A count data model for gamma waiting times

A new distribution for non-negative integers, or counts, is developed. It is based on the assumption that the waiting times separating consecutive events are independently and identically gamma distributed. Thus, the structural process generating the counts may exhibit duration dependence. In this framework, the frequently observed phenomenon of overdispersion, that is a variance that exceeds the mean, is caused by a decreasing hazard function of the gamma distributed waiting times, while an increasing hazard leads to underdispersion at the level of the counts. A Monte Carlo simulation and an application to fertility data illustrate the performance of the new distribution.     

Using panel methods to model waiting times for National Health Service surgery

Summary. Long waiting times for non-emergency (elective) procedures are a central feature of the UK's National Health Service, with about 1 million people waiting for surgery at any one time. This paper develops empirical models of the demand for and supply of elective surgery which simultaneously determine waiting times. The models are tested by using a panel of annual data for 5499 small areas from 1991 to 1998. Supply and demand functions are estimated for all specialties combined and for seven individual specialties, using panel data methods that incorporate simultaneously determined variables. The elasticity of demand with respect to waiting time varies between specialties but is always quite small. The results are discussed in the light of UK Government policy initiatives designed to reduce waiting times substantially. The analysis suggests that these initiatives will not stimulate demand markedly and therefore stand a good chance of succeeding provided that adequate additional resources are made available.     

Estimating avian dispersal distances from data on ringed birds

Data from birds ringed as chicks and recaptured during subsequent breeding seasons provide information on avian natal dispersal distances. However, national patterns of ring reports are influenced by recapture rates as well as by dispersal rates. While an extensive methodology has been developed to study survival rates using models that correct for recapture rates, the same is not true for dispersal. Here, we present such a method, showing how corrections for spatial heterogeneity in recapture rate can be built into estimates of dispersal rates if detailed atlas data and ringing totals can be combined with extensive data on birds ringed as chicks and recaptured as breeding adults. We show how the method can be implemented in the software package SURVIV (White, 1992).     

Estimating correlation from categorized bivariate normal data

The polychoric correlation and an approximate maximum likelihood estimator of correlation are compared for count data that are assumed to be derived from an underlying bivariate normal distribution. A related chi squared test for bivariate normality is also examined.     

Compound patterns and generating functions: From basic waiting times to counts of occurrence

We derive an explicit, closed form expression for the double generating function of the corresponding counts of occurrence, within a finite time horizon, of the single patterns contained in a compound pattern. The expression is in terms of a basic single, and a basic joint, generating functions for which exact solutions exist in the literature. The single generating function is associated with the basic waiting time for the first occurrence of the compound pattern. The joint generating function is that for the waiting time to reach a given single pattern and the associated counts of occurrence, within that waiting time, of the single patterns contained in the compound pattern. The literature on patterns is huge. Also, there are results that establish links between generating functions for counts of occurrence of the single patterns contained in a compound pattern with generating functions of some more complex waiting times associated with that compound pattern. The latter waiting times are known in the literature with names such as sooner, or later waiting times, or generalisations of such. On the other hand, our result fills a gap in the literature by providing a neat link connecting the generating functions of the basic quantities associated with occurrence of compound patterns.     

Estimating variation from data with varying element sizes

In some applications (repeated surveys, process control, monitoring use of resources etc), observations are often based on elements of varying size. This paper describes two methods of estimating the underlying variation. One method measures deviations from an overall estimate of a location parameter, while the other method is based on successive differences. Both are 'one-pass' estimates and are thus programmable without storing all the data.     

Estimating bird species richness from capture and count data

We used capture-recapture methods to estimate bird species richness from mist-net and point-count data from a study area in Campeche, Mexico. We estimated species richness separately for each survey technique for two habitats, forest and pasture, in six sampling periods. We then estimated richness based on species' detections by either technique, and estimated the proportion of species detected by each technique that are not part of the population sampled by the other technique. No consistent differences existed between richness estimates from count data and from capture data in the two habitats. In some sampling periods, over 50% of the richness estimate from one survey technique may be species that are not sampled by the other technique, suggesting that one technique may not be adequate to estimate total species richness and that comparing estimates from areas sampled by different techniques may not be valid.     

