FITTING THE NEGATIVE BINOMIAL DISTRIBUTION TO BIOLOGICAL DATA AND NOTE ON THE EFFICIENT FITTING OF THE NEGATIVE BINOMIAL

In studying the occurrence of plants and animals in nature, the number of individuals may be counted in each of many equal units of space or time. The original counts can be summarized in a frequency distribution, showing the number of units containing x = 0, 1, 2, 3, ... individuals of a given species. If every unit in the series were exposed equally to the chance of containing the organism, the distribution would follow the Poisson series, each unit having the population mean as its expected frequency. It is easy to test whether the variation in the number of individuals per unit agrees with this hypothesis. Since the expected variance of a Poisson distribution is equal to its mean, the observed variance s2, multiplied by the degrees of freedom n, may be divided by the sample mean x to obtain x2 = ns2/x. More often than not x2 is significantly larger than its expectation, not only in distributions of plants and animals in nature but even in the laboratory. A number of distributions have been devised for series in which the variance is significantly larger than the mean (2, 11, 21), frequently on the basis of more or less complex biological models. In the present paper this characteristic will be called "over dispersion". Perhaps the first of these was the negative binomial, which arose in deriving the Poisson series from the point binomial (27, 32) although it had been formulated in 1714 (2). Comparisons of expected and observed distributions have shown its wide applicability to biological data. The relative ea-se with which the negative binomial can be computed and

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