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1 A Brief Introduction and Overview of Applied Statistics

      CHAPTER OBJECTIVES

      The goal of this first chapter is to provide a global overview of the logic behind statistical inference and how it is the basis for analyzing data and addressing scientific problems. Statistical inference, in one form or another, has existed at least going back to the Greeks, even if it was only relatively recently formalized into a complete system. What unifies virtually all of statistical inference is that of probability. Without probability, statistical inference could not exist, and thus much of modern day statistics would not exist either (Stigler, 1986).

When we speak of the probability of an event occurring, we are seeking to know the likelihood of that event. Of course, that explanation is not useful, since all we have done is replace probability with the word likelihood. What we need is a more precise definition. Kolmogorov (1903–1987) established basic axioms of probability and was thus influential in the mathematics of modern-day probability theory. An axiom in mathematics is basically a statement that is assumed to be true without requiring any proof or justification. This is unlike a theorem in mathematics, which is only considered true if it can be rigorously justified, usually by other allied parallel mathematical results. Though the axioms help establish the mathematics of probability, they surprisingly do not help us define exactly what probability actually is. Some statisticians, scientists and philosophers hold that probability is a relative frequency, while others find it more useful to consider probability as a degree of belief. An example of a relative frequency would be flipping a coin 100 times and observing the number of heads that result. If that number is 40, then we might estimate the probability of heads on the coin to be 0.40, that is, 40/100. However, this number can also reflect our degree of belief in the probability of heads, by which we based our belief on a relative frequency. There are cases, however, in which relative frequencies are not so easily obtained or virtually impossible to estimate, such as the probability that COVID-19 will become a seasonal disease. Often, experts in the area have to provide good guesstimates based on prior knowledge and their clinical opinion. These probabilities are best considered subjective probabilities as they reflect a degree of belief or disbelief in a theory rather than a strict relative frequency. Historically, scholars who espouse that probability can be nothing more than a relative frequency are often called frequentists, while those who believe it is a degree of belief are usually called Bayesians, due to Bayesian statistics regularly employing subjective probabilities in its development and operations. A discussion of Bayesian statistics is well beyond the scope of this chapter and book. For an excellent introduction, as well as a general introduction to the rudiments of statistical theory, see Savage (1972).

      When you think about it for a moment, virtually all things in the world are probabilistic. As a recent example, consider the COVID-19 pandemic of 2020. Since the start of the outbreak, questions involving probability were front and center in virtually all media discussions. That is, the undertones of probability, science, and statistical inference were virtually everywhere where discussions of the pandemic were to be had. Concepts of probability could not be avoided. The following are just a few of the questions asked during the pandemic: