Showing papers in "Biometrika in 1986"

Longitudinal data analysis using generalized linear models

Abstract: SUMMARY This paper proposes an extension of generalized linear models to the analysis of longitudinal data. We introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence. The estimating equations are derived without specifying the joint distribution of a subject's observations yet they reduce to the score equations for multivariate Gaussian outcomes. Asymptotic theory is presented for the general class of estimators. Specific cases in which we assume independence, m-dependence and exchangeable correlation structures from each subject are discussed. Efficiency of the proposed estimators in two simple situations is considered. The approach is closely related to quasi-likelih ood. Some key ironh: Estimating equation; Generalized linear model; Longitudinal data; Quasi-likelihood; Repeated measures.

An improved Bonferroni procedure for multiple tests of significance

Abstract: SUMMARY A modification of the Bonferroni procedure for testing multiple hypotheses is presented. The method, based on the ordered p-values of the individual tests, is less conservative than the classical Bonferroni procedure but is still simple to apply. A simulation study shows that the probability of a type I error of the procedure does not exceed the nominal significance level, a, for a variety of multivariate normal and multivariate gamma test statistics. For independent tests the procedure has type I error probability equal to a. The method appears particularly advantageous over the classical Bonferroni procedure when several highly-correlated test statistics are involved.

A case-cohort design for epidemiologic cohort studies and disease prevention trials

Abstract: SUMMARY Suppose that a cohort of individuals is to be followed in order to relate failure rates to preceding covariate histories. A design is proposed which involves covariate data only for cases experiencing failure and for members of a randomly selected subcohort. Odds ratio and relative risk estimation procedures are presented for such a 'case-cohort' design. A small simulation study compares case-cohort relative risk estimation procedures to full-cohort and synthetic case-control analyses. Relevance to epidemiologic cohort studies and disease prevention trials is discussed.

Multilevel mixed linear model analysis using iterative generalized least squares

Abstract: SUMMARY Models for the analysis of hierarchically structured data are discussed. An iterative generalized least squares estimation procedure is given and shown to be equivalent to maximum likelihood in the normal case. There is a discussion of applications to complex surveys, longitudinal data, and estimation in multivariate models with missing responses. An example is given using educational data.

Survival models for heterogeneous populations derived from stable distributions

Abstract: SUMMARY A new three-parameter family of distributions on the positive numbers is proposed. It includes the stable distributions on the positive numbers, the gamma, the degenerate and the inverse Gaussian distributions. The family is characterized by the Laplace transform, from which moments, convolutions, infinite divisibility, unimodality and other properties are derived. The density is complicated, but a simple saddlepoint approximation is provided. Weibull and Gompertz distributions are naturally mixed over some of the distributions. The family is natural exponential in one of the parameters. The distributions are relevant for application as frailty distributions in life table methods for heterogeneous populations. Desirable properties of such distributions are discussed. As an example survival after myocardial infarction is considered.

A class of multivanate failure time distributions

Abstract: SUMMARY A class- of continuous multivariate lifetime distributions is proposed. The dependence between individuals in a group is modelled by a group specific quantity, which can be interpreted as an unobserved covariate common to the individuals in the group and assumed to follow a positive stable distribution. It is possible to include covariates in the model and discuss whether the dependence is still present after specific covariates are taken into account. If the conditional hazards are proportional, then the hazards in the marginal distributions are also proportional, but with different constants of proportionality. Also the hazard for the minimum in a group is proportional to the marginal hazards. If the conditional distributions given the group quantity are Weibull then the marginal distributions are also Weibull. This class can be used to test the hypothesis of independence of litter mates in the proportional hazards model.

Nonlinearity tests for time series

Abstract: SUMMARY This paper considers two nonlinearity tests for stationary time series. The idea of Tukey's one degree of freedom for nonadditivity test is generalized to the time series setting. The case of concurrent nonlinearity is discussed in detail. Simulation results show that the proposed tests are more powerful than that of Keenan (1985).

Residual variance and residual pattern in nonlinear regression

Abstract: SUMMARY A nonparametric estimator of residual variance in nonlinear regression is proposed. It is based on local linear fitting. Asymptotically the estimator has a small bias, but a larger variance compared with the parametric estimator in linear regression. Finite sample properties are investigated in a simulation study, including a comparison with other nonparametric estimators. The method is also useful for spotting heteroscedasticity and outliers in the residuals at an early stage of the data analysis. A further application is checking the fit of parametric models. This is illustrated for longitudinal growth data.

