A new serially correlated gamma-frailty process for longitudinal count data

We describe a new multivariate gamma distribution and discuss its implication in a Poisson-correlated gamma-frailty model. This model is introduced to account for between-subjects correlation occurring in longitudinal count data. For likelihood-based inference involving distributions in which high-dimensional dependencies are present, it may be useful to approximate likelihoods based on the univariate or bivariate marginal distributions. The merit of composite likelihood is to reduce the computational complexity of the full likelihood. A 2-stage composite-likelihood procedure is developed for estimating the model parameters. The suggested method is applied to a meta-analysis study for survival curves.     

On Bivariate Cumulative Damage Models With Application

Bivariate cumulative damage models are proposed where the responses given the damages are independent random variables. The bivariate damage process can be either bivariate Poisson or bivariate gamma. A bivariate continuous cumulative damage model is investigated in which the responses given the damages have gamma distributions. In this case evaluation of the joint density function and bivariate tail probability function is facilitated by expanding the gamma distributions of the conditional responses by Laguerre polynomials. This approach also leads to evaluation of associated survival models. Moments and estimating equations are discussed. In addition, a bivariate discrete cumulative damage model is investigated in which the responses given the damages have a distribution chosen from a class that includes the negative binomial, the Neyman Type‐A, the Polya‐Aeppli, and the Lagrangian Poisson. Probabilities are obtained from recursive formulas which do not involve cancellation error as all quantities are non‐negative. Moments and estimating equations are presented for these models also. The continuous and the discrete models are applied to describe the rise of systolic and diastolic blood pressure with age.     

A general noninteractive multiple toxicity model including probit, logit, and Weibull transformations

A multiple toxicity model for the quantal response of organisms is constructed based on an existing bivariate theory. The main assumption is that the tolerances follow a multivariate normal distribution function. However, any monotone tolerance distribution can be applied by mapping the integration region in the n-dimensional space of transforms on the n-dimensional space of normal equivalent deviates. General requirements to noninteractive bivariate tolerance distributions are discussed, and it is shown that bivariate logit and Weibull distributions, constructed according to the mapping procedure, meet these criteria. The univariate Weibull dose-response model is given a novel interpretation in terms of reactions between toxicant molecules and a hypothetical key receptor of the organism. The application of the multiple toxicity model is demonstrated using literature data for the action of gamma-benzene hexachloride and pyrethrins on flour beetles (Tribolium castaneum). Nonnormal tolerance distributions are needed when the mortality data include extreme response probabilities.     

On the Modified and Inflated Discrete Distributions

Firstly, a modified bivariate discrete distribution is considered where a set of counts are misreported as another set of counts with different modification rates. Variances and covariances are put in the closed form and for the case when all modification rates are the same, these variances and covariances are expressed as parabolic functions and they are actually evaluated for the bivariate negative binomial. Regarding the asymptotic distributions of the estimates, elements of variance-covariance matrix are obtained. Next, a multivariate inflated discrete distribution is taken up. For the case of inflated multivariate negative binomial, Bayesian estimates of inflation as well as those of parameters are given.     

Probability binning comparison: a metric for quantitating multivariate distribution differences

BACKGROUND: While several algorithms for the comparison of univariate distributions arising from flow cytometric analyses have been developed and studied for many years, algorithms for comparing multivariate distributions remain elusive. Such algorithms could be useful for comparing differences between samples based on several independent measurements, rather than differences based on any single measurement. It is conceivable that distributions could be completely distinct in multivariate space, but unresolvable in any combination of univariate histograms. Multivariate comparisons could also be useful for providing feedback about instrument stability, when only subtle changes in measurements are occurring. METHODS: We apply a variant of Probability Binning, described in the accompanying article, to multidimensional data. In this approach, hyper-rectangles of n dimensions (where n is the number of measurements being compared) comprise the bins used for the chi-squared statistic. These hyper-dimensional bins are constructed such that the control sample has the same number of events in each bin; the bins are then applied to the test samples for chi-squared calculations. RESULTS: Using a Monte-Carlo simulation, we determined the distribution of chi-squared values obtained by comparing sets of events from the same distribution; this distribution of chi-squared values was identical as for the univariate algorithm. Hence, the same formulae can be used to construct a metric, analogous to a t-score, that estimates the probability with which distributions are distinct. As for univariate comparisons, this metric scales with the difference between two distributions, and can be used to rank samples according to similarity to a control. We apply the algorithm to multivariate immunophenotyping data, and demonstrate that it can be used to discriminate distinct samples and to rank samples according to a biologically-meaningful difference. CONCLUSION: Probability binning, as shown here, provides a useful metric for determining the probability with which two or more multivariate distributions represent distinct sets of data. The metric can be used to identify the similarity or dissimilarity of samples. Finally, as demonstrated in the accompanying paper, the algorithm can be used to gate on events in one sample that are different from a control sample, even if those events cannot be distinguished on the basis of any combination of univariate or bivariate displays. Published 2001 Wiley-Liss, Inc.     

