A model for markers and latent health status

We extend the bivariate Wiener process considered by Whitmore and co-workers and model the joint process of a marker and health status. The health status process is assumed to be latent or unobservable. The time to reach the primary end point or failure (death, onset of disease, etc.) is the time when the latent health status process first crosses a failure threshold level. Inferences for the model are based on two kinds of data: censored survival data and marker measurements. Covariates, such as treatment variables, risk factors and base-line conditions, are related to the model parameters through generalized linear regression functions. The model offers a much richer potential for the study of treatment efficacy than do conventional models. Treatment effects can be assessed in terms of their influence on both the failure threshold and the health status process parameters. We derive an explicit formula for the prediction of residual failure times given the current marker level. Also we discuss model validation. This model does not require the proportional hazards assumption and hence can be widely used. To demonstrate the usefulness of the model, we apply the methods in analysing data from the protocol 116a of the AIDS Clinical Trials Group.     

A Latent Process Model for Joint Modeling of Events and Marker

The paper formulates joint modeling of a counting process and a sequence of longitudinal measurements, governed by a common latent stochastic process. The latent process is modeled as a function of explanatory variables and a Brownian motion process. The conditional likelihood given values of the latent process at the measurement times, has been drawn using Brownian bridge properties; then integrating over all possible values of the latent process at the measurement times leads to the desired joint likelihood. An estimation procedure using joint likelihood and a numerical optimization is described. The method is applied to the study of cognitive decline and Alzheimer's disease.     

A latent Markov model for detecting patterns of criminal activity

Summary.  The paper investigates the problem of determining patterns of criminal behaviour from official criminal histories, concentrating on the variety and type of offending convictions. The analysis is carried out on the basis of a multivariate latent Markov model which allows for discrete covariates affecting the initial and the transition probabilities of the latent process. We also show some simplifications which reduce the number of parameters substantially; we include a Rasch-like parameterization of the conditional distribution of the response variables given the latent process and a constraint of partial homogeneity of the latent Markov chain. For the maximum likelihood estimation of the model we outline an EM algorithm based on recursions known in the hidden Markov literature, which make the estimation feasible also when the number of time occasions is large. Through this model, we analyse the conviction histories of a cohort of offenders who were born in England and Wales in 1953. The final model identifies five latent classes and specifies common transition probabilities for males and females between 5-year age periods, but with different initial probabilities.     

A latent variable regression model for asymmetric bivariate ordered categorical data

In many areas of medical research, especially in studies that involve paired organs, a bivariate ordered categorical response should be analyzed. Using a bivariate continuous distribution as the latent variable is an interesting strategy for analyzing these data sets. In this context, the bivariate standard normal distribution, which leads to the bivariate cumulative probit regression model, is the most common choice. In this paper, we introduce another latent variable regression model for modeling bivariate ordered categorical responses. This model may be an appropriate alternative for the bivariate cumulative probit regression model, when postulating a symmetric form for marginal or joint distribution of response data does not appear to be a valid assumption. We also develop the necessary numerical procedure to obtain the maximum likelihood estimates of the model parameters. To illustrate the proposed model, we analyze data from an epidemiologic study to identify some of the most important risk indicators of periodontal disease among students 15-19 years in Tehran, Iran.     

A latent Gaussian Markov random-field model for spatiotemporal rainfall disaggregation

Summary. Rainfall data are often collected at coarser spatial scales than required for input into hydrology and agricultural models. We therefore describe a spatiotemporal model which allows multiple imputation of rainfall at fine spatial resolutions, with a realistic dependence structure in both space and time and with the total rainfall at the coarse scale consistent with that observed. The method involves the transformation of the fine scale rainfall to a thresholded Gaussian process which we model as a Gaussian Markov random field. Gibbs sampling is then used to generate realizations of rainfall efficiently at the fine scale. Results compare favourably with previous, less elegant methods.     

