Testing the Homogeneity of Two Survival Functions Against a Mixture Alternative Based on Censored Data

When the survival distribution in a treatment group is a mixture of two distributions of the same family, traditional parametric methods that ignore the existence of mixture components or the nonparametric methods may not be very powerful. We develop a modified likelihood ratio test (MLRT) for testing homogeneity in a two sample problem with censored data and compare the actual type I error and power of the MLRT with that nonparametric log-rank test and parametric test through Monte-Carlo simulations. The proposed test is also applied to analyze data from a clinical trial on early breast cancer.     

Comparison of dichotomized and distributional approaches in rare event clinical trial design: a fixed Bayesian design

This research was motivated by our goal to design an efficient clinical trial to compare two doses of docosahexaenoic acid supplementation for reducing the rate of earliest preterm births (ePTB) and/or preterm births (PTB). Dichotomizing continuous gestational age (GA) data using a classic binomial distribution will result in a loss of information and reduced power. A distributional approach is an improved strategy to retain statistical power from the continuous distribution. However, appropriate distributions that fit the data properly, particularly in the tails, must be chosen, especially when the data are skewed. A recent study proposed a skew-normal method. We propose a three-component normal mixture model and introduce separate treatment effects at different components of GA. We evaluate operating characteristics of mixture model, beta-binomial model, and skew-normal model through simulation. We also apply these three methods to data from two completed clinical trials from the USA and Australia. Finite mixture models are shown to have favorable properties in PTB analysis but minimal benefit for ePTB analysis. Normal models on log-transformed data have the largest bias. Therefore we recommend finite mixture model for PTB study. Either finite mixture model or beta-binomial model is acceptable for ePTB study.     

An application of the mixed-effects model and pattern mixture model to treatment groups with differential missingness suspected not-missing-at-random

Likelihood-based, mixed-effects models for repeated measures (MMRMs) are occasionally used in primary analyses for group comparisons of incomplete continuous longitudinal data. Although MMRM analysis is generally valid under missing-at-random assumptions, it is invalid under not-missing-at-random (NMAR) assumptions. We consider the possibility of bias of estimated treatment effect using standard MMRM analysis in a motivational case, and propose simple and easily implementable pattern mixture models within the framework of mixed-effects modeling, to handle the NMAR data with differential missingness between treatment groups. The proposed models are a new form of pattern mixture model that employ a categorical time variable when modeling the outcome and a continuous time variable when modeling the missingness-data patterns. The models can directly provide an overall estimate of the treatment effect of interest using the average of the distribution of the missingness indicator and a categorical time variable in the same manner as MMRM analysis. Our simulation results indicate that the bias of the treatment effect for MMRM analysis was considerably larger than that for the pattern mixture model analysis under NMAR assumptions. In the case study, it would be dangerous to interpret only the results of the MMRM analysis, and the proposed pattern mixture model would be useful as a sensitivity analysis for treatment effect evaluation.     

Efficient computation of nonparametric survival functions via a hierarchical mixture formulation

We propose a new algorithm for computing the maximum likelihood estimate of a nonparametric survival function for interval-censored data, by extending the recently-proposed constrained Newton method in a hierarchical fashion. The new algorithm makes use of the fact that a mixture distribution can be recursively written as a mixture of mixtures, and takes a divide-and-conquer approach to break down a large-scale constrained optimization problem into many small-scale ones, which can be solved rapidly. During the course of optimization, the new algorithm, which we call the hierarchical constrained Newton method, can efficiently reallocate the probability mass, both locally and globally, among potential support intervals. Its convergence is theoretically established based on an equilibrium analysis. Numerical study results suggest that the new algorithm is the best choice for data sets of any size and for solutions with any number of support intervals.     