The fitness effect of mutations across environments: Fisher's geometrical model with multiple optima

When are mutations beneficial in one environment and deleterious in another? More generally, what is the relationship between mutation effects across environments? These questions are crucial to predict adaptation in heterogeneous conditions in a broad sense. Empirical evidence documents various patterns of fitness effects across environments but we still lack a framework to analyze these multivariate data. In this article, we extend Fisher's geometrical model to multiple environments determining distinct peaks. We derive the fitness distribution, in one environment, among mutants with a given fitness in another and the bivariate distribution of random mutants’ fitnesses across two or more environments. The geometry of the phenotype‐fitness landscape is naturally interpreted in terms of fitness trade‐offs between environments. These results may be used to fit/predict empirical distributions or to predict the pattern of adaptation across heterogeneous conditions. As an example, we derive the genomic rate of substitution and of adaptation in a metapopulation divided into two distinct habitats in a high migration regime and show that they depend critically on the geometry of the phenotype‐fitness landscape.     

On Countable Mixtures of Bivariate Binomial Distributions

A unified treatment is given for mixtures of bivariate binomial distributions with respect to their index parameter(s). The use of probability generating functions is employed and a number of interesting properties including probabilities, factorial moments, factorial cumulants and conditional distributions are derived. Five classes of such mixtures are examined and several well known bivariate discrete distributions are used as illustrative examples. Biological applications are indicated including the fit of three bivariate distributions to an actual set of human family data.     

Conditionally in Bivariate Generalized Poisson Distributions

Three types of bivariate generalized Poisson distributions are defined and the structure of their conditional distributions is examined by using the Faa Di Bruno's formula. The resulting expressions involve Bell polynomials and can be interpreted in terms of convoluted random variables with one of the convolutes having the form of the marginal distribution. The three types of bivariate Neyman A distributions are used to illustrate the procedure.     

A class of goodness of fit tests for a copula based on bivariate right-censored data

The copula of a bivariate distribution, constructed by making marginal transformations of each component, captures all the information in the bivariate distribution about the dependence between two variables. For frailty models for bivariate data the choice of a family of distributions for the random frailty corresponds to the choice of a parametric family for the copula. A class of tests of the hypothesis that the copula is in a given parametric family, with unspecified association parameter, based on bivariate right censored data is proposed. These tests are based on first making marginal Kaplan-Meier transformations of the data and then comparing a non-parametric estimate of the copula to an estimate based on the assumed family of models. A number of options are available for choosing the scale and the distance measure for this comparison. Significance levels of the test are found by a modified bootstrap procedure. The procedure is used to check the appropriateness of a gamma or a positive stable frailty model in a set of survival data on Danish twins.     

Efficient Simulation of Multivariate Binomial and Poisson Distributions

Power investigations, for example, in statistical procedures for the assessment of agreement among multiple raters often require the simultaneous simulation of several dependent binomial or Poisson distributions to appropriately model the stochastical dependencies between the raters' results. Regarding the rather large dimensions of the random vectors to be generated and the even larger number of interactions to be introduced into the simulation scenarios to determine all necessary information on their distributions' dependence stucture, one needs efficient and fast algorithms for the simulation of multivariate Poisson and binomial distributions. Therefore two equivalent models for the multivariate Poisson distribution are combined to obtain an algorithm for the quick implementation of its multivariate dependence structure. Simulation of the multivariate Poisson distribution then becomes feasible by first generating and then convoluting independent univariate Poisson variates with appropriate expectations. The latter can be computed via linear recursion formulae. Similar means for simulation are also considered for the binomial setting. In this scenario it turns out, however, that exact computation of the probability function is even easier to perform; therefore corresponding linear recursion formulae for the point probabilities of multivariate binomial distributions are presented, which only require information about the index parameter and the (simultaneous) success probabilities, that is the multivariate dependence structure among the binomial marginals.     

