Why Are Proportions Important?

Inferences Regarding Proportions

WHY ARE PROPORTIONS IMPORTANT?

Chapter 9 covered statistical inferences with variables that represented interval or ratio-level measurement. Now we will discuss inferences with another type of variable—a proportion, which was introduced in Chapter 5. Let us review some of the terminology regarding variables, including a random variable, continuous and discrete variables, and binomial variables.

A random variable is a type of variable for which the specific value of each observation is determined by chance. For example, the systolic blood pressure measurement for each patient is a random value. Variables can be categorized further as continuous or discrete. Continuous variables can have an infinite number of values within a specified range. For example, weight is a continuous variable because it al-ways can be measured more precisely, depending on the precision of the measurement scale used. Discrete variables form data that can be arranged into specific groups or sets of values, e.g., blood type or race.

Bernoulli variables are discrete random variables that have only two possible values, e.g., success or failure. The binomial random variable is the number of suc-cesses in n trials. It can take on integer values from 0 to n. Let n = the number of ob-jects in a sample and p = the population proportion of a binomial characteristic, also known as a “success,” i.e., the proportion of successes; then, 1 – p = the proportion of failures. There are numerous examples of medical outcomes that represent bino-mial variables. Also, sometimes it is convenient to create a dichotomy from a con-tinuous variable. For example, we could look at the proportion of diabetic patients with hemoglobin A1C measurements above 7.5% versus the proportion with hemo-globin A1C below 7.5%.

Proportions are very important in medical studies, especially in research that uses dichotomous outcomes such as dead or alive, responding or not responding to a drug, or survival/nonsurvival for 5 years after treatment for a disease. Another example is the use of proportions to measure customer or patient satisfaction, for measures that have dichotomous responses: satisfied versus dissatisfied, yes/no, agree/disagree.

For example, a manufacturer of drugs to treat diabetes studied patients, physi-cians and nurses to see how well patients complied with their prescribed treatment and to see how well they understood the chronic nature of the disease. For this sur-vey, proportions of patients who gave certain responses in particular subgroups of the population were of primary interest. To illustrate, the investigators queried sub-jects as to complications from type II diabetes. Respondents’ knowledge about each type of complication—renal disease, retinopathy, peripheral neuropathy—was scored according to a yes/no format.

At medical device companies, the primary endpoint may be the success of a par-ticular medical or surgical procedure. The proportion of patients with successful out-comes may be a primary endpoint and that proportion for the treatment group may be compared to a proportion for a control group (i.e., a group that receives either a place-bo no treatment, or a competitor’s treatment). The groups receiving the treatment from a sponsoring company are generally referred to as the treatment groups and the group receiving the competitor’s treatment are called the active control groups. The term active control distinguishes them from control groups that receive placebo.

The sample proportion of successes is the number of successes divided by the number of patients who are treated. If we denote the total number of successes by S, then the estimated proportion pˆ = S/n, where n is the total number of patients treat-ed. This proportion in a clinical trial can be viewed as an estimate of a probability, namely, the probability of a success in the patient population being sampled. Detailed examples from clinical trials will be discussed later in this chapter.

The binomial model is usually appropriate for inferences that involve the use of clinical trial outcomes expressed as proportions. We can assume that patients have been selected randomly from a population of interest. We can view the success or failure of each patient’s treatment as the result of a Bernoulli trial with success probability equal to the population success probability p. In Chapter 5, a Bernoulli distribution was defined as a type of probability distribution associated with two mutually exclusive and exhaustive outcomes. Each patient can be viewed as being independent of the other patients. As we discussed in Section 5.6, the sample number of success-es out of n patients then has a binomial distribution with parameters n and p.