Bradford distribution and its application in modeling medical data: a suitable alternative to distributions defined on the unit interval.

Abstract

It has been observed that quite often, in health-related studies/clinical trials data, we frequently come across left/right skewed heavy-tailed survival data for which the available probability models from the existing literature are not adequate. In this article, we explore and study a new probability distribution with a bounded domain, namely the Bradford distribution with a single parameter θ, henceforth, in short BF(θ). For this distribution, several tractability advantages (in particular, for modeling medical data) have not been explored, to date, to the best of the knowledge of the author. The BF(θ) distribution represents the advantage of not involving any special function in its formulation. We provide a comprehensive study of general mathematical and statistical properties of this distribution that are desirable traits when fitting clinical trials/health data among many others. The utility of this distribution is highlighted via the application/(re-analysis of) in two different well-known datasets-(i) Arthritis pain relief data, and (ii) Tissue damage dataset for two different objectives. Using information theoretic criteria, we compare the performance of the BF(θ) distribution with several rival probability models defined on the unit interval, such as the beta, Kumaraswamy, bounded-weighted exponential, unit Weibull, and many more. The usefulness of the current paper is manifold, and can be summarized as follows: First, the BF(θ) distribution cannot be obtained via some routine transformation from any of the well-known absolutely continuous univariate random variables, and therefore, it augments the existing literature on statistical distribution theory.The BF(θ) distribution has simple and analytically tractable forms for the associated density and quantile function. Since the quantile function is in a closed form, this distribution can be utilized to model censored data which are an important feature. Specifically, this distribution has a clear advantage over the class of univariate distributions defined on the unit interval and that is, it has only one parameter. Needless to say, the associated inferential measures, such as the MLE (discussed in Section 3), are very easy to implement.Among several existing univariate absolutely continuous probability models, one of the major limitations is that they do not perform satisfactorily well for small samples, because of the presence of two or more model parameters. In contrast, the BF(θ) distribution appears to be working well in the case of small samples (such as for a sample of size 12, see Section 4.2). The above set of arguments warrant for a separate and detailed study for the BF(θ) distribution.