Unbalanced Repeated Measures with Random Coefficients

In a random coefficient repeated measures model, the regression coefficients relating the observations to some underlying variable, such as time, are themselves taken to be random distributed over experimental units. In this paper, a general approach to repeated measures analysis is extended to this wider model. In the model three specific error structures for the random regression coefficients have been studied, viz, the random coefficients variance matrix is considered to be (i) diagonal, (ii) proportional to the identity matrix and (iii) completely general. An example will be analyzed to illustrate the procedure.     

Testing the Equality of Dependent Variances

The likelihood ratio test for testing equality of vgE;2 correlated variables is developed. In general, evaluation of the test statistic involves the iterative optimization of a likelihood function with 1 + v(v – 1)/2 parameters. The explicit form of the test statistic is derived in the bivariate case, and an iterative algorithm for determining the maximum likelihood estimates is suggested. A limited Monte Carlo study determines the behavior of the proposed procedure under the null hypothesis and variety of parameter values.     

A Model for the Censoring Process in Incomplete Repeated Measurement Experiments

Premature terminations or dropouts occur often in repeated measurement experiments. A number of methods have been proposed to analyze such data but most of them assume that the censoring mechanism is, within each group, unaffected by the mechanism generating the response variables. In this paper, we propose a model for the censoring mechanism that generates dropouts. We then show how this model can be used to check whether the censoring mechanism is affected by the response variables and other covariates. Finally, the methods of the paper are applied to the “Halothane” data set.     

Testing Equality of Regressions When Error Variances are Heterogeneous to Study the Production Differentials of Rice in Bangladesh

Method to test the equality of p sets of regression coefficients in presence of heterogeneous error variances is suggested. The sets of regression coefficients are obtained from a sample where the sample observations are classified into p classes. The test statistic to test the significance of the regression is also suggested. As an application of the method, the equality of production functions of boro rice produced in different districts of Bangladesh in two different periods is tested.     

Categorical variables with many categories are preferentially selected in bootstrap‐based model selection procedures for multivariable regression models

Automated variable selection procedures, such as backward elimination, are commonly employed to perform model selection in the context of multivariable regression. The stability of such procedures can be investigated using a bootstrap‐based approach. The idea is to apply the variable selection procedure on a large number of bootstrap samples successively and to examine the obtained models, for instance, in terms of the inclusion of specific predictor variables. In this paper, we aim to investigate a particular important problem affecting this method in the case of categorical predictor variables with different numbers of categories and to give recommendations on how to avoid it. For this purpose, we systematically assess the behavior of automated variable selection based on the likelihood ratio test using either bootstrap samples drawn with replacement or subsamples drawn without replacement from the original dataset. Our study consists of extensive simulations and a real data example from the NHANES study. Our main result is that if automated variable selection is conducted on bootstrap samples, variables with more categories are substantially favored over variables with fewer categories and over metric variables even if none of them have any effect. Importantly, variables with no effect and many categories may be (wrongly) preferred to variables with an effect but few categories. We suggest the use of subsamples instead of bootstrap samples to bypass these drawbacks.     

A Variance Components Model for Analyzing Repeated Measures Designs with Non-Stationary Error Structure

When data are collected in the form of multiple measurements on several subjects, they are often analyzed as repeated measures data with some stationary error structure assumed for the errors. For data with non-stationary error structure, the multivariate model is often used. The multivariate model imposes restrictions that are often not met in practice by data of such type. At the same time, they ignore valuable information in the data that are related to time dependencies and time relations. In this paper, we propose a model that is a reparametrization of the multivariate model and is suitable to analyze general repeated measures designs with non-stationary error structure. The model is shown to be a variance components model whose components are estimated using the method of maximum likelihood. Several other properties of the model are derived and discussed including tests of significance. Finally, an example on neurological data is included to demonstrate its application in biological sciences.     

Joint Modelling of Repeated Measures and Survival Time Data

In many clinical trials both repeated measures data and event history data are simultaneously observed from the same subject. These two types of responses are usually correlated, because they are from the same subject. In this article, we propose a joint model for the combined analysis of repeated measures data and event history data in the framework of hierarchical generalized linear models. The correlation between repeated measures and event time is modelled by introducing a shared random effect. The model parameters are estimated using the hierarchical‐likelihood approach. The proposed model is illustrated using a real data set for the renal transplant patients.     

