Abstract
This paper proposes new linear regression models to deal with overdispersed binomial datasets. These new models, called tilted beta binomial regression models, are defined from the tilted beta binomial distribution, proposed assuming that the parameter of the binomial distribution follows a tilted beta distribution. As a particular case of this regression models, we propose the beta rectangular binomial regression models, defined from the binomial distribution assuming that their parameters follow a beta rectangular distribution. These new linear regression models, defined assuming that the parameters of these new distributions follow regression structures, are fitted applying Bayesian methods and using the OpenBUGS software. The proposed regression models are fitted to an overdispersed binomial dataset of the number of seeds that germinate depending on the type of chosen seed and root.
Highlights
The binomial distribution is normally used to model the number of successes obtained in a finite number of experiments
In these cases, it is often found that the variance of the response variable Y exceeds the theoretical variance of the binomial distribution
This phenomenon, known as extra-binomial variation, can lead to underestimation errors, lost efficiency of estimates and underestimation of the variance, wich that can in turn generate incorrect inferences about the regression parameters or the credible intervals (Collet, 1991; Cox, 1983; Williams, 1982)
Summary
The binomial distribution is normally used to model the number of successes obtained in a finite number of experiments. Hahn and Lopez Martın (2015) introduced tilted beta distribution, which has as particular cases the beta rectangular and the beta distributions. We generalize the beta binomial regression models for fitting overdispersed binomial count dataset (Cepeda-Cuervo and Cifuentes-Amado, 2017) by introducing the tilted beta binomial linear regression model. The tilted beta binomial probability is defined by assuming that the parameter of the binomial distribution follows the mean tilted beta distribution. We present a reparameterization of the tilted beta distribution proposed by Hahn and Lopez Martın (2015), in terms of the mean and the dispersion parameters of the beta distribution μb and φ, respectively and the mean of the tilted beta distribution μt.
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