Incorporating gene functional annotations in detecting differential gene expression

Summary.  The importance of incorporating existing biological knowledge, such as gene functional annotations in gene ontology, in analysing high throughput genomic and proteomic data is being increasingly recognized. In the context of detecting differential gene expression, however, the current practice of using gene annotations is limited primarily to validations. Here we take a direct approach to incorporating gene annotations into mixture models for analysis. First, in contrast with a standard mixture model assuming that each gene of the genome has the same distribution, we study stratified mixture models allowing genes with different annotations to have different distributions, such as prior probabilities. Second, rather than treating parameters in stratified mixture models independently, we propose a hierarchical model to take advantage of the hierarchical structure of most gene annotation systems, such as gene ontology. We consider a simplified implementation for the proof of concept. An application to a mouse microarray data set and a simulation study demonstrate the improvement of the two new approaches over the standard mixture model.     

Posterior Predictive p-values in Bayesian Hierarchical Models

Abstract.  The present work focuses on extensions of the posterior predictive p -value (ppp-value) for models with hierarchical structure, designed for testing assumptions made on underlying processes. The ppp-values are popular as tools for model criticism, yet their lack of a common interpretation limit their practical use. We discuss different extensions of ppp-values to hierarchical models, allowing for discrepancy measures that can be used for checking properties of the model at all stages. Through analytical derivations and simulation studies on simple models, we show that similar to the standard ppp-values, these extensions are typically far from uniformly distributed under the model assumptions and can give poor power in a hypothesis testing framework. We propose a calibration of the p -values, making the resulting calibrated p -values uniformly distributed under the model conditions. Illustrations are made through a real example of multinomial regression to age distributions of fish.     

A Bayesian hierarchical mixture model for platelet-derived growth factor receptor phosphorylation to improve estimation of progression-free survival in prostate cancer

Summary.  Advances in understanding the biological underpinnings of many cancers have led increasingly to the use of molecularly targeted anticancer therapies. Because the platelet-derived growth factor receptor (PDGFR) has been implicated in the progression of prostate cancer bone metastases, it is of great interest to examine possible relationships between PDGFR inhibition and therapeutic outcomes. We analyse the association between change in activated PDGFR (phosphorylated PDGFR) and progression-free survival time based on large within-patient samples of cell-specific phosphorylated PDGFR values taken before and after treatment from each of 88 prostate cancer patients. To utilize these paired samples as covariate data in a regression model for progression-free survival time, and be cause the phosphorylated PDGFR distributions are bimodal, we first employ a Bayesian hierarchical mixture model to obtain a deconvolution of the pretreatment and post-treatment within-patient phosphorylated PDGFR distributions. We evaluate fits of the mixture model and a non-mixture model that ignores the bimodality by using a supnorm metric to compare the empirical distribution of each phosphorylated PDGFR data set with the corresponding fitted distribution under each model. Our results show that first using the mixture model to account for the bimodality of the within-patient phosphorylated PDGFR distributions, and then using the posterior within-patient component mean changes in phosphorylated PDGFR so obtained as covariates in the regression model for progression-free survival time, provides an improved estimation.     

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.     

The Identification of “Unusual” Health-Care Providers From a Hierarchical Model

It has become common to adopt a hierarchical model structure when comparing the performance of multiple health-care providers. This structure allows some variation in such measures, beyond that explained by sampling variation, to be “normal,” in recognition of the fact that risk-adjustment is never perfect. The shrinkage estimates arising from such a model structure also have appealing properties.

It is not immediately clear, however, how “unusual” providers, that is, any with particularly high or low rates, can be identified based on such a model. Given that some variation in underlying rates is assumed to be the norm, we argue that it is not generally appropriate to identify a provider as interesting based only on evidence of it lying above or below the population mean. We note with concern, however, that this practice is not uncommon.