Misspecified proportional hazard models

Abstract: SUMMARY The properties of an estimator based on a proportional hazards model are investigated when the model is incorrect. The estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly. The results are used to investigate the effects on estimation if the true model is accelerated failure time, or if covariates are omitted from the proportional hazards model.

Infereni on full or partial parameters based on the standardized signed log likelihood ratio

Abstract: On etudie des tests et des limites de confiance bases sur la statistique du rapport de log vraisemblance signee, en ajustant cette statistique de telle sorte qu'a un ordre superieur d'approximation, la distribution normale standard soit valable

A maximum likelihood estimation method for random coefficient regression models

Abstract: SUMMARY A method for estimating the distribution of the parameters of a random coefficient regression model is proposed. This distribution, accounting for interindividual variability, is assumed to lie in a wide class of probability distributions rather than in a given parametric class. Estimation is based on observations from a sample of individuals and likelihood is the estimation criterion. Experimental designs may be different among individuals, allowing the method to apply to routinely collected data. The problem has strong connections with the theory of optimum design of experiments. Conditions are given under which the problem has a unique solution, which then corresponds to a discrete distribution. A simple pharmacokinetic model involving two parameters is used as an example; these parameters have a bimodal distribution as statistical specification. Moreover, only one observation is available per individual; thus the method applies even when the model is not identifiable.

Estimating distribution functions from survey data

Abstract: SUMMARY A simple method for estimating population distribution functions and associated quantiles from sample survey data is described and some asymptotic theory for it presented. The method assumes a model-based approach to survey estimation and allows auxiliary population information to be directly incorporated into the estimation process. Monte Carlo results comparing the proposed method with conventional design-based methods are given. These suggest that the model-based approach offers significant gains when the auxiliary population information is linearly related to the survey variables of interest.

Confidence regions and tests for a change-point in a sequence of exponential family random variables

Abstract: SUMMARY Maximum likelihood methods are used to test for a change in a sequence of independent exponential family random variables, with particular emphasis on the exponential distribution. The exact null and alternative distributions of the test statistics are found, and the power is compared with a test based on a linear trend statistic. Exact and approximate confidence regions for the change-point are based on the values accepted by a level x likelihood ratio test and a modification of the method proposed by Cox & Spj0tvoll (1982). The methods are applied to a classical data set on the time intervals between coal mine explosions, and the change in variation of stock market returns. In both cases the confidence regions for the change-point cover historical events that may have caused the changes.

Bayesian analysis of a Poisson process with a change-point

Abstract: SUMMARY A Bayesian approach to estimation and hypothesis testing for a Poisson process with a change-point is developed, and an example given.

On the bias of maximum likelihood estimation following a sequential test

Abstract: SUMMARY Tlhe bias of maximum likelihood estimates calculated at the end of a sequential procedure is investigated For the two sequential designs considered in detail, the sequential probability ratio test and the triangular test, this bias is appreciable A method of calculating an adjusted estimate with reduced bias is described, and an approximation to the standard error of the new estimate is provided Examples of the implementation of the method are given, and its advantages and disadvantages relative to alternative approaches are discussed

Efficient bootstrap simulation

Abstract: On donne deux idees pour renforcer une simulation: la premiere est d'equilibrer les echantillons simules, la seconde est de faire un usage explicite des approximations qui ne demandent pas de simulation

Reduced rank models for multiple time series

Abstract: SUMMARY This paper is concerned with the investigation of reduced rank coefficient models for multiple time series. In particular, autoregressive processes which have a structure to their coefficient matrices similar to that of classical multivariate reduced rank regression are studied in detail. The estimation of parameters and associated asymptotic theory are derived. The exact correspondence between the reduced rank regression procedure for multiple autoregressive processes and the canonical analysis of Box & Tiao (1977) is briefly indicated. To illustrate the methods, U.S. hog data are considered.

Semiparametric inference in a model for association in bivanate survival data

Abstract: SUMMARY Clayton's (1978) model for association in bivariate survival data is both of intrinsic importance and an interesting example of a semiparametric estimation problem, that is a problem where inference about a parameter is required in the presence of nuisance functions. We derive the asymptotic variance of Clayton's estimator, obtaining a simple explicit formula for uncensored data and indicating the modification required when the survival times are subject to arbitrary random censorship. Some comparisons are made with results derived by Oakes (1982) for other estimators within this model. In the absence of censoring the exact null variance of the score statistic corresponding to Clayton's estimator is derived and compared with that of the locally most powerful rank test given by Cuzick (1982).

