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.     

A Variation of the Method of Translation to Generate Bivariate and Multivariate Survival Distributions with Competing Risk or Reliability Applications

In the present paper, a variation of the methode of translation is discussed to generate bivariate or multivariate survival distributions starting a given bivariate or multivariate distribution which is not necessarily a life distribution. The new distribution has been called the dual of the given distribution. The duals of several bivariate and multivariate famílies of distributions are obtained, such as FRECHET'S , FARLIE -GUMBEL -MORGENSTERN'S , MARDIA'S and PLACKETT'S , among others. A necessary and sufficient condition is given for a bivariate distribution to be its own dual. Thus the present paper generates several survival distributions in addition to what are already available in the literature. These have important applications in competing risk theory or reliability of engineering systems.     

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).     

Four Applications of a Bivariate Pareto Distribution

The following model, of “latent structure” type, is considered: in each subpopulation, X and Y are random variables drawn independently from the same exponential distribution, and the parameter of the exponential distribution varies between subpopulations with a Gamma density. Over the whole population, X and Y are then positively correlated, and jointly have a bivariate PARETO distribution. Four examples show how this distribution is useful in analysing ordered contingency tables in which the two dimensions can be regarded as alternative measures of the same thing: the injuries to the two drivers in a road accident, or the severity of a lesion present in a patient as assessed by two physicians, for instance. Two extensions are considered: (a) allowing X and Y to have Gamma distributions, with each subpopulation having the same shape parameter but different scale parameters; (b) allowing the scale parameter for Y to be correlated with the scale parameter for X, rather than being identical to it. A new bivariate distribution with three shape parameters is derived, expressed in terms of a generalised hypergeometric function.     

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.     

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.     

Distribution patterns of flora and fauna in southern Chilean Coastal rain forests: Integrating Natural History and GIS

Knowledge of species richness centers is necessary for the design of conservation areas. In this study, we present a GIS analysis of two years of field data on animal and plant diversity distributions in evergreen, coastal rain forests of southern Chile (39°30′–41°25′ S). Despite their high endemism, these forests have remained largely unprotected. Field records were complemented with data from museum collections and scientific literature. We used selected environmental variables (evapotranspiration, altitude) and, in some cases, forest types as predictors of species distributions. Our study focused on the distribution of forest bryophytes, vascular plants, soil invertebrates, amphibians and birds. We generated distributional maps for each taxa based on their field records in the study area, complemented by natural history information, except in the case of bryophytes and soil invertebrates. In general, species richness was lower at 600 m elevation or above for all the taxa studied. Species richness tends to increase in the northern sector of the study area. We observed a greater richness of vascular plants near rivers and streams, and noted important floristic differences between west and east-facing slopes of the Coastal Range, with more species in the oriental side. Because species in high altitude forests are not a subset of those found at lower elevations, we propose that conservation strategies should prioritize the protection of the entire altitudinal gradient of the southern Coastal Range, especially in the more diverse oriental and northern sectors.     

Positive cooperativity induces multimodal site and thermodynamic affinity distributions in multivalent proteins

The characterization of the stoichiometric and site-affinity distributions for the reaction of hemoglobin with O(2) and CO is presented as an example of a multivalent receptor system which exhibits positive site-site interactions. The distributions of stoichiometric constants, T(i)(K(i))'s, are obtained assuming that the distribution of site constants, N(k), is known. The importance of these distributions is that they can be directly related to quantities measured experimentally and that they represent affinity distributions for each ligation step. In hemoglobin, positive site-site interactions generate both stoichiometric and site-affinity distributions with complex and previously unrecognized multimodal patterns that are very different from the theoretical distributions obtained in the absence of interactions. These distributions are related to the generation of heterogeneity during the ligand binding process. Experimental binding data show that these complex distributions can be related to the physiological functions of uptake, transport, and release of gaseous ligands by hemoglobin.     

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.     

Developing Univariate Distributions from Data for Risk Analysis

The importance of fitting distributions to data for risk analysis continues to grow as regulatory agencies, like the Environmental Protection Agency (EPA), continue to shift from deterministic to probabilistic risk assessment techniques. The use of Monte Carlo simulation as a tool for propagating variability and uncertainty in risk requires specification of the risk model's inputs in the form of distributions or tables of data. Several software tools exist to support risk assessors in their efforts to develop distributions. However, users must keep in mind that these tools do not replace clear thought about judgments that must be made in characterizing the information from data. This overview introduces risk assessors to the statistical concepts and physical reasons that support important judgments about appropriate types of parametric distributions and goodness-of-fit. In the context of using data to improve risk assessment and ultimately risk management, this paper discusses issues related to the nature of the data (representativeness, quantity, and quality, correlation with space and time, and distinguishing between variability and uncertainty for a set of data), and matching data and distributions appropriately. All data analysis (whether “Frequentist” or “Bayesian” or oblivious to the distinction) requires the use of subjective judgment. The paper offers an iterative process for developing distributions using data to characterize variability and uncertainty for inputs to risk models that provides incentives for collecting better information when the value of information exceeds its cost. Risk analysts need to focus attention on characterizing the information appropriately for purposes of the risk assessment (and risk management questions at hand), not on characterization for its own sake.     