We examine in detail three possible strategies for identifying unusual providers, carefully distinguishing between statistical “outliers” and “extremes.” A two-level normal model is used for mathematical simplicity, but we note that much of the discussion also applies to alternative data structures. Further, we emphasize throughout that each approach can be viewed as resulting from a Bayesian or a classical perspective. Three worked examples provide additional insight.     

A Bayesian hierarchical model for categorical longitudinal data from a social survey of immigrants

Summary.  The paper investigates a Bayesian hierarchical model for the analysis of categorical longitudinal data from a large social survey of immigrants to Australia. Data for each subject are observed on three separate occasions, or waves, of the survey. One of the features of the data set is that observations for some variables are missing for at least one wave. A model for the employment status of immigrants is developed by introducing, at the first stage of a hierarchical model, a multinomial model for the response and then subsequent terms are introduced to explain wave and subject effects. To estimate the model, we use the Gibbs sampler, which allows missing data for both the response and the explanatory variables to be imputed at each iteration of the algorithm, given some appropriate prior distributions. After accounting for significant covariate effects in the model, results show that the relative probability of remaining unemployed diminished with time following arrival in Australia.     

基于分层阿基米德Copula的金融时间序列的相关性分析

与阿基米德copula相比,分层阿基米德copula(HAC)的结构更具一般性,而相比于椭圆型copula它的待估参数个数更少。用两阶段极大似然法来估计HAC函数,主要的步骤是先估计出每个分量的边际分布,以此为基础再估计copula函数。实证分析中,采取Clayton和Gumbel型的HAC分析四只股票价格序列之间的相关性。在得出HAC的结构和估计其参数之前,运用ARMA-GARCH过程消除了序列的自相关性和条件异方差。通过比较赤迟信息准则,认为完全嵌套的Gumbel型HAC能更好地刻画这种相关性。     

A hierarchical modelling approach to analysing longitudinal data with drop-out and non-compliance, with application to an equivalence trial in paediatric acquired immune deficiency syndrome

Longitudinal clinical trials with long follow-up periods almost invariably suffer from a loss to follow-up and non-compliance with the assigned therapy. An example is protocol 128 of the AIDS Clinical Trials Group, a 5-year equivalency trial comparing reduced dose zidovudine with the standard dose for treatment of paediatric acquired immune deficiency syndrome patients. This study compared responses to treatment by using both clinical and cognitive outcomes. The cognitive outcomes are of particular interest because the effects of human immunodeficiency virus infection of the central nervous system can be more acute in children than in adults. We formulate and apply a Bayesian hierarchical model to estimate both the intent-to-treat effect and the average causal effect of reducing the prescribed dose of zidovudine by 50%. The intent-to-treat effect quantifies the causal effect of assigning the lower dose, whereas the average causal effect represents the causal effect of actually taking the lower dose. We adopt a potential outcomes framework where, for each individual, we assume the existence of a different potential outcomes process at each level of time spent on treatment. The joint distribution of the potential outcomes and the time spent on assigned treatment is formulated using a hierarchical model: the potential outcomes distribution is given at the first level, and dependence between the outcomes and time on treatment is specified at the second level by linking the time on treatment to subject-specific effects that characterize the potential outcomes processes. Several distributional and structural assumptions are used to identify the model from observed data, and these are described in detail. A detailed analysis of AIDS Clinical Trials Group protocol 128 is given; inference about both the intent-to-treat effect and average causal effect indicate a high probability of dose equivalence with respect to cognitive functioning.     

A hierarchical model for extreme wind speeds

Summary.  A typical extreme value analysis is often carried out on the basis of simplistic inferential procedures, though the data being analysed may be structurally complex. Here we develop a hierarchical model for hourly gust maximum wind speed data, which attempts to identify site and seasonal effects for the marginal densities of hourly maxima, as well as for the serial dependence at each location. A Gaussian model for the random effects exploits the meteorological structure in the data, enabling increased precision for inferences at individual sites and in individual seasons. The Bayesian framework that is adopted is also exploited to obtain predictive return level estimates at each site, which incorporate uncertainty due to model estimation, as well as the randomness that is inherent in the processes that are involved.     

