2. Frequentists and Bayesians

This chapter explains some elementary statistical concepts, including the distinction between a statistical estimator computed from data and the parameter that is being estimated. The process of making inferences from data will be discussed, including the importance of accounting for variability in data and one’s uncertainty when making statistical inferences. A detailed account of binomial confidence intervals will be presented, including a brief history of how Gauss, DeMoivre, and Laplace established important ideas that still are relevant today. The relationship between sample size and statistical reliability will be discussed and illustrated. I will introduce Bayesian statistics, which treats parameters as random quantities and thus is fundamentally different from frequentist statistics, which treats parameters as fixed but unknown. Graphical illustrations of posterior distributions of parameters will be given, including posterior credible intervals, illustrations of how a Bayesian analysis combines prior knowledge and data to form posterior distributions and make inferences, and how reliability improves with larger sample size. An example will be given of how being biomarker positive or negative may be related to the probability that someone has a particular disease.