and an exact distribution for efficient simulation of the test statistic. An observation is declared an outlier if it deviates significantly from the approximated distribution of the test statistic under the null hypothesis. We demonstrate the use of the LRT outlier detection framework on genetic data modeled by Chow–Liu trees.     

The effects of missing serial effects and/or heteroscedastic errors on mixed models using repeated growth data

When a two-level multilevel model (MLM) is used for repeated growth data, the individuals constitute level 2 and the successive measurements constitute level 1, which is nested within the individuals that make up level 2. The heterogeneity among individuals is represented by either the random-intercept or random-coefficient (slope) model. The variance components at level 1 involve serial effects and measurement errors under constant variance or heteroscedasticity. This study hypothesizes that missing serial effects or/and heteroscedasticity may bias the results obtained from two-level models. To illustrate this effect, we conducted two simulation studies, where the simulated data were based on the characteristics of an empirical mouse tumour data set. The results suggest that for repeated growth data with constant variance (measurement error) and misspecified serial effects (ρ > 0.3), the proportion of level-2 variation (intra-class correlation coefficient) increases with ρ and the two-level random-coefficient model is the minimum AIC (or AIC_c) model when compared with the fixed model, heteroscedasticity model, and random-intercept model. In addition, the serial effect (ρ > 0.1) and heteroscedasticity are both misspecified, implying that the two-level random-coefficient model is the minimum AIC (or AIC_c) model when compared with the fixed model and random-intercept model. This study demonstrates that missing serial effects and/or heteroscedasticity may indicate heterogeneity among individuals in repeated growth data (mixed or two-level MLM). This issue is critical in biomedical research.     

A comparison of regression models for ordinal data in an analysis of transplanted-kidney function

Three regression models for ordinal data, those of Fienberg, McCullagh, and Anderson, are applied to an analysis of kidney function among transplant recipients. The conclusions arising from each model are presented and contrasted.     

The likelihood ratio test for homogeneity in finite mixture models

The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re‐sampling procedure is recommended to obtain the asymptotic p‐value. Three kernel functions, normal, binomial and Poisson, are used in a simulation study which illustrates the procedure.     

The Analysis of Methadone Clinic Data Using Marginal and Conditional Logistic Models with Mixture or Random Effects

The paper develops methods for the statistical analysis of outcomes of methadone maintenance treatment (MMT). Subjects for this study were a cohort of patients entering MMT in Sydney in 1986. Urine drug tests on these subjects were performed weekly during MMT, and were reported as either positive or negative for morphine, the marker of recent heroin use. To allow correlation between the repeated binary measurements, a marginal logistic model was fitted using the generalized estimating equation (GEE) approach and the alternating logistic regression approach. Conditional logistic models are also considered. Results of separate fitting to each patient and score tests suggest that there is substantial between-patient variation in response to MMT. To account for the population heterogeneity and to facilitate subject-specific inference, the conditional logistic model is extended by introducing random intercepts. The two, three and four group mixture models are also investigated. The model of best fit is a three group mixture model, in which about a quarter of the subjects have a poor response to MMT, with continued heroin use independent of daily dose of methadone; about a quarter of the subjects have a very good response, with little or no heroin use, again independent of dose; and about half the subjects responded in a dose-dependent fashion, with reduced heroin use while receiving higher doses of methadone. These findings are consistent with clinical experience. There is also an association between reduced drug use and increased duration in treatment. The mixture model is recommended since it is quite tractable in terms of estimation and model selection as well as being supported by clinical experience.     

Shrinkage and pretest estimators for longitudinal data analysis under partially linear models

In this paper, we develop marginal analysis methods for longitudinal data under partially linear models. We employ the pretest and shrinkage estimation procedures to estimate the mean response parameters as well as the association parameters, which may be subject to certain restrictions. We provide the analytic expressions for the asymptotic biases and risks of the proposed estimators, and investigate their relative performance to the unrestricted semiparametric least-squares estimator (USLSE). We show that if the dimension of association parameters exceeds two, the risk of the shrinkage estimators is strictly less than that of the USLSE in most of the parameter space. On the other hand, the risk of the pretest estimator depends on the validity of the restrictions of association parameters. A simulation study is conducted to evaluate the performance of the proposed estimators relative to that of the USLSE. A real data example is applied to illustrate the practical usefulness of the proposed estimation procedures.