Estimating parametric semi-Markov models from panel data using phase-type approximations

Inference for semi-Markov models under panel data presents considerable computational difficulties. In general the likelihood is intractable, but a tractable likelihood with the form of a hidden Markov model can be obtained if the sojourn times in each of the states are assumed to have phase-type distributions. However, using phase-type distributions directly may be undesirable as they require estimation of parameters which may be poorly identified. In this article, an approach to fitting semi-Markov models with standard parametric sojourn distributions is developed. The method involves establishing a family of Coxian phase-type distribution approximations to the parametric distribution and merging approximations for different states to obtain an approximate semi-Markov process with a tractable likelihood. Approximations are developed for Weibull and Gamma distributions and demonstrated on data relating to post-lung-transplantation patients.     

Discrete distributions based on inter arrival times with application to football data

Motivated by McShane, Adrian, Bradlow and Fader [Journal of Business and Economic Statistics, 26, 2008, 369–378], we introduce thirteen discrete distributions. We give explicit expressions for their probability mass functions. We analyze two football data sets and show that some of the proposed distributions provide better fits than the distribution due to McShane, Adrian, Bradlow, and Fader.     

Estimating spatial means with an application to remotely sensed data

The paper examines alternative estimators for the mean of a spatial process where observations are not independent. Properties of the sample mean and its standard error are contrasted with those of maximum likelihood estimators derived for three spatial models. The information loss caused by spatial dependency in the data is examined. The distribution theory for the estimators is reviewed and the paper concludes with an empirical example illustrating the properties of the estimators and the practical benefits of the maximum likelihood procedure.     

Evidential inference based on record data and inter-record times

Science looks to statistics for an objective measure of the strength of evidence in a given body of observations. In this paper, we shall use a criterion defined as a combination of probabilities of weak and strong misleading evidence to do the comparison between only record values and the same number of record values and inter-record times. Also, a simulation is presented to illustrate the results.     

Joint analysis of longitudinal data comprising repeated measures and times to events

In biomedical and public health research, both repeated measures of biomarkers Y as well as times T to key clinical events are often collected for a subject. The scientific question is how the distribution of the responses [ T , Y | X ] changes with covariates X . [ T | X ] may be the focus of the estimation where Y can be used as a surrogate for T . Alternatively, T may be the time to drop-out in a study in which [ Y | X ] is the target for estimation. Also, the focus of a study might be on the effects of covariates X on both T and Y or on some underlying latent variable which is thought to be manifested in the observable outcomes. In this paper, we present a general model for the joint analysis of [ T , Y | X ] and apply the model to estimate [ T | X ] and other related functionals by using the relevant information in both T and Y . We adopt a latent variable formulation like that of Fawcett and Thomas and use it to estimate several quantities of clinical relevance to determine the efficacy of a treatment in a clinical trial setting. We use a Markov chain Monte Carlo algorithm to estimate the model's parameters. We illustrate the methodology with an analysis of data from a clinical trial comparing risperidone with a placebo for the treatment of schizophrenia.     

Estimating catch at age from market sampling data by using a Bayesian hierarchical model

Summary.  The paper develops a Bayesian hierarchical model for estimating the catch at age of cod landed in Norway. The model includes covariate effects such as season and gear, and can also account for the within-boat correlation. The hierarchical structure allows us to account properly for the uncertainty in the estimates.     

