J. F. Lawless

Tests for Detecting Overdispersion in Poisson Regression Models

Abstract Poisson regression models are widely used in analyzing count data. This article develops tests for detecting extra-Poisson variation in such situations. The tests can be obtained as score tests against arbitrary mixed Poisson alternatives and are generalizations of tests of Fisher (1950) and Collings and Margolin (1985). Accurate approximations for computing significance levels are given, and the power of the tests against negative binomial alternatives is compared with those of the Pearson and deviance statistics. One way to test for extra-Poisson variation is to fit models that parametrically incorporate and then test for the absence of such variation within the models; for example, negative binomial models can be used in this way (Cameron and Trivedi 1986; Lawless 1987a). The tests in this article require only the Poisson model to be fitted. Two test statistics are developed that are motivated partly by a desire to have good distributional approximations for computing significance levels. Simu...

Inference Based on Retrospective Ascertainment: An Analysis of the Data on Transfusion-Related AIDS

Abstract In some epidemiologic studies, identification of individuals for study is dependent on the occurrence of some event. Once an individual is identified, the time of a previous event, termed an initiating event, is determined retrospectively. This article considers problems of estimation when initiating events occur as a nonhomogeneous Poisson process, and the time s from the initiating event to the final event has pdf f(s) independent of the time of the initiating event. A simple form for the likelihood function is obtained and methods of parametric and nonparametric estimation are developed and considered. In particular, the model is related to a Poisson process in the plane, and for the parametric case simple algorithms are developed for parameter estimation. Regression models are also considered as well as various generalizations of the basic problem. Parallel to the theoretical development, data on patients diagnosed with acquired immune deficiency syndrome (AIDS) are considered and a detailed ...

Methods for the analysis and prediction of warranty claims

This article discusses methods whereby reports of warranty claims can be used to estimate the expected number of warranty claims per unit in service as a function of the time in service. These methods provide estimates that are adjusted for delays or lags corresponding to the time from the claim until it is entered into the data base used for analysis. Forecasts of the number and cost of claims on the population of all units in service are also developed, along with standard errors for these forecasts. The methods are based on a log-linear Poisson model for numbers of warranty claims. Both the case of a known distribution of reporting lag and simultaneous estimation of that distribution are considered. The use of residuals for model checking, extensions to allow for extra-Poisson variation, and the estimation of warranty costs are also considered.

Inference in the Generalized Gamma and Log Gamma Distributions

This paper is concerned with the problem of obtaining confidence intervals or tests of significance for the parameters or other characteristics of the generalized gamma distribution. Procedures are given whereby confidence intervals for the parameters, quantiles or the reliability (survivor) function of the distribution can be obtained, when the generalized gamma index parameter is known. Their application in studying robustness and model-dependence in lifetime distributions is also discussed.

Failure Inference From a Marker Process Based on a Bivariate Wiener Model

Many models have been proposed that relate failure times and stochastic time-varying covariates. In some of these models, failure occurs when a particular observable marker crosses a threshold level. We are interested in the more difficult, and often more realistic, situation where failure is not related deterministically to an observable marker. In this case, joint models for marker evolution and failure tend to lead to complicated calculations for characteristics such as the marginal distribution of failure time or the joint distribution of failure time and marker value at failure. This paper presents a model based on a bivariate Wiener process in which one component represents the marker and the second, which is latent (unobservable), determines the failure time. In particular, failure occurs when the latent component crosses a threshold level. The model yields reasonably simple expressions for the characteristics mentioned above and is easy to fit to commonly occurring data that involve the marker value at the censoring time for surviving cases and the marker value and failure time for failing cases. Parametric and predictive inference are discussed, as well as model checking. An extension of the model permits the construction of a composite marker from several candidate markers that may be available. The methodology is demonstrated by a simulated example and a case application.

Methods for the estimation of failure distributions and rates from automobile warranty data

We consider the occurrence of warranty claims for automobiles when both age and mileage accumulation may affect failure. The presence of both age and mileage limits on warranties creates interesting problems for the analysis of failures. We propose a family of models that relates failure to time and mileage accumulation. Methods for fitting the models based on warranty data and supplementary information about mileage accumulation are presented and illustrated on some real data. The general problem of modelling failures in equipment when both time and usage are factors is discussed.

A point-process model incorporating renewals and time trends, with application to repairable systems

We discuss models for recurrent events that incorporate both time trends and effects of past events, such as renewal-type behavior. Inference procedures, including tests for trend, are developed and illustrated on repairable-systems failure data. Simulations are used to examine the accuracy of large-sample approximations used for tests or interval estimation.

Estimation of reliability in field-performance studies

Likelihood-based methods are developed for the analysis of field-performance studies with particular attention centered on the estimation of regression coefficients in parametric models. Failure-record data are those in which the time to failure and the regressor variables are observed only for those items that fail in some prespecified follow-up or warranty period (0, T]. It is noted that for satisfactory inference about baseline failure rates or regression effects it is usually necessary to supplement the failure-record data either by incorporating specific prior information about x or by taking a supplementary sample of items that survive T ○. General methods are outlined and specific formulas for various likelihood-based methods are obtained when the failure-time model is exponential or Weibull. In these models the methods are compared with respect to asymptotic efftciency of estimation. Several extensions to more complicated sampling plans are considered.

Confidence Interval Estimation for the Weibull and Extreme Value Distributions

This paper reviews methods of constructing confidence intervals for parameters or other characteristics of the Weibull or extreme value distribution. The conditional method of obtaining confidence intervals is stressed, with emphasis on the flexibility of the method, and on the computations which are necessary to use it.

Construction of Tolerance Bounds for the Extreme-Value and Weibull Distributions

A method of finding confidence borunds on the percentiles of the two-parameter Weibull or extreme-value distributions is presented. The procedures, which arise from considering the distribution of parameter estimates given the observed value of an ancillary statistic, do not require the constuction of any tables, and are applicable whether the data are complete or Type II censored. The procedures also allow an investigation to be made into the adequacy of approximate confidence bounds based on an F approximation presented in Mann, Schafer and Singptuwalla [13], and those based on large sample theory for the maximum likelihood estimates.