Cell-Cycle Analysis Using A Monoclonal Antibody to Brdurd

The flow cytometric measurement of DNA distributions of cells has many applications in biomedical research. Phase fractions estimated (calculated) from such distributions are used to study the growth characteristics of various types of cells, particularly when the cells have been exposed to perturbing agents such as chemotherapeutic drugs. For more than 10 years many methods for resolving DNA distributions into the three cell subpopulations (G₁, S and G₂, + M) have been reported in the literature. A new method of analysis utilizing a monoclonal antibody to bromodeoxyuridine (BrdUrd) has been developed (Gratzner, 1982; Dolbeare et al., 1983) which makes it possible in most cases to accurately determine phase fractions without resorting to mathematical models. the procedure involves the incorporation of BrdUrd by growing (DNA synthesizing) S phase cells, labelling the BrdUrd with a fluorescent monoclonal antibody, and the bivariate measurement of the antibody and of total DNA content, the latter through propidium-iodide staining. the resulting bivariate distributions clearly and simply resolve the three subpopulations. This paper describes the method and illustrates its use in the analysis of various fractions of elutriated exponentially growing Chinese hamster ovary (CHO) cells.     

Bivariate flow karyotyping of human chromosomes: evaluation of variation in Hoechst 33258 fluorescence, chromomycin A3 fluorescence, and relative chromosomal DNA content.

The total variation of chromosome peak positions, in bivariate distributions of Hoechst 33258 and chromomycin A3 fluorescence of 19 healthy individuals, was compared with the experimental variation, determined from 23 bivariate distributions of chromosomes prepared separately from a single cell lineage. The experimental variation in Hoechst and chromomycin fluorescence and the relative chromosomal DNA content were determined from experiments performed over several days. The additional variance contributed by time was the same as the daily variance. The accuracy by which the relative chromosomal DNA content can be calculated from bivariate peak positions was investigated. A least squares method was used to fit the distributions of relative DNA content, obtained, respectively, from mono- and bivariate flow analyses of chromosomes from the same cell lineage. In general the DNA contents match quite well, but for a few chromosomes a difference was found, statistically discernible at the 5% level. The average relative chromosomal DNA content of the chromosomes from the 19 normal individuals, calculated from bivariate peak positions, showed a linear relation with the estimates published by other investigators.     

Use of mixture distributions to deconvolute the behavior of "hits" and controls in high-throughput screening data

The stochastic nature of high-throughput screening (HTS) data indicates that information may be gleaned by applying statistical methods to HTS data. A foundation of parametric statistics is the study and elucidation of population distributions, which can be modeled using modern spreadsheet software. The methods and results described here use fundamental concepts of statistical population distributions analyzed using a spreadsheet to provide tools in a developing armamentarium for extracting information from HTS data. Specific examples using two HTS kinase assays are analyzed. The analyses use normal and gamma distributions, which combine to form mixture distributions. HTS data were found to be described well using such mixture distributions, and deconvolution of the mixtures to the constituent gamma and normal parts provided insight into how the assays performed. In particular, the proportion of hits confirmed was predicted from the original HTS data and used to assess screening assay performance. The analyses also provide a method for determining how hit thresholds--values used to separate active from inactive compounds--affect the proportion of compounds verified as active and how the threshold can be chosen to optimize the selection process.     

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.     

Some Distributions of Induced Records

In this paper we discuss the properties of concomitants of record values. The distributions of the concomitants in the sequence of record values of bivariate distributions are obtained and their possible applications in the areas of biosciences have been pointed out.     

Characterization and Simulation of Uncertain Frequency Distributions: Effects of Distribution Choice,Variability, Uncertainty,and Parameter Dependence

We use bootstrap simulation to characterize uncertainty in parametric distributions, including Normal, Lognormal, Gamma, Weibull, and Beta, commonly used to represent variability in probabilistic assessments. Bootstrap simulation enables one to estimate sampling distributions for sample statistics, such as distribution parameters, even when analytical solutions are not available. Using a two-dimensional framework for both uncertainty and variability, uncertainties in cumulative distribution functions were simulated. The mathematical properties of uncertain frequency distributions were evaluated in a series of case studies during which the parameters of each type of distribution were varied for sample sizes of 5, 10, and 20. For positively skewed distributions such as Lognormal, Weibull, and Gamma, the range of uncertainty is widest at the upper tail of the distribution. For symmetric unbounded distributions, such as Normal, the uncertainties are widest at both tails of the distribution. For bounded distributions, such as Beta, the uncertainties are typically widest in the central portions of the distribution. Bootstrap simulation enables complex dependencies between sampling distributions to be captured. The effects of uncertainty, variability, and parameter dependencies were studied for several generic functional forms of models, including models in which two-dimensional random variables are added, multiplied, and divided, to show the sensitivity of model results to different assumptions regarding model input distributions, ranges of variability, and ranges of uncertainty and to show the types of errors that may be obtained from mis-specification of parameter dependence. A total of 1,098 case studies were simulated. In some cases, counter-intuitive results were obtained. For example, the point value of the 95th percentile of uncertainty for the 95th percentile of variability of the product of four Gamma or Weibull distributions decreases as the coefficient of variation of each model input increases and, therefore, may not provide a conservative estimate. Failure to properly characterize parameter uncertainties and their dependencies can lead to orders-of-magnitude mis-estimates of both variability and uncertainty. In many cases, the numerical stability of two-dimensional simulation results was found to decrease as the coefficient of variation of the inputs increases. We discuss the strengths and limitations of bootstrap simulation as a method for quantifying uncertainty due to random sampling error.     

