On the Use of the Negative Binomial in Epidemiology

The use of the negative binomial, or the method of inverse sampling (e.g. (1), (2)), is suggested for prospective studies. This method usually has the advantage of fostering an early decision on the statistical significance of some etiological factor in examining successive patient records. Several limiting situations for the distribution are discussed regarding the prevalence of a disease. An example illustrates the method.     

The Structure of Compounded Trivariate Binomial Distributions

A family of trivariate binomial mixtures with respect to their exponent parameter is introduced and its structure is studied by the use of probability generating functions. Expressions for probabilities, factorial moments and factorial cumulants are given. Conditional distributions are also examined. Illustrative examples include the trivariate Poisson, binomial, negative binomial and modified logarithmic series distributions. In addition, properties of the compounded trivariate Poisson distribution are discussed. Finally biological, medical and ecological applications are indicated.     

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 Non-Central Negative Binomial Distribution

The non-central negative binomial distribution is derived by mixing the POISSON distribution with a certain BESSEL function distribution, or the negative binomial distribution with the POISSON distribution. Some properties of the distribution are discussed, including a characterization. The fitting of the distribution to some data are given together with the efficiency for various procedures of estimation.     

The Negative Binomial Distribution of Order k and Some of Its Properties

The negative binomial distribution of order k is introduced and briefly studied. First it is shown that it is a proper probability distribution. Then its probability generating function, mean and variance are derived. Finally it is shown that the number of trials until the rth kth consecutive success (r ≧ 1, k ≧ 1) in independent trials with constant success probability p (0 < p < 1) is distributed as negative binomial distribution of order k. The present paper generalizes results of SHANE (1973), PHILIPPOU and MUWAFI (1982), and PHILIPPOU, GEORGHIOU and PHILIPPOU (1982).     

A Stochastic Process and Generalized Distributions for the Study of Oviposition Evolution of a Parasite

This paper considers a generalized birth process {X^m(t), t > 0} and presents a new stochastic model for the number of eggs laid by a parasite on a host. Also, given an underlying distribution for the number of visits between parasites and a host, this distribution is generalized by the distribution of the number of eggs per visit laid on the host. If a certain number of eggs are already present on the host, a parasite such as a Japanese weevil, may avoid oviposition in subsequent visits (see JANARDAN (1980)) to the same host. A class of generalized distributions are presented to model such situations. The case of a single egg laying parasite and a Poisson distribution for the number of visits of the parasite to the same host yields a distribution of particular interest. In order to develop this model, certain lemmas are derived. Finally a characteristic property of this stochastic model is presented.     

Some Approximations of the Borel-Tanner or the Generalized Poisson Distribution

This paper considers some approximations for the Borel-Tanner (Generalized Poisson) sums by using (i) Gram-Charlier Poisson expansion, (ii) Mixture of two Poisson distributions, (iii) Variance stabilizing technique, and (iv) negative binomial distribution. It has been found that the approximation obtained by using the negative binomial distribution seems to be more efficient than the other approximation.     

On designing the capacity of a reservoir

Two methods of computing the capacity of a reservoir are discussed. In the first method suppose that inputs into the reservoir during successive periods of fixed lengths are independently and identically distributed and that the supplies or releases from it are at the unit rate. Assume that the input is a scarce commodity and that the input rate is less than the supply rate. We then use the mean emptiness period of the reservoir for computing its capacity. In particular the capacity has been determined for geometric, binomial and exponential inputs. The second method is that of simulation for gamma type Markov autoregressive inputs and content-dependent supplies.     

On the efficiency of score tests for homogeneity in two-component parametric models for discrete data

Summary In many applications of two-component mixture models for discrete data such as zero-inflated models, it is often of interest to conduct inferences for the mixing weights. Score tests derived from the marginal model that allows for negative mixing weights have been particularly useful for this purpose. But the existing testing procedures often rely on restrictive assumptions such as the constancy of the mixing weights and typically ignore the structural constraints of the marginal model. In this article, we develop a score test of homogeneity that overcomes the limitations of existing procedures. The technique is based on a decomposition of the mixing weights into terms that have an obvious statistical interpretation. We exploit this decomposition to lay the foundation of the test. Simulation results show that the proposed covariate-adjusted test statistic can greatly improve the efficiency over test statistics based on constant mixing weights. A real-life example in dental caries research is used to illustrate the methodology.     

Transmission dynamics of lymphatic filariasis: density-dependence in the uptake of Wuchereria bancrofti microfilariae by vector mosquitoes

Gaining a better understanding of parasite infection dynamics in the vector mosquito (Diptera: Culicidae) population is central to improving knowledge regarding the transmission, persistence and hence control of lymphatic filariasis. Here, we use data on mosquito feeding experiments collated from the published literature to examine the available evidence regarding the functional form of the first component of this parasite-vector relationship for Wuchereria bancrofti (Filarioidea: Onchocercidae) causing Bancroftian filariasis, i.e. the rate of microfilariae (mf) uptake from the blood of infected humans by the feeding mosquito vector. Using a simple logarithmic regression model for describing the observed relationships between the mean numbers of mf ingested per mosquito and parasite load in humans in each study, and a linear mixed-effects meta-analytical framework for synthesizing the observed regressions across studies, we show here for the first time clear evidence for the existence of density-dependence in this process for all the three major filariasis transmitting mosquito vectors. An important finding of this study is that this regulation of mf uptake also varies significantly between the vector genera, being weakest in Culex, comparatively stronger in Aedes and most severe and occurring at significantly lower human mf loads in Anopheles mosquitoes. The analysis of the corresponding mf uptake prevalence data has further highlighted how density-dependence in mf uptake may influence the observed distributions of mf in vector populations. These results show that whereas strong regulation of mf uptake, especially when it leads to saturation in uptake at low human parasite intensities, can lead to static distributions of mf per mosquito with host parasite intensity, a weaker regulation of mf ingestion can give rise to changes in both mean mf loads and in the frequency distribution of parasites/mosquito with increasing human parasite intensity. These findings highlight the importance of considering local vector infection dynamics when attempting to predict the impacts of community-based filariasis control. They also emphasize the value of developing and applying robust meta-analytic methods for estimating functional relationships regarding parasitic infection from population ecological data.