Introduction to Probability Theory and Sampling Distributions

tatistical inference allows one to draw conclusions about the characteristics of a population on the basis of data collected from a sample of subjects from that population. Almost all the statistical inferences typically seen in the medical literature are based on probability models that connect summary statistics calculated using the observed data to estimates of parameter values in the population. This article will cover the basic principles behind probability theory and examine a few simple probability models that are commonly used, including the binomial, normal, and Poisson distributions. We will then see how sampling distributions are used as the basis for statistical inference and how they are related to simple probability models. Thus, this article forms the foundation for future articles in the series that will present the details of statistical inference in particular clinical situations. Making medical decisions on the basis of findings from various radiologic diagnostic tools is an everyday occurrence in clinical practice. In radiologic research, one often needs to draw conclusions about the relative performance of one diagnostic tool compared with another for the detection of a given condition of interest. Both of these tasks depend, in large part, on probability theory and its applications. In diagnosis, we are interested in calculating the probability that the condition of interest is present on the basis of results of a radiologic test. This probability depends on how sensitive and specific that test is in diagnosing the condition and on the background rate of the condition in the population. This calculation largely depends on a result from probability called Bayes’ theorem. Similarly, all statistical inferences, whether comparisons of two proportions representing diagnostic accuracies from two instruments or inferences from a more complex model, are based on probabilistic reasoning. Therefore, a thorough understanding of the meaning and proper interpretation of statistical inferences, crucial to daily decision making in a radiology department, depends on an understanding of probability and probability models. This article is composed of three main parts. We begin with an introduction to probability, including the definitions of probability, the different schools of thought about the interpretation of probabilities, and some simple examples. We continue by defining conditional probabilities and present Bayes’ theorem, which is used to manipulate conditional probabilities. The most common simple probability models, including the binomial, normal, and Poisson distributions, are presented next, along with the types of situations in which we would be most likely to use them. Finally, sampling strategies are examined. Armed with these basics of probability and sampling, we conclude with a discussion of how the outcome of interest defines the model parameter on which to focus inferences and how the sampling distribution of the estimator of that parameter enables valid inferences from the data collected in the sample about the population at large.