Optimally bounded score functions for generalized linear models with applications to logistic regression

Abstract: SUMMARY We study optimally bounded score functions for estimating regression parameters in a generalized linear model. Our work extends results obtained by Krasker & Welsch (1982) for the linear model and provides a simple proof of Krasker & Welsch's first-order condition for strong optimality. The application of these results to logistic regression is studied in some detail with an example given comparing the bounded-influence estimator with maximum likelihood. where h( . ), q( . ) and c( . ) are known functions and 0 is a vector of regression parameters. Models of this type include logistic and probit regression, Poisson regression, linear regression with known variance, and certain models for lifetime data. Our motivation for seeking robust estimators is the same as that in the linear model; maximum likelihood estimation is sometimes sensitive to outlying data. For logistic regression, Pregibon (1981,,1982) has documented the nonrobustness of the maximum likelihood estimator and expounded the benefits of diagnostics as well as robust or resistant fitting procedures; see also Johnson (1985). Much of the work on robust estimation concerns finding estimators which sacrifice little efficiency at the assumed model while providing protection against outliers and model violations. We follow this course finding bounded-influence estimators minimizing certain functionals of the asymptotic covariance matrix. Related work includes that of Hampel (1978), Krasker (1980) and Krasker & Welsch (1982). When fitting models to data, two important issues are identification of outliers and influential cases and accommodation of these observations. Frequently when influential cases are present, the fitted model is not representative of the bulk of the data. To rectify this, one can simply delete influential cases and refit via standard methods, but this

A note on A. Albert and J. A. Anderson's conditions for the existence of maximum likelihood estimates in logistic regression models

Abstract: SUMMARY This note expands the paper by Albert & Anderson (1984) on the existence and uniqueness of maximum likelihood estimates in logistic regression models. Their three possible mutually exclusive data pattems: (i) overlap, (ii) complete separation, and (iii) quasiseparation are considered. The maximum likelihood estimate exists only in (i). Modifications of the statement and proofs of Albert & Anderson's results are given for (ii) and (iii). The identifiability for a more general model arising in the study of (iii) is discussed together with the maximization of the corresponding likelihood. A linear program is presented which determines whether data is of type (i), (ii) or (iii), and in the case of (iii) identifies Albert & Anderson's minimal set Qm.

Bias in nonlinear regression

Abstract: : This document investigates the biases of the residuals and the maximum likelihood parameters estimates from standard, normal-theory nonlinear regression models. Emphasis is placed on determining the influence of individual cases on the biases and on understanding how the residual biases can affect the usefulness of standard diagnostic methods. It is shown that the various bias expressions in the literature are equivalent, that the biases in nonlinear regression can be studied usefully in the context of linear regression, and that diagnostic plots using residuals can be misleading because of substantial residual biases. For a class of partially nonlinear models, it is shown that the maximum intrinsic curvature is closely related to the residual expectations. Finally, the model associated with power transformations of single explanatory variables in linear regression is investigated in further detail and several numerical illustrations are presented. (Author)

A portmanteau test for self-exciting threshold autoregressive-type nonlinearity in time series

Abstract: SUMMARY A portmanteau test to detect self-exciting threshold autoregressive-type nonlinearity in time series data is proposed. The test is based on cumulative sums of standardized residuals from autoregressive fits to the data. Significance levels for the test under the hypothesis of linearity are obtain from the asymptotic distribution of the cumulative sums as Brownian motion. The performance of the test is evaluated for simulated data from linear, bilinear and self-exciting threshold autoregressive models. It is also compared with another test which has been suggested for detecting general nonlinearity. Features of the proposed test, which make it useful in identifying the autoregressive order and the lag in threshold models, are discussed.

Asymptotic properties of moment estimators for overdispersed counts and proportions

Abstract: SUMMARY Consistency and asymptotic normality of moment estimates of regression parameters and an overdispersion parameter are shown for data with extra-binomial and extraPoisson variation. The asymptotic covariance of the regression parameters is found to be unaffected by estimation of the overdispersion parameter. The moment estimates may be obtained using iterated weighted least squares.

A neighbour model for field experiments

Abstract: SUMMARY A neighbour-type model is proposed as an alternative to that of Wilkinson et al. (1983) for the analysis of field variety trials. The model assumes that the varieties are grouped into complete replicates and that the plots within a replicate are side by side. A method for the estimation of the variance parameters is developed as an extension of the procedure commonly used in incomplete block designs. The model is investigated using data from the 166 cereal variety trials studied by Patterson & Hunter (1983). Results show worthwhile gains in efficiency over conventional incomplete block analysis.

