Introduction

Counting in one form or another likely predates written human history. In modern society, counting is ubiquitous in many aspects of individual and collective human lives. One often-asked question of an individual is how many children are in his or her family. School systems are concerned with the related question of how many children are in a class or in an entire district, though the range of the counts in these cases is usually entirely different. A characteristic common to all counting systems is that counts are never less than zero, and some of these systems have maxima that may be treated as infinite: e.g., the number of grains of sand on all the beaches of the Earth, or the number stars in the observable universe. Some finite counting examples are the number of defects on an airplane, the number of radioactive decay products in a particle accelerator collision, or the number of students eligible for graduation term by term. Sports are teeming with examples of counting including the score from a round of golf, baseball's runs batted-in (RBI), and the number of goals averted in football. These examples demonstrate that count data permeate everyday human life.

The counting examples above suggest that count data have different types. The number of children in a nuclear family may be considered a tightly restricted range from zero to, say, twenty. However, the number of children in a school class is never zero. Batters on a baseball team may have a large number of players with no (zero) RBIs. The day-to-day sales of automobiles may range from zero and upwards based on a popular probability distribution such as the Poisson probability distribution or the negative binomial probability distribution. Count data statistical models rely on the probability distribution chosen for the parameter estimation method and reliability of the analysis outcomes.

Regardless of which probability distribution model is used to describe count data, one critical property of each model is that the counts are not transformed to approximate a normal distribution to allow the use of ordinary least squares estimation. Rather, the count distribution mean and variance are used to establish a nonlinear model link between the count response and the selected predictors.

Introduction

Model types vary widely depending on the type of data under consideration and the purpose of an investigation or analysis. In this chapter we give a short introduction to the statistical models used in this handbook. The reader may go directly to the chapter on how to use the model of interest. However, as the reader becomes more adept in model application, this chapter will be useful for acquiring additional model knowledge, and for finding literature to enhance the flexibility of and appreciation for the various models discussed.

Models, either empirical or mechanistic, allow us to gain insight into usually complicated processes. Models are used to represent particular aspects of a hypothesis, study subject, or process. As examples, a toy airplane represents the overall concept of a type of flying machine as it has the general shape and appearance of an aircraft. A hobby plane may represent intricate appearance details of a specific plane, but does not fly. A flying model may have the general appearance of a particular aircraft, but it generally lacks details of actual airplane appearance and instead is optimized specifically to fly. Then there are simulators which do not fly but allow for a facsimile of flight with performance characteristics intended to mimic the actual flying machine to train pilots.We see from these examples that each model has a different purpose, each with different relative advantages and disadvantages.

Models, then, have deficiencies relative to the object or process they represent. However, models do provide insight into a particular aspect of these objects or processes, and thereby they are useful. Models of gravitational attraction are mechanistic as nature dictates the functional relationship between, say, two masses. Stochastic or empirical models are often used to discover functional relationships among variables. They are used to provide summary information about sets of data. While it is possible to examine each observation in a data set, as the size of the data set increases, it is evermore difficult to assimilate the information held by each datum. A model matched to the data set can reveal processes and functional relationships otherwise inaccessible to individual observational examinations.