Semiparametric Estimation of a Two-component Mixture Model where One Component is known

Abstract.  We consider a two-component mixture model where one component distribution is known while the mixing proportion and the other component distribution are unknown. These kinds of models were first introduced in biology to study the differences in expression between genes. The various estimation methods proposed till now have all assumed that the unknown distribution belongs to a parametric family. In this paper, we show how this assumption can be relaxed. First, we note that generally the above model is not identifiable, but we show that under moment and symmetry conditions some 'almost everywhere' identifiability results can be obtained. Where such identifiability conditions are fulfilled we propose an estimation method for the unknown parameters which is shown to be strongly consistent under mild conditions. We discuss applications of our method to microarray data analysis and to the training data problem. We compare our method to the parametric approach using simulated data and, finally, we apply our method to real data from microarray experiments.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.     

Mixture modelling of recurrent event times with long-term survivors: Analysis of Hutterite birth intervals

We propose a mixture model that combines a discrete-time survival model for analyzing the correlated times between recurrent events, e.g. births, with a logistic regression model for the probability of never experiencing the event of interest, i.e., being a long-term survivor. The proposed survival model incorporates both observed and unobserved heterogeneity in the probability of experiencing the event of interest. We use Gibbs sampling for the fitting of such mixture models, which leads to a computationally intensive solution to the problem of fitting survival models for multiple event time data with long-term survivors. We illustrate our Bayesian approach through an analysis of Hutterite birth histories.     

Testing for the Presence of a Cure Fraction in Clustered Interval‐Censored Survival Data

Clustered interval‐censored survival data are often encountered in clinical and epidemiological studies due to geographic exposures and periodic visits of patients. When a nonnegligible cured proportion exists in the population, several authors in recent years have proposed to use mixture cure models incorporating random effects or frailties to analyze such complex data. However, the implementation of the mixture cure modeling approaches may be cumbersome. Interest then lies in determining whether or not it is necessary to adjust the cured proportion prior to the mixture cure analysis. This paper mainly focuses on the development of a score for testing the presence of cured subjects in clustered and interval‐censored survival data. Through simulation, we evaluate the sampling distribution and power behaviour of the score test. A bootstrap approach is further developed, leading to more accurate significance levels and greater power in small sample situations. We illustrate applications of the test using data sets from a smoking cessation study and a retrospective study of early breast cancer patients.     

On a mixture autoregressive model

We generalize the Gaussian mixture transition distribution (GMTD) model introduced by Le and co-workers to the mixture autoregressive (MAR) model for the modelling of non-linear time series. The models consist of a mixture of K stationary or non-stationary AR components. The advantages of the MAR model over the GMTD model include a more full range of shape changing predictive distributions and the ability to handle cycles and conditional heteroscedasticity in the time series. The stationarity conditions and autocorrelation function are derived. The estimation is easily done via a simple EM algorithm and the model selection problem is addressed. The shape changing feature of the conditional distributions makes these models capable of modelling time series with multimodal conditional distributions and with heteroscedasticity. The models are applied to two real data sets and compared with other competing models. The MAR models appear to capture features of the data better than other competing models do.     

An extension of the placebo‐based pattern‐mixture model

Pattern‐mixture models provide a general and flexible framework for sensitivity analyses of nonignorable missing data in longitudinal studies. The placebo‐based pattern‐mixture model handles missing data in a transparent and clinically interpretable manner. We extend this model to include a sensitivity parameter that characterizes the gradual departure of the missing data mechanism from being missing at random toward being missing not at random under the standard placebo‐based pattern‐mixture model. We derive the treatment effect implied by the extended model. We propose to utilize the primary analysis based on a mixed‐effects model for repeated measures to draw inference about the treatment effect under the extended placebo‐based pattern‐mixture model. We use simulation studies to confirm the validity of the proposed method. We apply the proposed method to a clinical study of major depressive disorders. Copyright © 2013 John Wiley & Sons, Ltd.     

Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach

The modeling and analysis of lifetime data in which the main endpoints are the times when an event of interest occurs is of great interest in medical studies. In these studies, it is common that two or more lifetimes associated with the same unit such as the times to deterioration levels or the times to reaction to a treatment in pairs of organs like lungs, kidneys, eyes or ears. In medical applications, it is also possible that a cure rate is present and needed to be modeled with lifetime data with long-term survivors. This paper presented a comparative study under a Bayesian approach among some existing continuous and discrete bivariate distributions such as the bivariate exponential distributions and the bivariate geometric distributions in presence of cure rate, censored data and covariates. In presence of lifetimes related to cured patients, it is assumed standard mixture cure rate models in the data analysis. The posterior summaries of interest are obtained using Markov Chain Monte Carlo methods. To illustrate the proposed methodology two real medical data sets are considered.     