Simultaneous confidexce intervals in an extended crow iii curve model

This paper deals with estimation problems under an extended growth curve model with two hierarchical within-individuals design matrices. The model in cludes the one whose mean structure consists of polynomial growth curves with two different degrees. First we propose certain simple estimators of the mean and covariance parameters which are closely related to the MEE's. Some basic properties of the estimators are given. Simultaneous confidence intervals are constructed, based on the estimators, for each and both of two growth curves. We give asymptotic approximations for the corresponding critical points. A numerical example is also given.     

Nonparametric survival regression using the beta-Stacy process

A novel class of hierarchical nonparametric Bayesian survival regression models for time-to-event data with uninformative right censoring is introduced. The survival curve is modeled as a random function whose prior distribution is defined using the beta-Stacy (BS) process. The prior mean of each survival probability and its prior variance are linked to a standard parametric survival regression model. This nonparametric survival regression can thus be anchored to any reference parametric form, such as a proportional hazards or an accelerated failure time model, allowing substantial departures of the predictive survival probabilities when the reference model is not supported by the data. Also, under this formulation the predictive survival probabilities will be close to the empirical survival distribution near the mode of the reference model and they will be shrunken towards its probability density in the tails of the empirical distribution.     

A note on Bayesian residuals as a hierarchical model diagnostic technique

When one wants to check a tentatively proposed model for departures that are not well specified, looking at residuals is the most common diagnostic technique. Here, we investigate the use of Bayesian standardized residuals to detect unknown hierarchical structure. Asymptotic theory, also supported by simulations, shows that the use of Bayesian standardized residuals is effective when the within group correlation, ρ, is large. However, we show that standardized residuals may not detect hierarchical structure when ρ is small. Thus, if it is important to detect modest hierarchical structure (i.e., ρ small) one should use other diagnostic techniques in addition to the standardized residuals. We use “quality of care” data from the Patterns of Care Study, a two-stage cluster sample of patients undergoing radiation therapy for cervix cancer, to illustrate the potential use of these residuals to detect missing hierarchical structure.     

A Bayesian hierarchical model for inference across related reverse phase protein arrays experiments

We consider inference for functional proteomics experiments that record protein activation over time following perturbation under different dose levels of several drugs. The main inference goal is the dependence structure of the selected proteins. A critical challenge is the lack of sufficient data under any one drug and dose level to allow meaningful inference on dependence structure. We propose a hierarchical model to implement the desired inference. The key element of the model is a shared dependence structure on (latent) binary indicators of protein activation.     

A study of R2 measure under the accelerated failure time models

For right-censored data, the accelerated failure time (AFT) model is an alternative to the commonly used proportional hazards regression model. It is a linear model for the (log-transformed) outcome of interest, and is particularly useful for censored outcomes that are not time-to-event, such as laboratory measurements. We provide a general and easily computable definition of the R² measure of explained variation under the AFT model for right-censored data. We study its behavior under different censoring scenarios and under different error distributions; in particular, we also study its robustness when the parametric error distribution is misspecified. Based on Monte Carlo investigation results, we recommend the log-normal distribution as a robust error distribution to be used in practice for the parametric AFT model, when the R² measure is of interest. We apply our methodology to an alcohol consumption during pregnancy data set from Ukraine.     

Modeling population and subject-specific growth in a latent trait measured by multiple instruments over time using a hierarchical Bayesian framework

Psychometric growth curve modeling techniques are used to describe a person’s latent ability and how that ability changes over time based on a specific measurement instrument. However, the same instrument cannot always be used over a period of time to measure that latent ability. This is often the case when measuring traits longitudinally in children. Reasons may be that over time some measurement tools that were difficult for young children become too easy as they age resulting in floor effects or ceiling effects or both. We propose a Bayesian hierarchical model for such a scenario. Within the Bayesian model we combine information from multiple instruments used at different age ranges and having different scoring schemes to examine growth in latent ability over time. The model includes between-subject variance and within-subject variance and does not require linking item specific difficulty between the measurement tools. The model’s utility is demonstrated on a study of language ability in children from ages one to ten who are hard of hearing where measurement tool specific growth and subject-specific growth are shown in addition to a group level latent growth curve comparing the hard of hearing children to children with normal hearing.KEYWORDS: Bayesian hierarchical models, psychometric modeling, language ability, growth curve modeling, longitudinal analysis     