The Doubly Noncentral Distribution of Wilks' Statistic in MANOVA

When plot effects are modeled in a randomized block design in multivariate analysis of variance the error and hypothesis matrices have independent noncentral Wishart distributions. This gives rise to doubly noncentral distribution of Wilks' statistic. The doubly noncentral distribution of Wilks' statistics, when the noncentrality matrices for the noncentral Wishart distributions in the above setting are of rank one or two, is investigated.     

Sample size estimation for survival outcomes in cluster-randomized studies with small cluster sizes

We present a method for computing sample size for cluster-randomized studies involving a large number of clusters with relatively small numbers of observations within each cluster. For multivariate survival data, only the marginal bivariate distribution is assumed to be known. The validity of this assumption is also discussed.     

Measuring agreement of multivariate discrete survival times using a modified weighted kappa coefficient

Summary .  Assessing agreement is often of interest in clinical studies to evaluate the similarity of measurements produced by different raters or methods on the same subjects. We present a modified weighted kappa coefficient to measure agreement between bivariate discrete survival times. The proposed kappa coefficient accommodates censoring by redistributing the mass of censored observations within the grid where the unobserved events may potentially happen. A generalized modified weighted kappa is proposed for multivariate discrete survival times. We estimate the modified kappa coefficients nonparametrically through a multivariate survival function estimator. The asymptotic properties of the kappa estimators are established and the performance of the estimators are examined through simulation studies of bivariate and trivariate survival times. We illustrate the application of the modified kappa coefficient in the presence of censored observations with data from a prostate cancer study.     

Regression analysis of multivariate interval-censored failure time data with application to tumorigenicity experiments

This paper discusses multivariate interval-censored failure time data that occur when there exist several correlated survival times of interest and only interval-censored data are available for each survival time. Such data occur in many fields. One is tumorigenicity experiments, which usually concern different types of tumors, tumors occurring in different locations of animals, or together. For regression analysis of such data, we develop a marginal inference approach using the additive hazards model and apply it to a set of bivariate interval-censored data arising from a tumorigenicity experiment. Simulation studies are conducted for the evaluation of the presented approach and suggest that the approach performs well for practical situations.     

A bivariate quantitative genetic model for a linear Gaussian trait and a survival trait

With the increasing use of survival models in animal breeding to address the genetic aspects of mainly longevity of livestock but also disease traits, the need for methods to infer genetic correlations and to do multivariate evaluations of survival traits and other types of traits has become increasingly important. In this study we derived and implemented a bivariate quantitative genetic model for a linear Gaussian and a survival trait that are genetically and environmentally correlated. For the survival trait, we considered the Weibull log-normal animal frailty model. A Bayesian approach using Gibbs sampling was adopted. Model parameters were inferred from their marginal posterior distributions. The required fully conditional posterior distributions were derived and issues on implementation are discussed. The two Weibull baseline parameters were updated jointly using a Metropolis-Hasting step. The remaining model parameters with non-normalized fully conditional distributions were updated univariately using adaptive rejection sampling. Simulation results showed that the estimated marginal posterior distributions covered well and placed high density to the true parameter values used in the simulation of data. In conclusion, the proposed method allows inferring additive genetic and environmental correlations, and doing multivariate genetic evaluation of a linear Gaussian trait and a survival trait.     

Merits of Modelling Multivariate Survival Data Using Random Effects Proportional Odds Model

