Modelling bivariate count series with excess zeros

Bivariate time series of counts with excess zeros relative to the Poisson process are common in many bioscience applications. Failure to account for the extra zeros in the analysis may result in biased parameter estimates and misleading inferences. A class of bivariate zero-inflated Poisson autoregression models is presented to accommodate the zero-inflation and the inherent serial dependency between successive observations. An autoregressive correlation structure is assumed in the random component of the compound regression model. Parameter estimation is achieved via an EM algorithm, by maximizing an appropriate log-likelihood function to obtain residual maximum likelihood estimates. The proposed method is applied to analyze a bivariate series from an occupational health study, in which the zero-inflated injury count events are classified as either musculoskeletal or non-musculoskeletal in nature. The approach enables the evaluation of the effectiveness of a participatory ergonomics intervention at the population level, in terms of reducing the overall incidence of lost-time injury and a simultaneous decline in the two mean injury rates.     

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.     

Bivariate POISSON Binomial Distributions

Three bivariate generalizations of the POISSON binomial distribution are introduced. The probabilities, moments, conditional distributions and regression functions for these distributions are obtained in terms of bipartitional polynomials. Recurrences for the probabilities and moments are also given. Parameter estimators are derived using the methods of moments and zero frequencies and the three distributions are fitted to some ecological data.     

Bivariate Modified Power Series Distribution Some Properties,Estimation and Applications

The class of bivariate modified power series probability distribution (BMPSD) has been defined by P(X = x, Y=y) =a(x, y) (g(?₁ ?₂))^x (h(?₁, ?₂))^xwhere a(x,y) is a sub-set of the Cartesian product of the set of non-negative integers and g(?₁, ?₂), h(?₁ ?₂) and f(?₁, (?₁, ?₂) are positive finite and successively differentiable functions of ?₁and ?₂. It includes a very large number of well known probability distributions. The recurrence relations for central moments and factorial moments have been determined. Also, the M.L. estimators for ?₁ and ?₂ and their asymptotic biases and variances are obtained. Some important properties are discussed. The results of an BMPSD have been applied to derive the corresponding results for the bivariate generalized negative bino-mial distribution and the bivariate Lagrangian Poisson distribution.     

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.     

On consistency of Kendall's tau under censoring

Necessary and sufficient conditions for consistency of a simpleestimator of Kendall's tau under bivariate censoring are presented.The results are extended to data subject to bivariate left truncationas well as right censoring.     

Semiparametric analysis of two-level bivariate binary data

In medical studies, paired binary responses are often observed for each study subject over timepoints or clusters. A primary interest is to investigate how the bivariate association and marginal univariate risks are affected by repeated measurements on each subject. To achieve this we propose a very general class of semiparametric bivariate binary models. The subject-specific effects involved in the bivariate log odds ratio and the univariate logit components are assumed to follow a nonparametric Dirichlet process (DP). We propose a hybrid method to draw model-based inferences. In the framework of the proposed hybrid method, estimation of parameters is done by implementing the Monte Carlo expectation-maximization algorithm. The proposed methodology is illustrated through a study on the effectiveness of tibolone for reducing menopausal problems experienced by Indian women. A simulation study is also conducted to evaluate the efficiency of the new methodology.     

Protein bioinformatics and mixtures of bivariate von Mises distributions for angular data

A fundamental problem in bioinformatics is to characterize the secondary structure of a protein, which has traditionally been carried out by examining a scatterplot (Ramachandran plot) of the conformational angles. We examine two natural bivariate von Mises distributions--referred to as Sine and Cosine models--which have five parameters and, for concentrated data, tend to a bivariate normal distribution. These are analyzed and their main properties derived. Conditions on the parameters are established which result in bimodal behavior for the joint density and the marginal distribution, and we note an interesting situation in which the joint density is bimodal but the marginal distributions are unimodal. We carry out comparisons of the two models, and it is seen that the Cosine model may be preferred. Mixture distributions of the Cosine model are fitted to two representative protein datasets using the expectation maximization algorithm, which results in an objective partition of the scatterplot into a number of components. Our results are consistent with empirical observations; new insights are discussed.     

A Guide to Bivariate Ideas of Quantal Response Analysis,as Applied in Biometrics and Econometrics

A review is provided of the several bivariate generalisations of quantal response analysis that have appeared in the biometric and econometric literatures. There are three main types: (i) where a binary outcome is the result of two stimulants, and thus the bivariate distribution of the thresholds for response is relevant; (ii) where three or more alternative outcomes may arise from a single stimulant; and (iii) where the response itself is bivariate (i.e., two types of response may simultaneously be observed).     

Bivariate Sign Tests Sensitive to Homeo- and Heteropoetic Treatment Effects

Starting from N paired differences of a continuous variable, BENNETT 's (1962) bivariate sign test is identified as a test against so called homeopoetic treatment effects (or location shifts in one or both variables X and Y). As a complement, a new sign test is identified to be sensitive to so called heteropoetic treatment effects (shifts in dispersion). Both tests are shown to be important in biomedical and psychosocial research. A numerical example is given from psychopharmacology using the new bivariate sign test.     