Testing for the maximum mean in a mixture of normals

We consider the test of the null hypothesis that the largest mean in a mixture of an unknown number of normal components is less than or equal to a given threshold. This test is motivated by the problem of assessing whether the Soviet Union has been operating in compliance with the Nuclear Test Ban Treaty. In our analysis, the number of normal components is determined using Akaike's Information Criterion while the hypothesis test itself is based on asymptotic results given by Behboodian for a mixture of two normal components. A bootstrap approach is also considered for estimating the standard error of the largest estimated mean. The performance of the testa are examined through the use of simulation.     

A method of power assessment for tests comparing several treatments with a control

A method of power assessment for the problem of comparing several treatments with a control is considered. Power assessment is based on the power function of a two-sided hypothesis test that none of the treatment is different from the control. Normally distributed data and binary response data are considered. Minimum power levels are found under certain easily interpretable range conditions on the treatment and control means or success probabilities. Expressions are provided allowing simple computer evaluation of minimum guaranteed power levels, and some illustrative tables of power levels are given.     

Randomized response model versus mixture model in the crossover design for clinical trials

ABSTRACT

The clinical trials are usually designed with the implicit assumption that data analysis will occur only after the trial is completed. It is a challenging problem if the sponsor wishes to evaluate the drug efficacy in the middle of the study without breaking the randomization codes. In this article, the randomized response model and mixture model are introduced to analyze the data, masking the randomization codes of the crossover design. Given the probability of treatment sequence, the test of mixture model provides higher power than the test of randomized response model, which is inadequate in the example. The paired t-test has higher powers than both models if the randomization codes are broken. The sponsor may stop the trial early to claim the effectiveness of the study drug if the mixture model concludes a positive result.     

Maximum likelihood estimation of bivariate logistic models for incomplete responses with indicators of ignorable and non-ignorable missingness

Summary. Missing observations are a common problem that complicate the analysis of clustered data. In the Connecticut child surveys of childhood psychopathology, it was possible to identify reasons why outcomes were not observed. Of note, some of these causes of missingness may be assumed to be ignorable , whereas others may be non-ignorable . We consider logistic regression models for incomplete bivariate binary outcomes and propose mixture models that permit estimation assuming that there are two distinct types of missingness mechanisms: one that is ignorable; the other non-ignorable. A feature of the mixture modelling approach is that additional analyses to assess the sensitivity to assumptions about the missingness are relatively straightforward to incorporate. The methods were developed for analysing data from the Connecticut child surveys, where there are missing informant reports of child psychopathology and different reasons for missingness can be distinguished.     

A nonparametric Bayesian model for inference in related longitudinal studies

Summary.  We discuss a method for combining different but related longitudinal studies to improve predictive precision. The motivation is to borrow strength across clinical studies in which the same measurements are collected at different frequencies. Key features of the data are heterogeneous populations and an unbalanced design across three studies of interest. The first two studies are phase I studies with very detailed observations on a relatively small number of patients. The third study is a large phase III study with over 1500 enrolled patients, but with relatively few measurements on each patient. Patients receive different doses of several drugs in the studies, with the phase III study containing significantly less toxic treatments. Thus, the main challenges for the analysis are to accommodate heterogeneous population distributions and to formalize borrowing strength across the studies and across the various treatment levels. We describe a hierarchical extension over suitable semiparametric longitudinal data models to achieve the inferential goal. A nonparametric random-effects model accommodates the heterogeneity of the population of patients. A hierarchical extension allows borrowing strength across different studies and different levels of treatment by introducing dependence across these nonparametric random-effects distributions. Dependence is introduced by building an analysis of variance (ANOVA) like structure over the random-effects distributions for different studies and treatment combinations. Model structure and parameter interpretation are similar to standard ANOVA models. Instead of the unknown normal means as in standard ANOVA models, however, the basic objects of inference are random distributions, namely the unknown population distributions under each study. The analysis is based on a mixture of Dirichlet processes model as the underlying semiparametric model.     

