Rejoinder: “Statistical disease mapping for heterogeneous neuroimaging studies”

We thank all the discussants for sharing their valuable viewpoints on the proposed statistical disease mapping (SDM) framework. In our article, we addressed the issue of imaging heterogeneity at both the global and local scales by efficiently borrowing common information shared among a large number of diseased and normal subjects. Understanding such imaging heterogeneity is critical in the development of urgently needed analytic approaches to the prevention, diagnosis, treatment, and prognosis of many diseases (e.g., Alzheimer's disease, brain cancer, and lung cancer), as well as precision medicine broadly. The discussants emphasized improvements to disease mapping by introducing some alternative modelling strategies and many possible future directions in this research topic. The sections of this rejoinder are organized by discussant to address each of their comments separately.     

Why a time effect often has a limited impact on capture‐recapture estimates in closed populations

The author is concerned with log‐linear estimators of the size N of a population in a capture‐recapture experiment featuring heterogeneity in the individual capture probabilities and a time effect. He also considers models where the first capture influences the probability of subsequent captures. He derives several results from a new inequality associated with a dispersive ordering for discrete random variables. He shows that in a log‐linear model with inter‐individual heterogeneity, the estimator N is an increasing function of the heterogeneity parameter. He also shows that the inclusion of a time effect in the capture probabilities decreases N in models without heterogeneity. He further argues that a model featuring heterogeneity can accommodate a time effect through a small change in the heterogeneity parameter. He demonstrates these results using an inequality for the estimators of the heterogeneity parameters and illustrates them in a Monte Carlo experiment     

Bayesian hierarchical models for adaptive basket trial designs

Basket trials evaluate a single drug targeting a single genetic variant in multiple cancer cohorts. Empirical findings suggest that treatment efficacy across baskets may be heterogeneous. Most modern basket trial designs use Bayesian methods. These methods require the prior specification of at least one parameter that permits information sharing across baskets. In this study, we provide recommendations for selecting a prior for scale parameters for adaptive basket trials by using Bayesian hierarchical modeling. Heterogeneity among baskets attracts much attention in basket trial research, and substantial heterogeneity challenges the basic assumption of exchangeability of Bayesian hierarchical approach. Thus, we also allowed each stratum-specific parameter to be exchangeable or nonexchangeable with similar strata by using data observed in an interim analysis. Through a simulation study, we evaluated the overall performance of our design based on statistical power and type I error rates. Our research contributes to the understanding of the properties of Bayesian basket trial designs.     

Inference for Epidemics with Three Levels of Mixing: Methodology and Application to a Measles Outbreak

Abstract. A stochastic epidemic model is defined in which each individual belongs to a household, a secondary grouping (typically school or workplace) and also the community as a whole. Moreover, infectious contacts take place in these three settings according to potentially different rates. For this model, we consider how different kinds of data can be used to estimate the infection rate parameters with a view to understanding what can and cannot be inferred. Among other things we find that temporal data can be of considerable inferential benefit compared with final size data, that the degree of heterogeneity in the data can have a considerable effect on inference for non‐household transmission, and that inferences can be materially different from those obtained from a model with only two levels of mixing. We illustrate our findings by analysing a highly detailed dataset concerning a measles outbreak in Hagelloch, Germany.     

Flexible modelling of simultaneously interval censored and truncated time‐to‐event data

This paper deals with the analysis of data from a HET‐CAM^VT experiment. From a statistical perspective, such data yield many challenges. First of all, the data are typically time‐to‐event like data, which are at the same time interval censored and right truncated. In addition, one has to cope with overdispersion as well as clustering. Traditional analysis approaches ignore overdispersion and clustering and summarize the data into a continuous score that can be analysed using simple linear models. In this paper, a novel combined frailty model is developed that simultaneously captures all of the aforementioned statistical challenges posed by the data. Copyright © 2015 John Wiley & Sons, Ltd.     

Loss functions for estimation of extrema with an application to disease mapping

It is often of interest to find the maximum or near maxima among a set of vector‐valued parameters in a statistical model; in the case of disease mapping, for example, these correspond to relative‐risk “hotspots” where public‐health intervention may be needed. The general problem is one of estimating nonlinear functions of the ensemble of relative risks, but biased estimates result if posterior means are simply substituted into these nonlinear functions. The authors obtain better estimates of extrema from a new, weighted ranks squared error loss function. The derivation of these Bayes estimators assumes a hidden‐Markov random‐field model for relative risks, and their behaviour is illustrated with real and simulated data.     