A shared component model for detecting joint and selective clustering of two diseases

The study of spatial variations in disease rates is a common epidemiological approach used to describe the geographical clustering of diseases and to generate hypotheses about the possible 'causes' which could explain apparent differences in risk. Recent statistical and computational developments have led to the use of realistically complex models to account for overdispersion and spatial correlation. However, these developments have focused almost exclusively on spatial modelling of a single disease. Many diseases share common risk factors (smoking being an obvious example) and, if similar patterns of geographical variation of related diseases can be identified, this may provide more convincing evidence of real clustering in the underlying risk surface. We propose a shared component model for the joint spatial analysis of two diseases. The key idea is to separate the underlying risk surface for each disease into a shared and a disease-specific component. The various components of this formulation are modelled simultaneously by using spatial cluster models implemented via reversible jump Markov chain Monte Carlo methods. We illustrate the methodology through an analysis of oral and oesophageal cancer mortality in the 544 districts of Germany, 1986–1990.     

A class of sequential tests for two‐sample composite hypotheses

The authors propose a class of statistics based on Rao's score for the sequential testing of composite hypotheses comparing two treatments (populations). Asymptotic approximations of the statistics lead them to propose sequential tests and to derive their monitoring boundaries. As special cases, they construct sequential versions of the two‐sample t‐test for normal populations and two‐sample z‐score tests for binomial populations. The proposed algorithms are simple and easy to compute, as no numerical integration is required. Furthermore, the user can analyze the data at any time regardless of how many inspections have been made. Monte Carlo simulations allow the authors to compare the power and the average stopping time (also known as average sample number) of the proposed tests to those of nonsequential and group sequential tests. A two‐armed comparative clinical trial in patients with adult leukemia allows them to illustrate the efficiency of their methods in the case of binary responses.     

On hypothesis tests in misspecified change-point problems for a Poisson process

Consider an inhomogeneous Poisson process X on [0, T] whose unk-nown intensity function “switches” from a lower function g_* to an upper function h_* at some unknown point ?_* that has to be identified. We consider two known continuous functions g and h such that g_*(t) ? g(t) < h(t) ? h_*(t) for 0 ? t ? T. We describe the behavior of the generalized likelihood ratio and Wald’s tests constructed on the basis of a misspecified model in the asymptotics of large samples. The power functions are studied under local alternatives and compared numerically with help of simulations. We also show the following robustness result: the Type I error rate is preserved even though a misspecified model is used to construct tests.     

Multivariate autoregressive extreme value process and its application for modeling the time series properties of the extreme daily asset prices

Abstract

In this article we suggest a new multivariate autoregressive process for modeling time-dependent extreme value distributed observations. The idea behind the approach is to transform the original observations to latent variables that are univariate normally distributed. Then the vector autoregressive DCC model is fitted to the multivariate latent process. The distributional properties of the suggested model are extensively studied. The process parameters are estimated by applying a two-stage estimation procedure. We derive a prediction interval for future values of the suggested process. The results are applied in an empirically study by modeling the behavior of extreme daily stock prices.     

Familles of confidence bands for the survival function under the general random censorship model and the Koziol-Green model

Familles of asymptotic 100(1 – α)% level confidence bands for the survival function under the general random right-censorship (GRC) model and the proportional-hazards model of random right-censorship, also known as the Koziol-Green (KG) model, are developed. The family of bands under the GRC model is based on the well-known product-limit estimator (PLE), and this family is rich in that it contains as special cases the bands of Hall and Wellner (1980) and Gillespie and Fisher (1979), and more generally, the GF-type and HW-type bands of Csörg? and Horváth (1986), as well as new bands not previously studied. The familles of bands under the KG model are based on the maximum-likelihood estimator of F under this particular model. We compare the PLE-based bands and the MLE-based bands under the KG model. This enables us to study the loss in efficiency of the former bands when used in a setting where they are not optimal. The notion of asymptotic relative width efficiency (ARWE), defined to be the limiting ratio of the sample sizes needed by the bands to achieve the same asymptotic widths, is employed to compare two bands. Through this efficiency measure it is shown that if the censoring parameter β is known, then the PLE-based bands are highly inefficient relative to the MLE-based bands when β is large. When β is not known, the MLE-based bands are asymptotically conservative. Despite their conservatism, they still dominate the PLE-based bands when β is not too small or equivalently when the degree of censoring is not too light. We also compare the various PLE-based bands under the GRC model. The resulting information is valuable for evaluating competing PLE-based bands. We illustrate the confidence bands by utilizing the well-known Channing House data.     