Recently, there has been a great deal of interest in the analysis of multivariate survival data. In most epidemiological studies, survival times of the same cluster are related because of some unobserved risk factors such as the environmental or genetic factors. Therefore, modelling of dependence between events of correlated individuals is required to ensure a correct inference on the effects of treatments or covariates on the survival times. In the past decades, extension of proportional hazards model has been widely considered for modelling multivariate survival data by incorporating a random effect which acts multiplicatively on the hazard function. In this article, we consider the proportional odds model, which is an alternative to the proportional hazards model at which the hazard ratio between individuals converges to unity eventually. This is a reasonable property particularly when the treatment effect fades out gradually and the homogeneity of the population increases over time. The objective of this paper is to assess the influence of the random effect on the within‐subject correlation and the population heterogeneity. We are particularly interested in the properties of the proportional odds model with univariate random effect and correlated random effect. The correlations between survival times are derived explicitly for both choices of mixing distributions and are shown to be independent of the covariates. The time path of the odds function among the survivors are also examined to study the effect of the choice of mixing distribution. Modelling multivariate survival data using a univariate mixing distribution may be inadequate as the random effect not only characterises the dependence of the survival times, but also the conditional heterogeneity among the survivors. A robust estimate for the correlation of the logarithm of the survival times within a cluster is obtained disregarding the choice of the mixing distributions. The sensitivity of the estimate of the regression parameter under a misspecification of the mixing distribution is studied through simulation. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the stochastic theory of compartments: Solution forn-compartment systems with irreversible,time-dependent transition probabilities

The bivariate distribution of a two-compartment stochastic system with irreversible, time-dependent transition probabilities is obtained for any point in time. The mean and variance of the number of particles in any compartment and the covariance between the number of particles in each of the two compartments are exhibited and compared to existing results. The two-compartment system is then generalized to ann-compartment catenary and to ann-compartment mammillary system. The multivariate distributions of these two systems are obtained under two sets of initial conditions: (1) the initial distribution is known; and (2) the number of particles in each compartment of the system at timet=0 is determined. The moments of these distributions are also produced and compared with existing results.     

Multivariate Polya and Inverse Polya Distributions of Order k

Multivariate Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the type II multinomial and multivariate negative binomial distributions of order k of PHILIPPOU , ANTZOULAKOS and TRIPSIANNIS (1990, 1988), respectively, with the Dirichlet distribution. It is noted that the above two distributions include as special cases a multivariate hypergeometric distribution of order k, a negative one, an inverse one, a negative inverse one and a discrete uniform of the same order. The probability generating functions, means, variances and covariances of the new distributions are obtained and five asymptotic results are established relating them to the above-mentioned multinomial and multivariate negative binomial distributions of order k, and to the type II negative binomial and the type I multivariate Poisson distributions of order k of PHILIPPOU (1983), and PHILIPPOU , ANTZOULAKOS and TRIPSIAN-NIS (1988), respectively. Potential applications are also indicated. The present paper extends to the multivariate case the work of PHILIPPOU , TRIPSIANNIS and ANTZOULAKOS (1989) on Polya and inverse Polya distributions of order k..     

A Multivariate Goodness-of-Fit Test for Stochastically Ordered Distributions

There are a number of nonparametric procedures known for testing goodness-of-fit in the univariate case. Similar procedures can be derived for testing goodness-of-fit in the multivariate case through an application of the theory of statistically equivalent blocks (SEB). The SEB transforms the data into coverages which are distributed as spacings from a uniform distribution on [0,1], under the null hypothesis. In this paper, we present a multivariate nonparametric test of goodness-of-fit based on the SEB when the multivariate distributions under the null hypothesis and the alternative hypothesis are “weakly” ordered. Empirical results are given on the performance of the proposed test in an application to the problem of assessing the reliability of a p-component system.     

Comparison of multivariate tests for genetic linkage

OBJECTIVES: Multivariate tests for linkage can provide improved power over univariate tests but the type I error rates and comparative power of commonly used methods have not previously been compared. Here we studied the behavior of bivariate formulations of the variance component (VC) and Haseman-Elston (H-E) approaches. METHODS: We compared through simulation studies the bivariate H-E test with the unconstrained bivariate VC approach and with a VC approach in which the major-gene correlation is constrained to +/-1. We also compared these methods to univariate methods. RESULTS: Bivariate approaches are more powerful than univariate analyses unless the traits are very highly positively correlated. The power of the bivariate H-E test was less than the VC procedures. The constrained test was often less powerful than the unconstrained test. The empirical distributions of the bivariate H-E test and the unconstrained bivariate VC test conformed with asymptotic distributions for samples of 100 or more sibships of size 4. CONCLUSIONS: The unconstrained VC test is valuable for testing for preliminary linkages using multivariate phenotypes. The bivariate H-E test was less powerful than the bivariate VC tests.