Hesitograms

Hesitograms are a theoretical concept in probability and statistics, representing a probability distribution that is not continuous or discrete but rather a mixture of both. They arise in situations where an observation can take on a continuous range of values or can land exactly on a specific value with a non-zero probability.

The term "hesitogram" was coined by Dr. Donald Berry in the context of medical research, particularly in cancer clinical trials. It describes a variable that might have a continuous distribution, like tumor size, but also has a chance of being precisely zero, indicating the complete absence of the disease. This "hesitation" at zero is what gives the distribution its name.

Mathematically, a hesitogram can be represented as a sum of a continuous distribution function and a discrete probability mass at one or more specific points. For example, a hesitogram could be composed of a uniform distribution over an interval and a discrete probability of 0.2 at the value of zero.

The analysis of hesitograms requires statistical methods that can accommodate both continuous and discrete components. Standard techniques for purely continuous or discrete data may not be appropriate. Special regression models and hypothesis testing procedures are often employed to model and interpret data exhibiting this mixed distribution. While the term is not universally adopted, the concept of mixed distributions is fundamental in various applied fields.