Contagious Distributions Research Articles

Density Estimation of Corn Rootworm Egg Populations Based upon Frequency of Occurrence

A new approach is developed for the estimation of insect population densities which eliminates the necessity of counting every individual per sample. Although the method was developed for populations of corn rootworm eggs, it seems as applicable to other organisms featuring contagious distributions. Estimation by the proposed method is based upon the proportion (p) of samples containing at least t insects each, where t, termed threshold density, can be any positive integer specified. For example, if t is 1, counting is entirely unnecessary since the only information required is the proportion of samples in which the organism is present. The prediction equation which links mean density to the variables ° and t must be estimated beforehand from a series of samples extending over a wide range of population densities. Expressing the model in terms of logarithms premits the estimation of its parameters by linear least squares. The study indicates that there is an optimal threshold level associated with every population density and specified allowable error. Estimation near these optima is apparently more efficient than conventional estimation involving complete sample enumeration. By enlisting a two—stage sampling scheme, one should consistently be able to approach the optimum threshold and thereby produce estimates comparable in efficiency to direct counts but free from much of the labor.

Pattern analysis of Scleropyll trees aggregated to different degrees

Pattern analysis data are presented for a number of tree species populations sampled from two sites located in dry sclerophyll forest within the Ku-ring-gai Chase park of New South Wales. The distributions proved to be predominantly contagious or random, with regularity occurring only occasionally. Observed variations in the degree of aggregation exhibited by a species were taken into account in interpreting pattern analysis curves. The relation of pattern analysis data to sample quadrat data fitted to known mathematical models is extremely variable, and it is shown that pattern at block sizes other than the one under consideration may suppress the appearance of deviations from randomness at block sizes which do show contagion when sampled with randomly placed quadrats. The possible origins of contagious distributions in eucalypt forest are briefly discussed.

The ecology of a natural population of Queensland fruit fly, Dacus tryoni II. The distribution of eggs and its relation to behaviour

The oviposition of Queensland fruit fly is described and the distribution of eggs within and between fruits is analysed. Olfactory and visual cues direct females to fruit. A brief exploratory phase then follows after which the female applies the ventral surface of her head to areas of fruit surface and then moves forward to probe these same areas with her ovipositor. The female lays her eggs into the fruit either after piercing the cuticle with her ovipositor or through existing holes such as previous oviposition holes or holes made by codling moth larvae. As many as 70% of the eggs laid into hard, shiny fruits were laid into existing holes, but in softer fruits with protuberances over the surface less than 20% were laid in this way. The difference is related to a backward component in the push by the ovipositor, leading to difficulty in piercing hard, smooth unbroken skin. Ovipositions are further grouped on single fruits owing to the favourability of certain areas in relation to such physical factors as wind and illumination. In the laboratory it was not possible under any circumstances to achieve a uniform dispersion of ovipositions when a succession of single females was given the choice of a number of oviposition sites. Contagious distributions were the rule, although randomness of oviposition was exhibited in certain experimental designs. In the field, under natural and experimental conditions, oviposition holes were generally contagiously distributed between fruit. The data offer no support to the hypothesis that oviposition by D. tryoni is affected, in a limiting way, by previous acts of oviposition by other females. An alternative hypothesis for egg limitation at high density, based on aggressive interactions between females on fruit, is suggested.

Studies on structure in plant communities. VI. The significance of pattern evaluation in some Australian dry-land vegetation types

The analysis of density data from a variety of Australian dry-land vegetation types provides evidence of widespread randomness when the component distributions are compared with a Poissonian model. While such randomness may be reflecting traditionally held beliefs concerning structure in colonizing and "relict" populations, randomness may develop as a result of a differential elimination of individuals between high and low density phases in an originally patterned population. It is suggested that structure in vegetation cannot be estimated reliably by simple observation alone, and this statement is evidenced by the ubiquity of random (and to a lesser extent contagious) distributions which are found in Australian dry-land communities, in contradistinction to the widely held view that such distributions are necessarily essentially regular in their character.