Non‐stationary Cross‐Covariance Models for Multivariate Processes on a Globe

Abstract. In geophysical and environmental problems, it is common to have multiple variables of interest measured at the same location and time. These multiple variables typically have dependence over space (and/or time). As a consequence, there is a growing interest in developing models for multivariate spatial processes, in particular, the cross‐covariance models. On the other hand, many data sets these days cover a large portion of the Earth such as satellite data, which require valid covariance models on a globe. We present a class of parametric covariance models for multivariate processes on a globe. The covariance models are flexible in capturing non‐stationarity in the data yet computationally feasible and require moderate numbers of parameters. We apply our covariance model to surface temperature and precipitation data from an NCAR climate model output. We compare our model to the multivariate version of the Matérn cross‐covariance function and models based on coregionalization and demonstrate the superior performance of our model in terms of AIC (and/or maximum loglikelihood values) and predictive skill. We also present some challenges in modelling the cross‐covariance structure of the temperature and precipitation data. Based on the fitted results using full data, we give the estimated cross‐correlation structure between the two variables.     

Maximum Likelihood Estimators in Regression Models for Error‐prone Group Testing Data

Since Dorfman's seminal work on the subject, group testing has been widely adopted in epidemiological studies. In Dorfman's context of detecting syphilis, group testing entails pooling blood samples and testing the pools, as opposed to testing individual samples. A negative pool indicates all individuals in the pool free of syphilis antigen, whereas a positive pool suggests one or more individuals carry the antigen. With covariate information collected, researchers have considered regression models that allow one to estimate covariate‐adjusted disease probability. We study maximum likelihood estimators of covariate effects in these regression models when the group testing response is prone to error. We show that, when compared with inference drawn from individual testing data, inference based on group testing data can be more resilient to response misclassification in terms of bias and efficiency. We provide valuable guidance on designing the group composition to alleviate adverse effects of misclassification on statistical inference.     

Statistical studies of infectious disease incidence

Methods for the analysis of data on the incidence of an infectious disease are reviewed, with an emphasis on important objectives that such analyses should address and identifying areas where further work is required. Recent statistical work has adapted methods for constructing estimating functions from martingale theory, methods of data augmentation and methods developed for studying the human immunodeficiency virus–acquired immune deficiency syndrome epidemic. Infectious disease data seem particularly suited to analysis by Markov chain Monte Carlo methods. Epidemic modellers have recently made substantial progress in allowing for community structure and heterogeneity among individuals when studying the requirements for preventing major epidemics. This has stimulated interest in making statistical inferences about crucial parameters from infectious disease data for such community settings.     

A review of statistical methods in imaging genetics

With the rapid growth of modern technology, many biomedical studies are being conducted to collect massive datasets with volumes of multi‐modality imaging, genetic, neurocognitive and clinical information from increasingly large cohorts. Simultaneously extracting and integrating rich and diverse heterogeneous information in neuroimaging and/or genomics from these big datasets could transform our understanding of how genetic variants impact brain structure and function, cognitive function and brain‐related disease risk across the lifespan. Such understanding is critical for diagnosis, prevention and treatment of numerous complex brain‐related disorders (e.g., schizophrenia and Alzheimer's disease). However, the development of analytical methods for the joint analysis of both high‐dimensional imaging phenotypes and high‐dimensional genetic data, a big data squared (BD²) problem, presents major computational and theoretical challenges for existing analytical methods. Besides the high‐dimensional nature of BD², various neuroimaging measures often exhibit strong spatial smoothness and dependence and genetic markers may have a natural dependence structure arising from linkage disequilibrium. We review some recent developments of various statistical techniques for imaging genetics, including massive univariate and voxel‐wise approaches, reduced rank regression, mixture models and group sparse multi‐task regression. By doing so, we hope that this review may encourage others in the statistical community to enter into this new and exciting field of research. The Canadian Journal of Statistics 47: 108–131; 2019 © 2019 Statistical Society of Canada     

Deterministic Evolution of Strength in Multiple Comparisons Models: Who is the Greatest Golfer?

