Applied Biostatistics for the Health Sciences. Richard J. Rossi

risk factor is present, and the larger the odds ratio is the more likely it is for an individual to have the disease when the risk factor is present.

      Note that the odds ratio takes on values between 0 and infinity, it is equal to 1 when the disease is independent of the risk factor, it is larger than 1 when the disease is more likely when the risk factor is present, and it is less than 1 when the disease is less likely when the risk factor is present.

       Example 2.28

      In a retrospective study of the health problems associated with smoking, a researcher might be interested in the relationship between whether an individual has lung cancer and whether or not the individual smokes more than 20 cigarettes a day. In this case, the risk factor is whether an individual smokes more than 20 cigarettes a day, and the odds ratio is

      In this case, the odds ratio provides important information on whether or not smoking 20 or more cigarettes per day is associated with the incidence of lung cancer.

      Now suppose, the odds that an individual having lung cancer given the individual smoked at least 20 cigarettes a day is 0.25, and the odds that an individual having lung cancer given the individual smoked fewer than 20 cigarettes a day is 0.08. Then, the odds ratio is OR=0.250.08=3.125, and thus, an individual who smokes 20 or more cigarettes a day is 3.125 times as likely to have lung cancer than an individual who smokes fewer than 20 cigarettes a day.

       Example 2.29

      In the article “Videofluoroscopic evidence of aspiration predicts pneumonia and death but not dehydration following stroke” published in Dysphagia (Schmidt, et. al, 1994), the authors reported the results of a retrospective study where the odds of developing pneumonia for those aspirating thickened liquids or more solid consistencies of 1, and an odds of developing pneumonia for those not aspirating of 0.178. Based on the reported odds, the odds ratio for developing pneumonia for those aspirating thickened liquids or more solid consistencies is OR=10.178=5.62. Thus, an individual aspirating thickened liquids or more solid consistencies was 5.62 times as likely to develop pneumonia as an individual not aspirating.

      Some final notes on the relative risk and odds ratio are listed below.

       The relative risk can only be computed in prospective studies such as clinical trials and cohort studies.

       The odds ratio can be computed for both prospective and retrospective studies.

       The relative risk and odds ratio always agree on being equal to one, greater than one, or less than one.

       When the risk of a disease is small, the odds ratio is approximately the same as the relative risk; however, the relative risk and odds ratio can be very different when the risk of a disease is large.

       Example 2.30

In a prospective study on a particular condition there is a treatment group and a control group with each group having 100 subjects. The treatment group receives the treatment and the control group receives a placebo. Suppose 12 subjects in the treatment group were cured, and in the control group four were cured. Based on this study, the relative risk for the condition is

      Hence, the treatment group is three times as likely to be cured as is the control group.

      The odds ratio is

      which is relatively similar to the relative risk since the probability of being cured is small.

      2.4 Probability Models

      The distribution of a quantitative variable is often modeled with a probability distribution. There is a wide range of mathematical probability models that are available for modeling the distribution of a quantitative variable; however, the particular probability model that is best used to model the distribution of a quantitative variable will depend on whether the variable of interest is discrete or continuous, the distribution of the variable, and any theoretical conditions or assumptions made about the population being modeled. Two of the most commonly used probability models in biomedical research are the Binomial Probability Model, which is associated with a discrete counting variable, and the Normal Probability Model, which is often used to model the distribution of a continuous variable.

      2.4.1 The Binomial Probability Model

      The binomial probability model can be used for modeling the number of times a particular event occurs in a sequence of repeated trials. In particular, a binomial random variable is a discrete variable that is used to model chance experiments involving repeated dichotomous trials. That is, the binomial model is used to model repeated trials where the outcome of each trial is one of the two possible outcomes. The conditions under which the binomial probability model can be used are given below.

       THE BINOMIAL CONDITIONS

      The binomial distribution can be used to model the number of successes in n trials when

      1 each trial of the experiment results in one of the two outcomes, denoted by S for success and F for failure,

      2 the trials will be repeated n times under identical conditions and each trial is independent of the others,

      3 the probability of a success is the same on each of the n trials,

      4 the random variable of interest, say X, is the number of successes in the n trials.

A random variable satisfying the above conditions is called a binomial random variable. Note that a binomial random variable X simply counts the number of successes that occurred in n trials. The probability distribution for a binomial random variable X is given by the mathematical expression

      where p(x) is the probability that X is equal to the value x. In this formula

       n!x!(n−x)! is the number of ways for there to be x successes in n trials,

       n!=n(n−1)(n−2)⋯3⋅2⋅1 and 0!=1 by definition,

       p is the probability of a success on any of the n trials,

       px is the probability of having x successes in n trials,

       1−p is the probability of a failure on any of the n trials,

       (1−p)n−x is the probability of getting n − x failures in n trials.

Examples of the binomial distribution are given in Figure 2.24. Note that a binomial distribution will have a longer tail to the right when p < 0.5, a longer tail to the left when p > 0.5, and is symmetric when p = 0.5.