Recursive nonparametric testing for dose-response relaflonships subject to downturns at high doses

Abstract: SUMMARY A general recursive strategy is introduced to test nonparametrically for an increasing dose-response relationship when a downturn in response at high doses is possible. A specific implementation involving a test of Jonckheere (1954) and Terpstra (1952) is considered, and its size and power properties are investigated via Monte Carlo methods. Data from mutation research illustrate the phenomenon of concern.

On the design of experiments under spatial correlation

Abstract: On considere tout d'abord le cas ou les effets du traitement sont estimes par les moindres carres. On considere ensuite le cas ou l'analyse a utiliser est les moindres carres generalises utilisant une matrice de correlation connue

Approximate predictive likelihood

Abstract: SUMMARY A predictive likelihood is given which approximates both Bayes and maximum likeli- hood predictive inference by expansion of a posterior likelihood. This synthesizes and extends previous results and is widely applicable. The approximation usually differs from exact Bayes posterior predictive density by Op(n-2), and from exact predictive likelihood by Op(n-1), but does not depend on the availability of prior information and is applicable when exact predictive likelihood cannot be found. The results are applied to the prediction Various treatments of the likelihood function form the basis of parametric inference, but until recently few non-Bayesian attempts have been made to define versions of it suitable for prediction. However Lauritzen (1974), Hinkley (1979), and more recently Butler (1986) have proposed rather similar definitions of predictive likelihood based on the idea of conditioning on the value of a sufficient statistic. Unfortunately they apply only when sufficiency provides a genuine reduction of the data. Moreover, even if for a given problem an exact predictive likelihood exists, the calculations needed to derive it can be onerous. This paper provides approximations to Hinkley-Lauritzen predictive likelihood accur- ate to Op(n-1) in many cases, which are applicable in situations where maximum likelihood estimation is regular. They also apply to problems which admit no exact predictive likelihood because sufficiency provides no real reduction of the data. The approximations are derived from a posterior predictive density and may if prior informa- tion is available be regarded as Bayesian procedures. Their construction requires only the repeated maximization of a likelihood and evaluation of the observed information matrix at the maximum, and is fast and accurate using usual numerical maximization methods. There is a close connexion with the approximations to posterior moments and marginal densities given by Tierney & Kadane (1986), who suggest the same approxima- tion as used here but give no detailed results for prediction. Section 2 of this paper shows how Laplace's method for integrals may be applied to a posterior predictive density to yield approximate predictive likelihood. Section 3 discusses a connection with conditional inference: essentially the same approximation

Diagnostic testing of univariate time series models

Abstract: SUMMARY The portmanteau statistic for testing the adequacy of an autoregressive-moving average model is based on the first m autocorrelations of the residuals from the fitted model. This paper examines the properties of this test for various choices of m. A modification which allows the use of small values of m is shown to result in a more powerful test. The Lagrange multiplier statistic (Godfrey, 1979) and a test statistic examined by Newbold (1980) are also discussed.

Prediction mean squared error for state space models with estimated parameters

Abstract: SUMMARY We obtain a conditional prediction mean squared error for a state space model with estimated parameters. An important application of our results is the derivation of conditional forecast and interpolation mean squared errors for autoregressive-moving average models with estimated parameters. We also obtain the conditional mean squared error for filtered and smoothed estimates of the state vector. of z(t) and S(t I N; 0) = var {z(t) - z(t I N; 0)} is the mean squared error. Both z(t I N; 0) and S(t I N; 0) can be computed efficiently using the Kalman filter and the smoothing algorithms in Chapters 3 and 7 respectively of Anderson & Moore (1979). If 0 is unknown, then we can estimate it by maximum likelihood to obtain 0, say, and A A then use z(t I N; 0) and S(t IN; 0) in place of z(t I N; 0) and S(t I N; 0). However, the estimate S(t I N; 0) of mean squared error underestimates the true mean squared error because it does not take into account the extra variation due to estimating the parameters. One important example of this problem is the adjustment of the forecast mean squared error of an ARMA process to allow for the variability due to parameter estimation. With the exception of Phillips (1979) and Fuller & Hasza (1981), whose work is confined to the autoregressive model, results in the literature on ARMA and ARMAX models with estimated parameters have dealt with the unconditional forecast mean squared error. See, for example, Bloomfield (1972), Yamamoto (1976, 1981) and Baillie (1979, 1980). Their unconditional forecast mean squared error is an average over all realizations of y, whereas the actual prediction error depends on the particular realization y we observe. This will be made clear for the first-order autoregressive model in ? 2. Therefore the