Testing the proportional odds assumption in multiply imputed ordinal longitudinal data

A popular choice when analyzing ordinal data is to consider the cumulative proportional odds model to relate the marginal probabilities of the ordinal outcome to a set of covariates. However, application of this model relies on the condition of identical cumulative odds ratios across the cut-offs of the ordinal outcome; the well-known proportional odds assumption. This paper focuses on the assessment of this assumption while accounting for repeated and missing data. In this respect, we develop a statistical method built on multiple imputation (MI) based on generalized estimating equations that allows to test the proportionality assumption under the missing at random setting. The performance of the proposed method is evaluated for two MI algorithms for incomplete longitudinal ordinal data. The impact of both MI methods is compared with respect to the type I error rate and the power for situations covering various numbers of categories of the ordinal outcome, sample sizes, rates of missingness, well-balanced and skewed data. The comparison of both MI methods with the complete-case analysis is also provided. We illustrate the use of the proposed methods on a quality of life data from a cancer clinical trial.     

Doubly periodic non-homogeneous Poisson models for hurricane data

Non-homogeneous Poisson processes with periodic claim intensity rate have been proposed as claim counts in risk theory. Here a doubly periodic Poisson model with short- and long-term trends is studied. Beta-type intensity functions are presented as illustrations. The likelihood function and the maximum likelihood estimates of the model parameters are derived.Doubly periodic Poisson models are appropriate when the seasonality does not repeat exactly the same short-term pattern every year, but has a peak intensity that varies over a longer period. This reflects periodic environments like those forming hurricanes, in alternating El Niño/La Niña years. An application of the model to the data set of Atlantic hurricanes affecting the United States (1899–2000) is discussed in detail.     

One-sided EWMA control charts for monitoring Poisson processes with varying sample sizes

ABSTRACT

Recently considerable research has been devoted to monitoring increases of incidence rate of adverse rare events. This paper extends some one-sided upper exponentially weighted moving average (EWMA) control charts from monitoring normal means to monitoring Poisson rate when sample sizes are varying over time. The approximated average run length bounds are derived for these EWMA-type charts and compared with the EWMA chart previously studied. Extensive simulations have been conducted to compare the performance of these EWMA-type charts. An illustrative example is given.     

Hierarchical Bayesian modeling of marked non-homogeneous Poisson processes with finite mixtures and inclusion of covariate information

We investigate marked non-homogeneous Poisson processes using finite mixtures of bivariate normal components to model the spatial intensity function. We employ a Bayesian hierarchical framework for estimation of the parameters in the model, and propose an approach for including covariate information in this context. The methodology is exemplified through an application involving modeling of and inference for tornado occurrences.     

Data-driven smooth tests of the proportional hazards assumption

A new test of the proportional hazards assumption in the Cox model is proposed. The idea is based on Neyman’s smooth tests. The Cox model with proportional hazards (i.e. time-constant covariate effects) is embedded in a model with a smoothly time-varying covariate effect that is expressed as a combination of some basis functions (e.g., Legendre polynomials, cosines). Then the smooth test is the score test for significance of these artificial covariates. Furthermore, we apply a modification of Schwarz’s selection rule to choosing the dimension of the smooth model (the number of the basis functions). The score test is then used in the selected model. In a simulation study, we compare the proposed tests with standard tests based on the score process.     

On tests for spatial trend in a non-homogeneous Poisson process

Certain spatial point patterns may be associated with a single fixed point. Some simple tests for spatial trend in radial and angular distribution are recommended when no interaction between the radial and angular distribution is found. Interaction tests for more complex rate finctions are discussed and a practical example of the analysis applied to the mortality experience around a steel plant is provided.     

A Bayesian approach for non-homogeneous gamma degradation processes

A Bayesian approach based on the Markov Chain Monte Carlo technique is proposed for the non-homogeneous gamma process with power-law shape function. Vague and informative priors, formalized on some quantities having a “physical” meaning, are provided. Point and interval estimation of process parameters and some functions thereof are developed, as well as prediction on some observable quantities that are useful in defining the maintenance strategy is proposed. Some useful approximations are derived for the conditional and unconditional mean and median of the residual life to reduce computational time. Finally, the proposed approach is applied to a real dataset.     

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.     

Parameter estimation for fractional Poisson processes

The paper proposes a formal estimation procedure for parameters of the fractional Poisson process (fPp). Such procedures are needed to make the fPp model usable in applied situations. The basic idea of fPp, motivated by experimental data with long memory is to make the standard Poisson model more flexible by permitting non-exponential, heavy-tailed distributions of interarrival times and different scaling properties. We establish the asymptotic normality of our estimators for the two parameters appearing in our fPp model. This fact permits construction of the corresponding confidence intervals. The properties of the estimators are then tested using simulated data.     