Using Median Lines in Robust Bivariate Data Analysis

Methods for robust comparison of bivariate errors-in-variables are considered. The concept of median lines is introduced for the robust estimation of principal components. Median lines separate the bivariate sample space into two equally sized parts. Statistical properties of the model parameters are derived. Robust residual analysis assesses linear relationships as well as goodness of fit and allows for the detection of potential outliers. Special emphasis is laid on graphical methods. A bivariate box-plot is proposed for exploratory data analysis. The median lines procedure is illustrated by a real example.     

Nonparametric One‐way Analysis of Variance of Replicated Bivariate Spatial Point Patterns

A common problem in neuropathological studies is to assess the spatial patterning of cells on tissue sections and to compare spatial patterning between disorder groups. For a single cell type, the cell positions constitute a univariate point process and interest focuses on the degree of spatial aggregation. For two different cell types, the cell positions constitute a bivariate point process and the degree of spatial interaction between the cell types is of interest. We discuss the problem of analysing univariate and bivariate spatial point patterns in the one‐way design where cell patterns have been obtained for groups of subjects. A bootstrapping procedure to perform a nonparametric one‐way analysis of variance of the spatial aggregation of a univariate point process has been suggested by Diggle, Lange and Bene? (1991). We extend their replication‐based approach to allow the comparison of the spatial interaction of two cell types between groups, to include planned comparisons (contrasts) and to assess whole groups against complete spatial randomness and spatial independence. We also accommodate several replicate tissue sections per subject. An advantage of our approach is that it can be applied when processes are not stationary, a common problem in brain tissue sections since neurons are arranged in cortical layers. We illustrate our methods by applying them to a neuropathological study to investigate abnormalities in the functional relationship between neurons and astrocytes in HIV associated dementia. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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 Characterization of the Bivariate Exponential Distribution

In this article a characterisation of the Gumbel's Bivariate Exponential Distribution is established on the basis of the properties of the conditional expectation of the component variables. The characterising property is proposed as the definition of lack of memory in the bivariate case.     

Testing Equality of Means in the Presence of Correlation and Missing Data

A statistic, derived from the combination of two dependent tests, is proposed for testing the hypothesis of equality of the means of a bivariate normal distribution with unknown common variance and correlation coefficient when observations are missing on one or both variates. The null distribution of the statistic is approximated by a well-known distribution. The empirical powers of the statistic are computed and compared with some of the known statistics. The comparisons support the use of the proposed test.     

Bayesian dose-finding in phase I/II clinical trials using toxicity and efficacy odds ratios

A Bayesian adaptive design is proposed for dose-finding in phase I/II clinical trials to incorporate the bivariate outcomes, toxicity and efficacy, of a new treatment. Without specifying any parametric functional form for the drug dose-response curve, we jointly model the bivariate binary data to account for the correlation between toxicity and efficacy. After observing all the responses of each cohort of patients, the dosage for the next cohort is escalated, deescalated, or unchanged according to the proposed odds ratio criteria constructed from the posterior toxicity and efficacy probabilities. A novel class of prior distributions is proposed through logit transformations which implicitly imposes a monotonic constraint on dose toxicity probabilities and correlates the probabilities of the bivariate outcomes. We conduct simulation studies to evaluate the operating characteristics of the proposed method. Under various scenarios, the new Bayesian design based on the toxicity-efficacy odds ratio trade-offs exhibits good properties and treats most patients at the desirable dose levels. The method is illustrated with a real trial design for a breast medical oncology study.     

A note on the statistical properties of animal locations

A statistical model of the successive locations of an animal in the plane induces a statistical model of the relative positions of successive locations. A common locational model is that the Cartesian coordinates of successive locations in the plane are independent bivariate normal random variables. This note gives the statistical properties of the direction and length of the vector joining successive locations.     

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.     

The estimation of a bivariate fitness function from two samples taken from a population

Summary The fitness of animals subjected to natural selection can be defined as the probability of surviving selection for a given interval of time, or some convenient multiple of this. If the fitness is related to some measurable variablesX, Y, Z,… then the relationship is expressed mathematically in the fitness functionw(x, y, z,…) and this function can be estimated by comparing the joint distribution ofX, Y, Z,… in samples taken before and after selection. In an earlier paper (Manly, 1975) the problems involved in estimating a fitness function of one variable were discussed. In the present paper various methods for estimating a bivariate fitness function are proposed and compared on some semiartificial sample data. It is concluded that either a generalized version ofO’Donald’s (1968) method of moments or a weighted multiple regression method will be most satisfactory. Alternative methods involving assumptions of normality will need to be used with great care.     

Testing the Mean Vector and the Correlation Coefficient in Life-Testing Under Bivariate Normality

In life-testing situations under bivariate normality of (X, U), a few smallest or a few largest Y-observations may not be available. Tests for μ = 0 (mean vector) and o = 0 (correlation coefficient) are developed from the available Y-observations and their concomitant X-observations. The robustness of these tests to departures from normality is investigated.