Capturing the Spillover Effect With Multiplicative Error Models

Recent statistical models for the analysis of volatility in financial markets serve the purpose of incorporating the effect of other markets in their structure, in order to study the spillover or the contagion phenomena. Extending the Multiplicative Error Model we are able to capture these characteristics, under the assumption that the conditional mean of the volatility can be decomposed into the sum of one component representing the proper volatility of the time series analyzed, and other components, each representing the volatility transmitted from one other market. Each component follows a proper dynamics with elements that can be usefully interpreted. This particular decomposition allows to establish, each time, the contribution brought by each individual market to the global volatility of the market object of the analysis. We experiment this model with four stock indices.     

Identifying intraclass correlations necessitating hierarchical modeling

Hierarchical binary outcome data with three levels, such as disease remission for patients nested within physicians, nested within clinics are frequently encountered in practice. One important aspect in such data is the correlation that occurs at each level of the data. In parametric modeling, accounting for these correlations increases the complexity. These models may also yield results that lead to the same conclusions as simpler models. We developed a measure of intraclass correlation at each stage of a three-level nested structure and identified guidelines for determining when the dependencies in hierarchical models need to be taken into account. These guidelines are supported by simulations of hierarchical data sets, as well as the analysis of AIDS knowledge in Bangladesh from the 2011 Demographic Health Survey. We also provide a simple rule of thumb to assist researchers faced with the challenge of choosing an appropriately complex model when analyzing hierarchical binary data.     

Extensions of a Conflict Measure of Inconsistencies in Bayesian Hierarchical Models

Abstract.  In a recent paper we extended and refined some tools introduced by O'Hagan for criticism of Bayesian hierarchical models. Especially, avoiding double use of data by a data-splitting approach was a main concern. Such tools can be applied at each node of the model, with a view to diagnosing problems of model fit at any point in the model structure. As O'Hagan, we investigated a Gaussian model of one-way analysis of variance. Through extensive Markov chain Monte Carlo simulations it was shown that our method detects model misspecification about as well as the one of O'Hagan, when this is properly calibrated, while retaining the desired false warning probability for data generated from the assumed model. In the present paper, we suggest some new measures of conflict based on tail probabilities of the so-called integrated posterior distributions introduced in our recent paper. These new measures are equivalent to the measure applied in the latter paper in simple Gaussian models, but seem more appropriately adjusted to deviations from normality and to conflicts not concerning location parameters. A general linear normal model with known covariance matrices is considered in detail.     

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.     

A hierarchical model for space–time surveillance data on meningococcal disease incidence

Summary. We describe a model-based approach to analyse space–time surveillance data on meningococcal disease. Such data typically comprise a number of time series of disease counts, each representing a specific geographical area. We propose a hierarchical formulation, where latent parameters capture temporal, seasonal and spatial trends in disease incidence. We then add—for each area—a hidden Markov model to describe potential additional (autoregressive) effects of the number of cases at the previous time point. Different specifications for the functional form of this autoregressive term are compared which involve the number of cases in the same or in neighbouring areas. The two states of the Markov chain can be interpreted as representing an 'endemic' and a 'hyperendemic' state. The methodology is applied to a data set of monthly counts of the incidence of meningococcal disease in the 94 départements of France from 1985 to 1997. Inference is carried out by using Markov chain Monte Carlo simulation techniques in a fully Bayesian framework. We emphasize that a central feature of our model is the possibility of calculating—for each region and each time point—the posterior probability of being in a hyperendemic state, adjusted for global spatial and temporal trends, which we believe is of particular public health interest.