Regression Analysis of Quantity Data with Exact Zeroes

Measurements of the magnitude or duration of physical phenomenon have the property that they are positive and continuous, except for the possibility of exact zeroes when the phenomenon does not occur. Such data cannot be transformed to normality by power transformations or any other means, and special treatment of the zero observations is usually required. The approach of this paper is to model quantity data using a family of exponential family distributions intermediate between the Poisson and the gamma families. These families have the feature of power mean-variance relationships with exponent between one and two. Regression modelling is possible using the established framework of generalized linear models. With suitable assumptions this approach allows the information in both the zero and positive observations to contribute to the estimation of all parts of the model.