Nonparametric estimation of waiting time distributions in a Markov model based on current status data

We construct nonparametric estimators of state waiting time distribution functions in a Markov multistate model using current status data. This is a particularly difficult problem since neither the entry nor the exit times of a given state are directly observed. These estimators are obtained, using the Markov property, from estimators of counting processes of state entry and exit times, as well as, the size of “at risk” sets of state entry and transitions out of that state. Consistency of our estimators is established. Finite-sample behavior of our estimators is studied by simulation, in which we show that our estimators based on current status data compare well with those based on complete data. We also illustrate our method using a pubertal development data set obtained from the NHANES III [1997. NHANES III Reference Manuals and Reports (CD-ROM). Analytic and Reporting Guidelines: The Third National Health and Nutrition Examination Survey (1988–94). National Center for Health Statistics, Centers for Disease Control and Prevention, Hyattsville, MD] study.     

Review and implementation of cure models based on first hitting times for Wiener processes

The development of models and methods for cure rate estimation has recently burgeoned into an important subfield of survival analysis. Much of the literature focuses on the standard mixture model. Recently, process-based models have been suggested. We focus on several models based on first passage times for Wiener processes. Whitmore and others have studied these models in a variety of contexts. Lee and Whitmore (Stat Sci 21(4):501–513, 2006) give a comprehensive review of a variety of first hitting time models and briefly discuss their potential as cure rate models. In this paper, we study the Wiener process with negative drift as a possible cure rate model but the resulting defective inverse Gaussian model is found to provide a poor fit in some cases. Several possible modifications are then suggested, which improve the defective inverse Gaussian. These modifications include: the inverse Gaussian cure rate mixture model; a mixture of two inverse Gaussian models; incorporation of heterogeneity in the drift parameter; and the addition of a second absorbing barrier to the Wiener process, representing an immunity threshold. This class of process-based models is a useful alternative to the standard model and provides an improved fit compared to the standard model when applied to many of the datasets that we have studied. Implementation of this class of models is facilitated using expectation-maximization (EM) algorithms and variants thereof, including the gradient EM algorithm. Parameter estimates for each of these EM algorithms are given and the proposed models are applied to both real and simulated data, where they perform well.     

Estimating the proportion of true null hypotheses, with application to DNA microarray data

Summary.  We consider the problem of estimating the proportion of true null hypotheses, π ₀, in a multiple-hypothesis set-up. The tests are based on observed p -values. We first review published estimators based on the estimator that was suggested by Schweder and Spjøtvoll. Then we derive new estimators based on nonparametric maximum likelihood estimation of the p -value density, restricting to decreasing and convex decreasing densities. The estimators of π ₀ are all derived under the assumption of independent test statistics. Their performance under dependence is investigated in a simulation study. We find that the estimators are relatively robust with respect to the assumption of independence and work well also for test statistics with moderate dependence.     

The IPW estimator for the joint distribution function of the gap times from recurrent event data

Lee et al. in 2016 proposed a nonparametric estimator of the joint distribution of the gap time between transplant and the first infection and the following gap times between consecutive infections. In this article, we propose an alternative estimator based on the inverse-probability weighted (IPW) approach. Asymptotic properties of the proposed estimator are established . Simulation results indicate that the IPW estimator performs as well as the estimator proposed by Lee et al. We also propose an IPW estimator for estimating the joint distribution function of the gap times between consecutive recurrent events beyond the first episode.     

Bayesian regression analysis of data with censored initiating and terminating times: applications to aids

Data with censored initiating and terminating times arises quite frequently in acquired immunodeficiency syndrome (AIDS) epidemiologic studies. Analysis of such data involves a complicated bivariate likelihood, which is difficult to deal with computationally. Bayesian analysis, op the other hand, presents added complexities that have yet to be resolved. By exploiting the simple form of a complete data likelihood and utilizing the power of a Markov Chain Monte Carlo (MCMC) algorithm, this paper presents a methodology for fitting Bayesian regression models to such data. The proposed methods extend the work of Sinha (1997), who considered non-parametric Bayesian analysis of this type of data. The methodology is illustiated with an application to a cohort of HIV infected hemophiliac patients.