A joint model for longitudinal data profiles and associated event risks with application to a depression study

Summary.  In many longitudinal studies, a subject's response profile is closely associated with his or her risk of experiencing a related event. Examples of such event risks include recurrence of disease, relapse, drop-out and non-compliance. When evaluating the effect of a treatment, it is sometimes of interest to consider the joint process consisting of both the response and the risk of an associated event. Motivated by a prevention of depression study among patients with malignant melanoma, we examine a joint model that incorporates the risk of discontinuation into the analysis of serial depression measures. We present a maximum likelihood estimator for the mean response and event risk vectors. We test hypotheses about functions of mean depression and withdrawal risk profiles from our joint model, predict depression from updated patient histories, characterize associations between components of the joint process and estimate the probability that a patient's depression and risk of withdrawal exceed specified levels. We illustrate the application of our joint model by using the depression data.     

Estimation in a model for a semi-Markov process with covariates under right-censoring

A semi-Markov model with covariates is proposed for a multi-state process with a finite number of states such that the transition probabilities between the states and the distribution functions of the duration times between the occurrence of two states depend on a discrete covariate. The hazard rates for the time elapsed between two successive states depend on the covariate through a proportional hazards model involving a set of regression parameters, while the transition probabilities depend on the covariate in an unspecified way. We propose estimators for these parameters and for the cumulative hazard functions of the sojourn times. A difficulty comes from the fact that when a sojourn time in a state is right-censored, the next state is unknown. We prove that our estimators are consistent and asymptotically Gaussian under the model constraints.     

A functional marked point process model for lupus data

Modelling for marked point processes is an important problem, but has received remarkably little attention in the statistical literature. The authors developed a marked point process model that incorporates the use of functional data analysis in a joint estimation of the frequency function of the point process and the intensity of the mark, with application to data from 22 lupus patients consisting of times of flares in symptom severity combined with a quantitative assessment of the severity. The data indicate that a rapid decrease in drug dose is significantly associated with a decrease in flare frequency. Experiments with simulated data designed to model the actual data further support this conclusion. The Canadian Journal of Statistics 40: 517–529; 2012 © 2012 Statistical Society of Canada     

A Bayesian latent spatial model for mapping the cortical signature of progression to Alzheimer's disease

Prior studies have shown that atrophy in vulnerable cortical regions is associated with an increased risk of progression to clinical dementia. In this work, we utilize the longitudinal structural magnetic resonance imaging (MRI) data from the Alzheimer's Disease Neuroimaging Initiative (ADNI) to investigate the relationship between the temporally changing spatial topography of cortical thickness and conversion from mild cognitive impairment to Alzheimer's disease (AD). We develop a novel Bayesian latent spatial model that employs the spatial information underlying the thickness effects across the cortical surface. The proposed method facilitates the development of imaging markers by reliably quantifying and mapping the regional vulnerability to AD progression across the cortical surface. Simulation results showed substantial gains in statistical power and estimation performance by accounting for the spatial structure of the association. Using MRI data from ADNI, we examined the topographic patterns of anatomic regions where cortical thinning is associated with an increased risk of developing AD.     

A Poisson geometric process approach for predicting drop-out and committed first-time blood donors

A Poisson geometric process (PGP) model is proposed to study individual blood donation patterns for a blood donor retention program. Extended from the geometric process (GP) model of Lam [16 Y. Lam, Geometric process and replacement problem, Acta Math. Appl. Sin. 4 (1988), pp. 366–377. doi: 10.1007/BF02007241[Crossref] , [Google Scholar]], the PGP model captures the rather pronounced trend patterns across clusters of donors via the ratio parameters in a mixture setting. Within the state-space modeling framework, it allows for overdispersion by equating the mean of the Poisson data distribution to a latent GP. Alternatively, by simply setting, the mean of the Poisson distribution to be the mean of a GP, it has equidispersion. With the group-specific mean and ratio functions, the mixture PGP model facilitates classification of donors into committed, drop-out and one-time groups. Based on only two years of observations, the PGP model nicely predicts donors’ future donations to foster timely recruitment decision. The model is implemented using a Bayesian approach via the user-friendly software WinBUGS.     

