Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 67

Many biomedical research studies concern the incidence of a particular disease or condition. The absolute risk (AR) of a disease is the probability that an individual develops the disease. Most conditions/diseases are affected by risk factors that increase the incidence of the disease. For example, it is well known that the risk factor smoking cigarettes increases an individual’s chance of developing lung cancer.

An important research question that is often asked in the study of a disease is whether or not the disease is independent of a particular risk factor. When the disease is independent of the risk factor, the risk factor does not increase or decrease the incidence of the disease. On the other hand, when the disease is dependent on the risk factor, the risk factor does affect the chance of having this disease, and in this case, the disease is said to be associated with the risk factor.

In a prospective study where the risk factor is either labeled present or absent, exposed or unexposed, or treated or untreated, an important measure of the strength of the association between the disease and the risk factor is the relative risk (RR), which is also known as the risk ratio. The relative risk is the ratio of the probability of having the disease when the risk factor is present and the probability of having the disease when the risk factor is absent. In particular, the formula for the relative risk is

Note that the relative risk is only appropriate for prospective studies, it takes on values between 0 and ∞, and it is one when the disease is independent of the risk factor. When the relative risk is close to 1, there is only a small difference in the probabilities of having the disease when the risk factor is present or absent, and therefore, the relative risk suggests that the risk factor does not have much influence on the incidence of the disease. A risk ratio larger than 1 suggests there is an increased risk of the disease when the risk factor is present, and a risk ratio smaller than 1, suggests there is decreased risk of the disease when the risk factor is absent.

The larger the relative risk is, the more likely it is for an individual to have the disease when the risk factor is present. For example, if the relative risk is RR = 5.8, then the disease is 5.8 times as likely when the risk factor is present than when it is absent.

Example 2.27

Suppose in a prospective study, the probability of having the disease given a particular risk factor is present is 0.10, and the probability of having the disease when the risk factor is absent is 0.02. Then, the relative risk of the disease for this risk factor is RR=0.100.02=5. Thus, the disease is five times as likely when the risk factor is present.

Finally, in the presence of a significant relative risk, it is also important to look at the absolute risk to assess the practical significance of the risk factor. For example, if the relative risk is RR = 10, but the absolute risk is AR = 0.000001, then the disease is very rare and even with the presence of the risk factor the risk is only 1 in 100,000. Thus, when the absolute risk of the disease is small, large values of relative risk may not truly indicate significant effects of having the risk factor. Also, when the absolute risk of the disease is large, a relative risk close slightly larger than 1 can indicate a significant effect due to the risk factor. Therefore, it is recommended that both the absolute risk and the relative risk be reported with the results of the study.

Because the relative risk can only be used in prospective studies, an alternative measure of the association of the disease and risk factor is required for retrospective studies. The odds ratio (OR) is an alternative measure of association that can be used in both prospective and retrospective studies.

The odds ratio is based on the odds of having the disease rather than the probability of having the disease. The odds of having the disease is ratio of the probability of having the disease to the probability of not having the disease. The formula for computing the odds of having the disease is

The odds of a disease is between 0 and ∞. Furthermore, when the odds = 1, having the disease is just as likely as not having the disease, when the odds < 1, the disease is less likely than not having the disease, and when the odds > 1, the disease is more likely than not having the disease.

For example, if the probability of having the disease, which is the absolute risk of the disease, is 0.2, then the odds of having the disease is odds(Disease)=0.20.8=0.25. Thus, the disease is one-fourth as likely as not having the disease.

In most cases the odds of having a disease will be different for the presence or absence of a particular risk factor. Thus, it is often useful to compare the odds when the risk factor is present to the odds when the risk factor is absent. The odds ratio is one method used to compare these two odds. In particular, the odds ratio for a disease is the ratio of the odds of the disease when the risk factor is present to the odds when the risk factor is absent, and the formula for computing the odds ratio is

The odds ratio measures the degree to which the disease is more likely when the 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.