Analysis of infant growth curves using multivariate adaptive splines

In this paper, we study the effect of cocaine use by a pregnant woman on the growth of her infant after birth. Using a data set from a retrospective study, we found that cocaine use was a marginally significant contributor to the infant growth as measured by bodyweight. From a statistical point of view, the data represent a common, though complex, structure that has received little attention in the statistical literature. To analyze these data, we adopt and further enhance an approach developed recently called MASAL (multivariate adaptive splines for the analysis of longitudinal data). In addition to the fitting of growth curves, we demonstrate particularly how to explore and estimate the underlying covariance structures for the longitudinal data that were collected from a rather irregular schedule.     

The Wald Statistic in Proportional Hazards Hypothesis Testing

In survivorship modelling using the proportional hazards model of Cox (1972, Journal of the Royal Statistical Society, Series B, 34, 187–220), it is often desired to test a subset of the vector of unknown regression parameters β in the expression for the hazard rate at time t. The likelihood ratio test statistic is well behaved in most situations but may be expensive to calculate. The Wald (1943, Transactions of the American Mathematical Society 54, 426–482) test statistic is easier to calculate, but has some drawbacks. In testing a single parameter in a binomial logit model, Hauck and Donner (1977, Journal of the American Statistical Association 72, 851–853) show that the Wald statistic decreases to zero the further the parameter estimate is from the null and that the asymptotic power of the test decreases to the significance level. The Wald statistic is extensively used in statistical software packages for survivorship modelling and it is therefore important to understand its behavior. The present work examines empirically the behavior of the Wald statistic under various departures from the null hypothesis and under the presence of Type I censoring and covariates in the model. It is shown via examples that the Wald statistic's behavior is not as aberrant as found for the logistic model. For the single parameter case, the asymptotic non-null distribution of the Wald statistic is examined.     

Information-theoretic indices and an approximate significance test for testing the molecular clock hypothesis with genetic distances

Distance-based phylogenetic methods are widely used in biomedical research. However, distance-based dating of speciation events and the test of the molecular clock hypothesis are relatively underdeveloped. Here I develop an approximate test of the molecular clock hypothesis for distance-based trees, as well as information-theoretic indices that have been used frequently in model selection, for use with distance matrices. The results are in good agreement with the conventional sequence-based likelihood ratio test. Among the information-theoretic indices, AICu is the most consistent with the sequence-based likelihood ratio test. The confidence in model selection by the indices can be evaluated by bootstrapping. I illustrate the usage of the indices and the approximate significance test with both empirical and simulated sequences. The tests show that distance matrices from protein gel electrophoresis and from genome rearrangement events do not violate the molecular clock hypothesis, and that the evolution of the third codon position conforms to the molecular clock hypothesis better than the second codon position in vertebrate mitochondrial genes. I outlined evolutionary distances that are appropriate for phylogenetic reconstruction and dating.     

Some contributions to the analysis of multivariate data

In this paper, we provide an overview of recently developed methods for the analysis of multivariate data that do not necessarily emanate from a normal universe. Multivariate data occur naturally in the life sciences and in other research fields. When drawing inference, it is generally recommended to take the multivariate nature of the data into account, and not merely analyze each variable separately. Furthermore, it is often of major interest to select an appropriate set of important variables. We present contributions in three different, but closely related, research areas: first, a general approach to the comparison of mean vectors, which allows for profile analysis and tests of dimensionality; second, non‐parametric and parametric methods for the comparison of independent samples of multivariate observations; and third, methods for the situation where the experimental units are observed repeatedly, for example, over time, and the main focus is on analyzing different time profiles when the number p of repeated observations per subject is larger than the number n of subjects.     

Testing for Poisson Zero Inflation in Disease Mapping

When analyzing mortality data due to rare diseases in small areas, it is common to find several health zones with no mortality cases. In these circumstances, the classical homogeneous model based on the Poisson distribution used to estimate the relative risks within each area may encounter lack of fit due to a disproportionately large frequency of zeros. To cope with these zeros, the zero inflated Poisson model can be used. In this paper, we propose a test for detecting zero inflation in the context of disease mapping which is based on bootstrap techniques. The test is illustrated using male mortality data due to brain cancer in Navarra, Spain. In addition, comparisons with other tests for Poisson zero inflation such as the score test and the likelihood ratio test are carried out in terms of empirical power and size using the brain cancer scenario. The proposed bootstrap test has good power and size and works well when detecting the excess of zeros in small area data sets. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)