Repeated samples from a deep midwater planktonic ostracod community

Ten repeated horizontal hauls with a modified Indian Ocean net fitted with a catch dividing device were taken at a depth of 550 m off Fuertaventura. The planktonic ostracods from these hauls have been examined. Thirty-one species were identified, eleven in average numbers of 10/haul. Coefficients of variation were higher for the individual species than for the total ostracods. The coefficients of variation were substantially reduced if the percentage species compositions were analysed, suggesting that the patchiness of the constituent species is related in part to the patchiness of the overall population. Analyses showed that the distributions were non-random, but it was considered that too few samples had been taken to fit contagious distributions. Correlation coefficients showed that the ostracods were divided into three main groups of species which were inter-correlated at the 1% level. Species diversity indices indicated that the population showed a very close fit to MacArthur's ‘broken stick’ model, which suggests that, despite the effects of diurnal vertical migration, the community was ecologically highly organized and stable. Spearman Rank tests on the order of species abundance in each haul showed that the order of species abundance was a significantly consistent measure of the community sampled. Order of species abundance may, therefore, prove a useful method of analysing plankton communities, even from single plankton samples.

A Method of Analysing Untransformed Data from the Negative Binomial and Other Contagious Distributions

We give a method of analysing data from the Negative Binomial and other generalized Poisson distributions which is free of certain disadvantages that arise when the data are transformed and then subjected to an ordinary analysis of variance. The method of analysis proposed is based on a statistic called XU2 which is constructed from estimators given by Hinz & Gurland (1967 a). This technique can be used for testing a general linear hypothesis relating to untransformed data and it is illustrated here by means of an example based on a two-way layout.

A method of analysing untransformed data from the negative binomial and other contagious distributions

We give a method of analysing data from the Negative Binomial and other generalized Poisson distributions which is free of certain disadvantages that arise when the data are transformed and then subjected to an ordinary analysis of variance. The method of analysis proposed is based on a statistic called XU2 which is constructed from estimators given by Hinz & Gurland (1967 a). This technique can be used for testing a general linear hypothesis relating to untransformed data and it is illustrated here by means of an example based on a two-way layout.

Some Developments in 'Distance Sampling'

Frequency distributions of the distance from a randomly located point to the nearest (or nth nearest) individual plant or animal are described. Results applicable to systematic and contagious (negative binomial) distributions are derived as supplements to the considerable existing literature on randomly distributed individuals. A new 'index of nonrandomness' is suggested, and demonstrated on some previously published data. Efficiency of distance methods relative to plot sampling is considered in terms of number of individuals enumerated.

On a general class of contagious distributions: The Pascal-Poisson distribution

In the paper entitled “A Generalized Class of Contagious Distributions”, Gurland (1958) has presented a brief review of the classical approach to the development of Neyman Type A distribution. By imposing a distribution on one of the intermediate parameters involved, he obtains the generalized class. We have obtained a completely different class of contagious distributions by altering the classical approach. In particular, we have obtained a distribution which arises as a limiting form of the general distribution under certain conditions and which does not seem to have been referred to in the literature. We call this distribution the Pascal-Poisson distribution in view of the form of its p.g.f. We have studied the problem of the fitting of this distribution.

Measurement of non‐randomness in spatial distributions

SummaryThe measurement of departure from randomness in spatial distributions has widespread application in ecological work. Several “indices of non‐randomness” are compared with regard to their dependence on sample number, sample size and density. Criteria for the best choice of index for specific situations are discussed. A new coefficient Cx is proposed for use with positively contagious distributions and tests of significance are given. When Cx and another index (S2/m−1) are used for positive and negative contagion respectively, values ranging from −1 through 0 (random) to +1 are obtained, regardless of sample number, sample size or density.