Some estimators and tests for accelerated hazards model using weighted cumulative hazard difference

For a censored two-sample problem, Chen and Wang [Y.Q. Chen and M.-C. Wang, Analysis of accelerated hazards models, J. Am. Statist. Assoc. 95 (2000), pp. 608–618] introduced the accelerated hazards model. The scale-change parameter in this model characterizes the association of two groups. However, its estimator involves the unknown density in the asymptotic variance. Thus, to make an inference on the parameter, numerically intensive methods are needed. The goal of this article is to propose a simple estimation method in which estimators are asymptotically normal with a density-free asymptotic variance. Some lack-of-fit tests are also obtained from this. These tests are related to Gill–Schumacher type tests [R.D. Gill and M. Schumacher, A simple test of the proportional hazards assumption, Biometrika 74 (1987), pp. 289–300] in which the estimating functions are evaluated at two different weight functions yielding two estimators that are close to each other. Numerical studies show that for some weight functions, the estimators and tests perform well. The proposed procedures are illustrated in two applications.     

Exact and Monte Carlo calculations of integrated likelihoods for the latent class model

The latent class model or multivariate multinomial mixture is a powerful approach for clustering categorical data. It uses a conditional independence assumption given the latent class to which a statistical unit is belonging. In this paper, we exploit the fact that a fully Bayesian analysis with Jeffreys non-informative prior distributions does not involve technical difficulty to propose an exact expression of the integrated complete-data likelihood, which is known as being a meaningful model selection criterion in a clustering perspective. Similarly, a Monte Carlo approximation of the integrated observed-data likelihood can be obtained in two steps: an exact integration over the parameters is followed by an approximation of the sum over all possible partitions through an importance sampling strategy. Then, the exact and the approximate criteria experimentally compete, respectively, with their standard asymptotic BIC approximations for choosing the number of mixture components. Numerical experiments on simulated data and a biological example highlight that asymptotic criteria are usually dramatically more conservative than the non-asymptotic presented criteria, not only for moderate sample sizes as expected but also for quite large sample sizes. This research highlights that asymptotic standard criteria could often fail to select some interesting structures present in the data.     

Regions of alternatives with high and low power for goodness-of-fit tests

In the present paper we find finite dimensional spaces W of alternatives with high power for a given class of tests and non-parametric alternatives. On the orthogonal complement of W the power function is flat. These methods can be used to reduce the dimension of interesting alternatives. We sketch a device how to calculate (approximately) an alternative with maximum power of a fixed test on a given ball of certain non-parametric alternatives.

The calculations are done within different asymptotic models specified by signal detection tests. Specific tests are Kolmogorov–Smirnov type tests, integral tests (like the Anderson and Darling test) and Rényi tests for hazard based models. The statistical meaning and interpretation of the spaces of alternatives with high power is discussed. These alternatives belong to least favorable directions of a class of statistical functionals which are linear combinations of quantile functions. For various cases their meaning is explained for parametric submodels, in particular for location alternatives.     

A class of tests of proportional hazards assumption for left-truncated and right-censored data

In this paper, we proposed a class of tests of proportional hazards assumption for left-truncated and right-censored data based on a pair of estimators of the hazard ratio constant. Using counting process and martingale theory, the asymptotically normal distribution of the test statistic is derived and a family of consistent estimators of variance are also provided. Extensive simulation studies were conducted to evaluate the performance of the proposed test statistics under finite sample situations. Two real data sets are analyzed to illustrate our method.     

An integrated model based on statistical process control and maintenance for two-stage dependent processes

The close relationship between quality and maintenance of manufacturing systems has contributed to the development of integrated models which use the concept of statistical process control (SPC) and maintenance. This article demonstrates the integration of the Shewhart individual-residual (Z_X ? Z_e) joint control chart and maintenance for two-stage dependent processes by jointly optimizing their policies to minimize the expected total costs associated with quality, maintenance and inspection. To evaluate the effectiveness of the proposed model, two stand-alone models—a maintenance model and an SPC model—are proposed. Then a numerical example is given to illustrate the application of the proposed integrated model. The results show that the integrated model outperforms the two stand-alone models with regard to the expected cost per unit time. Finally, a sensitivity analysis is conducted to develop insights into time parameters and cost parameters that influence the integration efforts.