Parameter estimation in longitudinal studies with outcome-dependent follow-up

In many observational studies, individuals are measured repeatedly over time, although not necessarily at a set of prespecified occasions. Instead, individuals may be measured at irregular intervals, with those having a history of poorer health outcomes being measured with somewhat greater frequency and regularity; i.e., those individuals with poorer health outcomes may have more frequent follow-up measurements and the intervals between their repeated measurements may be shorter. In this article, we consider estimation of regression parameters in models for longitudinal data where the follow-up times are not fixed by design but can depend on previous outcomes. In particular, we focus on general linear models for longitudinal data where the repeated measures are assumed to have a multivariate Gaussian distribution. We consider assumptions regarding the follow-up time process that result in the likelihood function separating into two components: one for the follow-up time process, the other for the outcome process. The practical implication of this separation is that the former process can be ignored when making likelihood-based inferences about the latter; i.e., maximum likelihood (ML) estimation of the regression parameters relating the mean of the longitudinal outcomes to covariates does not require that a model for the distribution of follow-up times be specified. As a result, standard statistical software, e.g., SAS PROC MIXED (Littell et al., 1996, SAS System for Mixed Models), can be used to analyze the data. However, we also demonstrate that misspecification of the model for the covariance among the repeated measures will, in general, result in regression parameter estimates that are biased. Furthermore, results of a simulation study indicate that the potential bias due to misspecification of the covariance can be quite considerable in this setting. Finally, we illustrate these results using data from a longitudinal observational study (Lipshultz et al., 1995, New England Journal of Medicine 332, 1738-1743) that explored the cardiotoxic effects of doxorubicin chemotherapy for the treatment of acute lymphoblastic leukemia in children.     

Semiparametric estimation in copula models for bivariate sequential survival times

Sequentially observed survival times are of interest in many studies but there are difficulties in analyzing such data using nonparametric or semiparametric methods. First, when the duration of followup is limited and the times for a given individual are not independent, induced dependent censoring arises for the second and subsequent survival times. Non-identifiability of the marginal survival distributions for second and later times is another issue, since they are observable only if preceding survival times for an individual are uncensored. In addition, in some studies a significant proportion of individuals may never have the first event. Fully parametric models can deal with these features, but robustness is a concern. We introduce a new approach to address these issues. We model the joint distribution of the successive survival times by using copula functions, and provide semiparametric estimation procedures in which copula parameters are estimated without parametric assumptions on the marginal distributions. This provides more robust estimates and checks on the fit of parametric models. The methodology is applied to a motivating example involving relapse and survival following colon cancer treatment.     

Bayesian Reliability Estimation of a Two-parameter Cauchy Distribution

This paper gives an approximate Bayes procedure for the estimation of the reliability function of a two-parameter Cauchy distribution using Jeffreys' non-informative prior with a squared-error loss function, and with a log-odds ratio squared-error loss function. Based on a Monte Carlo simulation study, two such Bayes estimators of the reliability are compared with the maximum likelihood estimator.     

Boolean Models: Maximum Likelihood Estimation from Circular Clumps

This paper deals with the problem of making inferences on the maximum radius and the intensity of the Poisson point process associated to a Boolean Model of circular primary grains with uniformly distributed random radii. The only sample information used is observed radii of circular clumps (DUPAC, 1980). The behaviour of maximum likelihood estimation has been evaluated by means of Monte Carlo methods.     

Common canonical variates

Canonical correlation analysis measures the linear relationshipbetween two random vectors X₁ and X₂ as the maximum correlationbetween linear combinations of X₁ and linear combinations ofX₂. Several generalisations of canonical correlation analysisto k2 random vectors X₁ ..., X_k have been proposed in the literature(Kettenring, 1971, 1985), based on the principle of maximisingsome generalised measure of correlation. In this paper we proposean alternative generalisation, called common canonical variates,based on the assumption that the canonical variates have thesame coefficients in all k sets of variables. This generalisationis applicable in situations where all X_i have the same dimension.We present normal theory maximum likelihood estimation of commoncanonical variates, and illustrate their use on a morphometricdata set.     

A Rapid and Efficient Estimation Procedure for the Negative Binomial Distribution

This paper presents a new noniterative procedure for estimating the parameters of a negative binomial distribution. The procedure uses the first moment equation and an equation based on the weighted sample mean, with weights ω_x∝ α^z. The selection of a value for α is examined. A simulation study has been carried out and also the method has been applied to the 35 data sets analysed by Martin and Katti (1965, Biometrics) in order to compare it with the method of moments and with the method of maximum likelihood (ML). We conclude that the new procedure has greater relative efficiency than the method of moments; it gives estimates which are consistently close to ML and are easy to calculate.     

Accurate critical constants for the one-sided approximate likelihood ratio test of a normal mean vector when the covariance matrix is estimated

Tang, Gnecco, and Geller (1989, Biometrika 76, 577-583) proposed an approximate likelihood ratio (ALR) test of the null hypothesis that a normal mean vector equals a null vector against the alternative that all of its components are nonnegative with at least one strictly positive. This test is useful for comparing a treatment group with a control group on multiple endpoints, and the data from the two groups are assumed to follow multivariate normal distributions with different mean vectors and a common covariance matrix (the homoscedastic case). Tang et al. derived the test statistic and its null distribution assuming a known covariance matrix. In practice, when the covariance matrix is estimated, the critical constants tabulated by Tang et al. result in a highly liberal test. To deal with this problem, we derive an accurate small-sample approximation to the null distribution of the ALR test statistic by using the moment matching method. The proposed approximation is then extended to the heteroscedastic case. The accuracy of both the approximations is verified by simulations. A real data example is given to illustrate the use of the approximations.     

On easily interpretable multivariate reference regions of rectangular shape

Till now, multivariate reference regions have played only a marginal role in the practice of clinical chemistry and laboratory medicine. The major reason for this fact is that such regions are traditionally determined by means of concentration ellipsoids of multidimensional Gaussian distributions yielding reference limits which do not allow statements about possible outlyingness of measurements taken in specific diagnostic tests from a given patient or subject. As a promising way around this difficulty we propose to construct multivariate reference regions as p-dimensional rectangles or (in the one-sided case) rectangular half-spaces whose edges determine univariate percentile ranges of the same probability content in each marginal distribution. In a first step, the corresponding notion of a quantile of a p-dimensional probability distribution of any type and shape is made mathematically precise. Subsequently, both parametric and nonparametric procedures of estimating such a quantile are described. Furthermore, results on sample-size calculation for reference-centile studies based on the proposed definition of multivariate quantiles are presented generalizing the approach of Jennen-Steinmetz and Wellek.