Correction: Fitting of Some Contagious Distributions to Some Available Data by the Maximum Likelihood Method

Correction: Fitting of Some Contagious Distributions to Some Available Data by the Maximum Likelihood Method

Fitting of Certain Contagious Distributions to Some Available Data by the Maximum Likelihood Method

Several distributions have been proposed for fitting to a large class of biological data which have been characterized as being contagious. Two of the earliest distributions used in fitting this type of data are the Negative Binomial and the Neyman Type A (cf. Greenwood and Yule [1920] and Neyman [1939]). More recently the Poisson Binomial and the Poisson with zeroes have been suggested (cf. McGuire et al. [1957] and A. C. Cohen [1960]). The selection of the best distribution for a particular set of data is complicated by the interrelations of these distributions and their occasionally ambiguous relationship to the models used in formulating them. Some of the distributions are limiting forms of the others for extreme parameter values. For example, the Neyman Type A and the Negative Binomial are limits of the Poisson Binomial as n -co and n -0 respectively, while both the Poisson and Poisson with zeroes are limiting forms of the Neyman Type A. Feller [1943] and Gurland [1958] have shown that different models can be used in deriving the same distributions. Apparently because of the complexity of the N.A. distribution, it is often fitted by the method of moments, and then compared with the negative binomial distribution which is fitted by the maximum likelihood method (c.f. Bliss [1953] and McGuire, et al. [1957]). This results in the confounding of the fitting process with the comparison of the distributions. The same is true about the fitting of the Poisson Bionomial. We note for reference that certain simplifications have been obtained by Douglas [1955] for the Neyman Type A fitting and by Sprott [1958] for the Poisson Binomial fitting. In spite of the simplification of various methods of fitting some of the contagious distributions the task of setting up computer routines or of using desk calculators to fit these distributions is relatively tedious. This is particularly true when the investigator is not directly concerned with the distributions fitted in this paper but wishes to use them for a comparison with some new distribution. In view of this, it was decided to set up programs

A Model for the Kinetics of Phage Attachment to Bacteria in Suspension

In this paper a model is proposed for some aspects of the kinetics of phage attachment to bacteria in suspension. The model proposed is based on a construction of a set of differential equations describing the rates of change of concentrations of the various possible particles, each consisting of a bacterium with i (0 < i < r) attached phage particles. The recursive solution of this set of equations yields a discrete distribution of the bacteria according to the number of phage particles attached to each of them. This distribution, although it is timedependent and has a finite set of values, is similar to Greenwood-YulePolya's [1920, 1930] Negative Binomial. Contagious distributions have been classified by Gurland [1958], Beall and Rescia [1953] and others, mainly, through the use of compounding procedures. The fact that our assumptions lead to a hypothesis of true contagion, as defined by Feller [1943], has suggested the desirability of direct derivation of the descriptive equations. Certain assumptions regarding the parameters permit the formulation of simplified equations among which the non-degenerate ones are similar to the Binomial, the Geometric and the Poisson distributions. Experimental data will be given in a future paper in which fitting the distribution will be discussed.

A fitting procedure for some generalized poisson distributions

Abstract Many of the contagious distributions considered in the biological sciences are members of the generalized Poisson family. Four distributions which belong to this family and have been used frequently are the Negative Binomial (cf. Bliss [2]), Neyman Type A (cf. Beall and Rescia [1]), Poisson Binomial (cf. McGuire et al. [10]) and the generalized Polya-Aeppli (cf. Skellam [14]).

Some applications of the negative binomial and other contagious distributions.

Some applications of the negative binomial and other contagious distributions.

Numerical studies on some contagious distributions.

A numerical investigation of three contagious distributions characteristic of epidemic data has been made using an electronic computer.All three show a modified threshold effect such that the proportion of a community affected tends to 100% as its size increases though there is no abrupt change of form.We would like to thank Mr M. J. R. Healy for assistance with one of the programmes, and Miss Molly Fretton for her help with the computing.