We present a statistical methodology for fitting time‐varying rankings, by estimating the strength parameters of the Plackett–Luce multiple comparisons model at regularly spaced times for each ranked item. We use the little‐known method of barycentric rational interpolation to interpolate between the strength parameters so that a competitor's strength can be evaluated at any time. We chose the time‐varying strengths to evolve deterministically rather than stochastically, a preference that we reason often has merit. There are many statistical and computational problems to overcome on fitting anything beyond ‘toy’ data sets. The methodological innovations here include a method for maximizing a likelihood function for many parameters, approximations for modelling tied data and an approach to the elimination of secular drift of the estimated ‘strengths’. The methodology has obvious applications to fields such as marketing, although we demonstrate our approach by analysing a large data set of golf tournament results, in search of an answer to the question ‘who is the greatest golfer of all time?’     

大数据的整合分析方法

大数据具有数据来源差异性、高维性及稀疏性等特点,如何挖掘数据集间的异质性和共同性并降维去噪是大数据分析的目标与挑战之一。整合分析（Integrative Analysis）同时分析多个独立数据集,避免因地域、时间等因素造成的样本差异而引起模型不稳定,是研究大数据差异性的有效方法。它的特点是将每个解释变量在所有数据集中的系数视为一组,通过惩罚函数对系数组进行压缩,研究变量间的关联性并实现降维。本文从同构数据整合分析、异构数据整合分析以及考虑网络结构的整合分析三方面梳理了惩罚整合分析方法的原理、算法和研究现状。统计模拟发现,在弱相关、一般相关和强相关三种情形下, Group Bridge、 Group MCP、Composite MCP都表现良好,其中 Group Bridge的假阳数最低且最稳定。最后,将整合分析用于研究具有来源差异性的新农合家庭医疗支出,以及具有超高维、小样本等大数据典型特征的癌症基因数据,得到了一些有意义的结论。     

Fast Inference for Network Models of Infectious Disease Spread

Models of infectious disease over contact networks offer a versatile means of capturing heterogeneity in populations during an epidemic. Highly connected individuals tend to be infected at a higher rate early during an outbreak than those with fewer connections. A powerful approach based on the probability generating function of the individual degree distribution exists for modelling the mean field dynamics of outbreaks in such a population. We develop the same idea in a stochastic context, by proposing a comprehensive model for 1‐week‐ahead incidence counts. Our focus is inferring contact network (and other epidemic) parameters for some common degree distributions, in the case when the network is non‐homogeneous ‘at random’. Our model is initially set within a susceptible–infectious–removed framework, then extended to the susceptible–infectious–removed–susceptible scenario, and we apply this methodology to influenza A data.     

Shared frailty models with baseline generalized Pareto distribution

Abstract

In this article, we have considered three different shared frailty models under the assumption of generalized Pareto Distribution as baseline distribution. Frailty models have been used in the survival analysis to account for the unobserved heterogeneity in an individual risks to disease and death. These three frailty models are with gamma frailty, inverse Gaussian frailty and positive stable frailty. Then we introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters. We applied these three models to a kidney infection data and find the best fitted model for kidney infection data. We present a simulation study to compare true value of the parameters with the estimated values. Model comparison is made using Bayesian model selection criterion and a well-fitted model is suggested for the kidney infection data.     

Shared frailty models based on reversed hazard rate for modified inverse Weibull distribution as baseline distribution

The unknown or unobservable risk factors in the survival analysis cause heterogeneity between individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the shared frailty models were suggested. The most common shared frailty model is a model in which frailty act multiplicatively on the hazard function. In this paper, we introduce the shared gamma frailty model and the inverse Gaussian frailty model with the reversed hazard rate. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. We also apply the proposed models to the Australian twin data set and a better model is suggested.     

Segmentation,alignment and statistical analysis of biosignals with application to disease classification

We present a novel methodology for a comprehensive statistical analysis of approximately periodic biosignal data. There are two main challenges in such analysis: (1) the automatic extraction (segmentation) of cycles from long, cyclostationary biosignals and (2) the subsequent statistical analysis, which in many cases involves the separation of temporal and amplitude variabilities. The proposed framework provides a principled approach for statistical analysis of such signals, which in turn allows for an efficient cycle segmentation algorithm. This is achieved using a convenient representation of functions called the square-root velocity function (SRVF). The segmented cycles, represented by SRVFs, are temporally aligned using the notion of the Karcher mean, which in turn allows for more efficient statistical summaries of signals. We show the strengths of this method through various disease classification experiments. In the case of myocardial infarction detection and localization, we show that our method compares favorably to methods described in the current literature.     

