A random variable is a type of variable for which the specific value of each observation is determined by chance.

**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 *p*ˆ = *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*.

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