Spatial order as a source of kinetic cooperativity in organized bound enzyme systems.

When enzyme molecules are distributed within a negatively charged matrix, the kinetics of the conversion of a negatively charged substrate into a product depends on the organization of fixed charges and bound enzyme molecules. Organization is taken to mean the existence of macroscopic heterogeneity in the distribution of fixed charge density, or of bound enzyme density, or of both. The degree of organization is quantitatively expressed by the monovariate moments of charge and enzyme distributions as well as by the bivariate moments of these two distributions. The overall reaction rate of the bound enzyme system may be expressed in terms of the monovariate moments of the charge density and of the bivariate moments of charge and enzyme densities. The monovariate moments of enzyme density do not affect the reaction rate. With respect to the situation where the fixed charges and enzyme molecules are randomly distributed in the matrix, the molecular organization, as expressed by these two types of moments, generates an increase or decrease of the overall reaction rate as well as a cooperativity of the kinetic response of the system. Thus both the alteration of the rate and the modulation of cooperativity are the consequence of a spatial organization of charges with respect to the enzyme molecules. The rate equations have been derived for different types of organization of fixed charges and enzyme molecules, namely, clustered charges and homogeneously distributed enzyme molecules, clustered enzyme molecules and homogeneously distributed charges, clusters of charges and clusters of enzymes that partly overlap, and clusters of enzymes and clusters of charges that are exactly superimposed. Computer simulations of these equations show how spatial molecular organization may modulate the overall reaction rate.     

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.     

Analysis of bivariate flow karyotypes

Bivariate flow karyotype analysis is performed using data from chromosomes stained with two fluorescent dyes, typically chromomycin A3 and Hoechst-33258, and measured in a flow cytometer or cell sorter (Carrano et al.: Proceedings of the National Academy of Sciences of the United States of America 76:1382-1384, 1979; Gray et al.: Proceedings of the National Academy of Sciences of the United States of America 72:1231-1234, 1975; Langlois et al.: Proceedings of the National Academy of Sciences of the United States of America 79:7876-7880, 1982). In the resulting bivariate histogram, most chromosome types appear as individual peaks. In sorting of chromosomes to purify a specific chromosomal type, its corresponding peak in the bivariate histogram is delineated by a rectangular region which surrounds it. All events (objects) that fall within this region trigger the sorting process. In most cases, peaks for different chromosomal types overlap to some extent, and in addition there is always an underlying background due to chromosome fragments and clumps. Thus the sorted population will not be pure; it may include more than one chromosome type and will include debris. To determine the purity of a sort, i.e., the percentage of the sorted material that is of the actual chromosomal type desired, two methods of mathematical analysis have been developed. In the more general method, the bivariate data within an analysis region that includes the sort region, are fit with a series of bivariate Gaussian functions, one for each peak. In a simplified method, the data within the analysis region are transformed into a univariate distribution of either chromomycin A3 or Hoechst-33258 fluorescence. The peaks in these univariate distributions are fit with univariate Gaussian functions. In both methods the purity is determined mathematically. The results of both methods agree well with independent methods of analysis.     

Alu and LINE1 distributions in the human chromosomes: evidence of global genomic organization expressed in the form of power laws

Spatial distribution and clustering of repetitive elements are extensively studied during the last years, as well as their colocalization with other genomic components. Here we investigate the large-scale features of Alu and LINE1 spatial arrangement in the human genome by studying the size distribution of interrepeat distances. In most cases, we have found power-law size distributions extending in several orders of magnitude. We have also studied the correlations of the extent of the power law (linear region in double-logarithmic scale) and of the corresponding exponent (slope) with other genomic properties. A model has been formulated to explain the formation of the observed power laws. According to the model, 2 kinds of events occur repetitively in evolutionary time: random insertion of several types of intruding sequences and occasional loss of repeats belonging to the initial population due to "elimination" events. This simple mechanism is shown to reproduce the observed power-law size distributions and is compatible with our present knowledge on the dynamics of repeat proliferation in the genome.