A Graphical Diagnostic for Identifying Influential Model Choices in Bayesian Hierarchical Models

Abstract. Real‐world phenomena are frequently modelled by Bayesian hierarchical models. The building‐blocks in such models are the distribution of each variable conditional on parent and/or neighbour variables in the graph. The specifications of centre and spread of these conditional distributions may be well motivated, whereas the tail specifications are often left to convenience. However, the posterior distribution of a parameter may depend strongly on such arbitrary tail specifications. This is not easily detected in complex models. In this article, we propose a graphical diagnostic, the Local critique plot, which detects such influential statistical modelling choices at the node level. It identifies the properties of the information coming from the parents and neighbours (the local prior) and from the children and co‐parents (the lifted likelihood) that are influential on the posterior distribution, and examines local conflict between these distinct information sources. The Local critique plot can be derived for all parameters in a chain graph model.     

A test for detecting etiologic heterogeneity in epidemiological studies

Current statistical methods for analyzing epidemiological data with disease subtype information allow us to acquire knowledge not only for risk factor-disease subtype association but also, on a more profound account, heterogeneity in these associations by multiple disease characteristics (so-called etiologic heterogeneity of the disease). Current interest, particularly in cancer epidemiology, lies in obtaining a valid p-value for testing the hypothesis whether a particular cancer is etiologically heterogeneous. We consider the two-stage logistic regression model along with pseudo-conditional likelihood estimation method and design a testing strategy based on Rao's score test. An extensive Monte Carlo simulation study is carried out, false discovery rate and statistical power of the suggested test are investigated. Simulation results indicate that applying the proposed testing strategy, even a small degree of true etiologic heterogeneity can be recovered with a large statistical power from the sampled data. The strategy is then applied on a breast cancer data set to illustrate its use in practice where there are multiple risk factors and multiple disease characteristics of simultaneous concern.     

Exact Statistical Tests for Heterogeneity of Frequencies Based on Extreme Values

Sophisticated statistical analyses of incidence frequencies are often required for various epidemiologic and biomedical applications. Among the most commonly applied methods is the Pearson's χ² test, which is structured to detect non specific anomalous patterns of frequencies and is useful for testing the significance for incidence heterogeneity. However, the Pearson's χ² test is not efficient for assessing the significance of frequency in a particular cell (or class) to be attributed to chance alone. We recently developed statistical tests for detecting temporal anomalies of disease cases based on maximum and minimum frequencies; these tests are actually designed to test of significance for a particular high or low frequency. The purpose of this article is to demonstrate merits of these tests in epidemiologic and biomedical studies. We show that our proposed methods are more sensitive and powerful for testing extreme cell counts than is the Pearson's χ² test. This feature could provide important and valuable information in epidemiologic or biomeidcal studies. We elucidated and illustrated the differences in sensitivity among our tests and the Pearson's χ² test by analyzing a data set of Langerhans cell histiocytosis cases and its hypothetical sets. We also computed and compared the statistical power of these methods using various sets of cell numbers and alternative frequencies. The investigation of statistical sensitivity and power presented in this work will provide investigators with useful guidelines for selecting the appropriate tests for their studies.     

Some statistical considerations in the clinical development of cancer immunotherapies

Immuno‐oncology has emerged as an exciting new approach to cancer treatment. Common immunotherapy approaches include cancer vaccine, effector cell therapy, and T‐cell–stimulating antibody. Checkpoint inhibitors such as cytotoxic T lymphocyte–associated antigen 4 and programmed death‐1/L1 antagonists have shown promising results in multiple indications in solid tumors and hematology. However, the mechanisms of action of these novel drugs pose unique statistical challenges in the accurate evaluation of clinical safety and efficacy, including late‐onset toxicity, dose optimization, evaluation of combination agents, pseudoprogression, and delayed and lasting clinical activity. Traditional statistical methods may not be the most accurate or efficient. It is highly desirable to develop the most suitable statistical methodologies and tools to efficiently investigate cancer immunotherapies. In this paper, we summarize these issues and discuss alternative methods to meet the challenges in the clinical development of these novel agents. For safety evaluation and dose‐finding trials, we recommend the use of a time‐to‐event model‐based design to handle late toxicities, a simple 3‐step procedure for dose optimization, and flexible rule‐based or model‐based designs for combination agents. For efficacy evaluation, we discuss alternative endpoints/designs/tests including the time‐specific probability endpoint, the restricted mean survival time, the generalized pairwise comparison method, the immune‐related response criteria, and the weighted log‐rank or weighted Kaplan‐Meier test. The benefits and limitations of these methods are discussed, and some recommendations are provided for applied researchers to implement these methods in clinical practice.