On the superposition of overlapping Poisson processes and nonparametric estimation of their intensity function

It is shown that, when measuring time in the Total Time on Test scale, the superposition of overlapping realizations of a nonhomogeneous Poisson process is also a Poisson process and is sufficient for inferential purposes. Hence, many nonparametric procedures which are available when only one realization is observed can be easily extended for the overlapping realizations setup. These include, for instance, the constrained maximum likelihood estimator of a monotonic intensity and bootstrap confidence bands based on Kernel estimates of the intensity. The kernel estimate proposed here performs the smoothing in the Total Time on Test scale and it is shown to behave approximately as a usual kernel estimate but with a variable bandwidth which is inversely proportional to the number of realizations at-risk. Likewise, bootstrap samples can be obtained from the single realization of the superimposed process. The methods are illustrated using a real data set consisting of the failure histories of 40 electrical power transformers.     

Comparisons of tests of distributional assumption in Poisson regression model

Count data consists of discrete non-negative integer values. Poisson regression model is one of the most popular model used to model count data. This model assumes that response variable has Poisson distribution. The purpose of this article is to assess distributional assumption of this model by using some goodness of fit tests. These tests are compared in respect to type I error and power rates of tests with different samples, parameters and sample sizes. Simulation study suggests that the most powerful tests are generally Dean–Lawless and Cameron–Trivedi score tests.     

Testing the proportional odds model for interval-censored data

This paper discusses the goodness-of-fit test for the proportional odds model for K-sample interval-censored failure time data, which frequently occur in, for example, periodic follow-up survival studies. The proportional odds model has a feature that allows the ratio of two hazard functions to be monotonic and converge to one and provides an important tool for the modeling of survival data. To test the model, a procedure is proposed, which is a generalization of the method given in Dauxois and Kirmani [Dauxois JY, Kirmani SNUA (2003) Biometrika 90:913–922]. The asymptotic distribution of the procedure is established and its properties are evaluated by simulation studies     

Poisson bounds for moments of shot noise processes

Starting from general moment formulas for shot noise processes (v(t)), for which a simple proof is given by means of the CAMPBELL theorem concerning marked point pro¬cesses, and using certain monotonicity property, bounds are obtained for E[v(t)]² resp. for the variance Var v(t) under the additional assumption that the intershot times are inde¬pendent random variables with distribution having the property 1SFBXJE (NWTJE)     

Win-probabilities for comparing two Poisson variables

ABSTRACT

This article considers the problem of choosing between two possible treatments which are each modeled with a Poisson distribution. Win-probabilities are defined as the probabilities that a single potential future observation from one of the treatments will be better than, or at least as good as, a potential future observation from the other treatment. Using historical data from the two treatments, it is shown how estimates and confidence intervals can be constructed for the win-probabilities. Extensions to situations with three or more treatments are also discussed. Some examples and illustrations are provided, and the relationship between this methodology and standard inference procedures on the Poisson parameters is discussed.     

On minimum Lp-distance estimation for inhomogeneous Poisson processes

ABSTRACT

We consider the problem of parameter estimation by the observations of the inhomogeneous Poisson processes. We suppose that the intensity function of these processes is a smooth function of the unknown parameter and as a method of estimation we take the minimum distance approach. We are interested by the behavior of estimators in non Hilbertian situation and we define the minimum distance estimation (MDE) with the help of the L^p metrics. We show that (under regularity conditions) the MDE is consistent and we describe its limit distribution.     

Accelerated life testing models for nonhomogeneous Poisson processes

In this paper an axiomatic approach is used to construct accelerated life testing (ALT) models for Nonhomogeneous Poisson Processes (NHPPs). First, the models of random lifetime variables and Nonhomogeneous Poisson Processes used for modeling non-repairable and repairable systems are compared. Then, an axiomatic approach for the construction of ALT models for NHPPs is given. Particular models are considered that can be constructed by this method.     

Testing the homogeneous marginal utility of income assumption

We develop a test for the hypothesis that every agent from a population of heterogeneous consumers has the same marginal utility of income function. This homogeneous marginal utility of income (HMUI) assumption is often (implicitly) used in applied demand studies because it has nice aggregation properties and facilitates welfare analysis. If the HMUI assumption holds, we can also identify the common marginal utility of income function. We apply our results using a U.S. cross sectional dataset on food consumption.     

Asymptotic deficiency of score test for inhomogeneous Poisson processes

In this work, based on a realization of an inhomogeneous Poisson process whose intensity function depends on an unknown real parameter, we test a simple null hypothesis against a sequence of close (contiguous) one-sided alternatives. The main object is to obtain the asymptotic deficiency of the score test with respect to the Neyman–Pearson test.     

A non-parametric estimator for the doubly periodic Poisson intensity function

In a series of papers, J. Garrido and Y. Lu have proposed and investigated a doubly periodic Poisson model, and then applied it to analyze hurricane data. The authors have suggested several parametric models for the underlying intensity function. In the present paper we construct and analyze a non-parametric estimator for the doubly periodic intensity function. Assuming that only a single realization of the process is available in a bounded window, we show that the estimator is consistent and asymptotically normal when the window expands indefinitely. In addition, we calculate the asymptotic bias and variance of the estimator, and in this way gain helpful information for optimizing the performance of the estimator.