On Bayesian analyses and finite mixtures for proportions

When the results of biological experiments are tested for a possible difference between treatment and control groups, the inference is only valid if based upon a model that fits the experimental results satisfactorily. In dominant-lethal testing, foetal death has previously been assumed to follow a variety of models, including a Poisson, Binomial, Beta-binomial and various mixture models. However, discriminating between models has always been a particularly difficult problem. In this paper, we consider the data from 6 separate dominant-lethal assay experiments and discriminate between the competing models which could be used to describe them. We adopt a Bayesian approach and illustrate how a variety of different models may be considered, using Markov chain Monte Carlo (MCMC) simulation techniques and comparing the results with the corresponding maximum likelihood analyses. We present an auxiliary variable method for determining the probability that any particular data cell is assigned to a given component in a mixture and we illustrate the value of this approach. Finally, we show how the Bayesian approach provides a natural and unique perspective on the model selection problem via reversible jump MCMC and illustrate how probabilities associated with each of the different models may be calculated for each data set. In terms of estimation we show how, by averaging over the different models, we obtain reliable and robust inference for any statistic of interest.     

Practical approaches for design and analysis of clinical trials of infertility treatments: crossover designs and the Mantel–Haenszel method are recommended

Crossover designs have some advantages over standard clinical trial designs and they are often used in trials evaluating the efficacy of treatments for infertility. However, clinical trials of infertility treatments violate a fundamental condition of crossover designs, because women who become pregnant in the first treatment period are not treated in the second period. In previous research, to deal with this problem, some new designs, such as re‐randomization designs, and analysis methods including the logistic mixture model and the beta‐binomial mixture model were proposed. Although the performance of these designs and methods has previously been evaluated in large‐scale clinical trials with sample sizes of more than 1000 per group, the actual sample sizes of infertility treatment trials are usually around 100 per group. The most appropriate design and analysis for these moderate‐scale clinical trials are currently unclear. In this study, we conducted simulation studies to determine the appropriate design and analysis method of moderate‐scale clinical trials for irreversible endpoints by evaluating the statistical power and bias in the treatment effect estimates. The Mantel–Haenszel method had similar power and bias to the logistic mixture model. The crossover designs had the highest power and the smallest bias. We recommend using a combination of the crossover design and the Mantel–Haenszel method for two‐period, two‐treatment clinical trials with irreversible endpoints. Copyright © 2015 John Wiley & Sons, Ltd.     

A Finite Mixture Three-Parameter Weibull Model for the Analysis of Wind Speed Data

This article presents a mixture three-parameter Weibull distribution to model wind speed data. The parameters are estimated by using maximum likelihood (ML) method in which the maximization problem is regarded as a nonlinear programming with only inequality constraints and is solved numerically by the interior-point method. By applying this model to four lattice-point wind speed sequences extracted from National Centers for Environmental Prediction (NCEP) reanalysis data, it is observed that the mixture three-parameter Weibull distribution model proposed in this paper provides a better fit than the existing Weibull models for the analysis of wind speed data under study.     

Bayesian analysis for mixtures of discrete distributions with a non-parametric component

Bayesian finite mixture modelling is a flexible parametric modelling approach for classification and density fitting. Many areas of application require distinguishing a signal from a noise component. In practice, it is often difficult to justify a specific distribution for the signal component; therefore, the signal distribution is usually further modelled via a mixture of distributions. However, modelling the signal as a mixture of distributions is computationally non-trivial due to the difficulties in justifying the exact number of components to be used and due to the label switching problem. This paper proposes the use of a non-parametric distribution to model the signal component. We consider the case of discrete data and show how this new methodology leads to more accurate parameter estimation and smaller false non-discovery rate. Moreover, it does not incur the label switching problem. We show an application of the method to data generated by ChIP-sequencing experiments.