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This article is motivated by the challenge of analysing an agricultural field experiment with observations that are positive on a continuous scale or zero. Such data can be analysed using two-part models, where the distribution is a mixture of a positive distribution and a Bernoulli distribution. However, traditional two-part models do not include any dependencies between the two parts of the model. Since the probability of zero is anticipated to be high when the expected value of the positive part is low, and the other way around, this article introduces dependency-extended two-part models. In addition, these extensions allow for modelling the median instead of the mean, which has advantages when distributions are skewed. The motivating example is an incomplete block trial comparing ten treatments against weed. Gamma and lognormal distributions were used for the positive response, although any density on the support of real numbers can be accommodated. In a cross-validation study, the proposed new models were compared with each other and with a baseline model without dependencies. Model performance and sensitivity to choice of priors were investigated through simulation. A dependency-extended two-part model for the median of the lognormal distribution performed best with regard to mean square error in prediction. Supplementary materials accompanying this paper appear online.

Bayesian Analysis of Nonnegative Data Using Dependency-Extended Two-Part Models