A Generalized Class of Contagious Distributions

Frequently the data arising in studies of entomology and bacteriology cannot be described by the usual distribution functions but rather by some type of contagious distribution. As Feller [2] has pointed out there are essentially two kinds of contagion involved in such distributions. One type, true contagion, is due to the fact that each event increases (or decreases) the probability of succeeding favorable events. The other type, apparent contagion, is due to an inhomogeneity of the population. Some distributions, such as the negative binomial, can apparently be interpreted on the basis of both types of contagion. In a paper by Neyman [3] a class of contagious distributions is derived from a certain biological model which takes into account the fact that the distribution of larvae over the plots of a field depends upon the fact that the larvae are hatched from egg-masses which appear at random over the field. This class of distributions has been successful in accounting for the distribution of some insect populations (cf. Beall [21]). The negative binomial has also been found useful as a possible underlying distribution for insect populations (cf. Bliss and Fisher [23]). Contagious distributions have also been used in the study of accident and medical statistics (cf. Dubourdieu [24], Greenwood and Yule [14], Lundberg [25], Newbold [261.) A family of contagious distributions which provides a generalization of Neyman's types A, B, C, was recently presented by Beall and Rescia [1]. The p.g.f. (probability generating function) was shown to be

Particulate Iron in Offshore Waters of the Panama Bight and in the Gulf of Panama

Particulate iron is reported from replicate samples taken at two depths at several locations in the Panama Bight during November 1955, and in replicate samples taken at two depths at a single inshore location in the Gulf of Panama at bimonthly intervals from December 1954 through December 1955. In addition, a few samples were taken at other locations in the Gulf of Panama and from a river flowing into the Gulf. The content of particulate iron, both in the Panama Bight and in the Gulf, shows highly contagious distributions, necessitating a logarithmic transformation to make the data amenable to statistical analysis. All samples from the Panama Bight could have arisen from a single log‐normal distribution with a mean of 12.2 µg/L. In the Gulf of Panama, there is significantly higher iron in the surface (115.2 µg/L) than at 30 meters (62.1 µg/L) during the early part of the year when the standing crops of plankton organisms are high. Particulate iron content is higher inshore in the Gulf of Panama than offshore in the Panama Bight, and two samples from the Rio Escota, containing 692 and 863 µg/L, were much higher than in the Gulf, indicating that possibly the iron is being brought in by runoff. From comparison of iron content in inshore and offshore waters, in both of which biological production is high, it is inferred that particulate iron cannot be a limiting factor on phytoplankton growth in the inshore waters.

Effect of Area of Search on Reproduction of Encarsia formosa Gahan (Hymenoptera: Chalcidoidea)

In an earlier laboratory study (Burnett, 1958) influences of various types of host distribution on the efficiency of the chalcid parasite Encarsia formosa Gahan were determined. A constant number of parasites searched among five host densities for scales of the greenhouse whitefly, Trialeurodes vaporariorum (Westw.), that were distributed in three patterns over a 25-square-inch surface. In addition to the increased parasitism resuItIng from contagious distributions of the host, it was found that approximately the same percentage of each of the five different host densities within the same distribution was attacked by the parasite.

The Method of Maximum Likelihood Applied to the Poisson Binomal Distribution

Much has been written about the application of contagious distributions to plant and insect populations, for a discussion of the methods and a review of the literature see McGuire, Brindley, and Bancroft [1957]. Three distributions considered in the past are the negative binomial, the Neyman Type A, and the Poisson binomial. The method of maximum likelihood has been used to fit the negative binomial, as described by Fisher and Bliss [1953], and to fit the Neyman Type A, as described by Douglas [1955], and by Shenton [1949]. It is the purpose of this paper to describe a procedure for fitting the Poisson binomial distribution by the method of maximum likelihood, and to consider the efficiencies of the method of moments and the method of sample zero frequency. Since the Neyman Type A distribution is a limiting form of the Poisson binomial, the formulae developed here will include as special cases some of the formulae developed previously for the Neyman Type A.