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What's the denominator? A lesson on risk.


If you had acne, would you use a medication that can cause depression, stomach pain, blurred vision, diarrhea, and rectal bleeding? Perhaps you would answer, "Depends on how bad the acne is." What about a medication that caused scores of infants to be born with severe birth defects, including malformations of the face, skull, heart, and central nervous system? Still interested? All these side effects can be caused by isotretinoin (trade name Accutane). In the late 1980s and early 1990s, there was great concern over the side effects of this widely used and effective therapy for the treatment of disfiguring nodular acne.

If those dangers seem alarming, you might be asking, "How could the Food and Drug Administration have allowed Accutane to remain on the market?" An accurate answer to that question requires more information than given. In fact, it is impossible to evaluate the safety of new drugs-or household products or toys or anything else-until you have a great deal of information. Most often the missing information is the population size from which the mortality and morbidity data are being drawn or the baseline incidence of complications in the absence of the potential treatment. Without those numbers-the denominators against which meaningful comparisons can be made-the data are not helpful in rational risk analysis.

Mathematics is at the heart of all science and is central to the interpretation of research findings in settings ranging from the clinical application of new medications to personal decisions about acceptable levels of risk. The National Science Education Standards (NRC, 1996) include the following statement as part of "understandings about scientific inquiry:" Mathematics is essential in scientific inquiry. Mathematical tools and models guide and improve the posing of questions, gathering data, constructing explanations and communicating results.

Unfortunately, many Americans, including most of our students, have a poor grasp of mathematics, and rational risk analysis is one casualty. People who lack the skill of relating one number to another may overreact to relatively minor risks, perceiving certain situations or activities as dangerous when, in fact, they are not. Conversely, these people may disregard real risks. The tear of flying by someone who never wears a seat belt in his car is one example. Consider that, on average for the U.S. population, airline accidents will subtract four days off a person's life whereas auto accidents subtract an average of 182 days (Morgan, 1993). A variety of psychosocial factors affect how one perceives risk. For example, genetic counselors have discovered that the "perceived burden" of a genetic illness is more significant to parents than the empirical risk of being affected, which can influence pregnancy decisions (Ekwo, 1987; Meryash, 1989, 1992). For example, parents might be more reluctant to take the relatively low risk recurrence (e.g., 3%-5%) for a neural tube defect, which can be a very burdensonre disorder, than to take the 25% recurrence risk of phenylketonuria, which is perceived as less burdensome.

Likewise, most people view the risk of acute but unlikely illness (e.g., mercury poisoning) as more disturbing than the risk of chronic and common illness (e.g., heart disease or diabetes). This is consistent with our tendency to evaluate a risk, in part, according to our perception of "dread" (Morgan, 1993). But this is not the whole story. Ignorance about how to analyze the numbers used to describe risk is yet another reason for faulty risk assessment. Many people have never been challenged to answer and internalize the question: What is the risk being compared to? This is a question that teachers can address with their students, and this brief activity may help students begin to evaluate risk more thoughtfully.

* Background

Expressions of risk fall into two categories: absolute and relative. Absolute risks are measured in terms of simple, unqualified relationships or likelihoods, such as four persons per 100,000 or 22% of a given population. Relative risks are expressed as changes in relationships or percentages, increases and decreases reported in terms of "x-fold" or "y-times" or other comparisons that draw their meaning from some absolute risk (including, somewhat confusingly, an "x percent increase" as compared with "x percent off). For example, the risk of lung cancer increases roughly 20-fold for a long-term, heavy smoker compared to someone who has never smoked (Alberg & Samet, 2003). This is a relative risk. Without understanding the absolute risk, however, there is no making sense of the relative risk.

If, hypothetically, the likelihood of lung cancer for a nonsmoker is one in 20 million (the absolute risk for the nonsmoker), then a 20-fold increase in relative risk for the smoker would translate to an adjusted absolute risk of lung cancer of one in a million. At one in a million, many people might (rationally) decide to smoke (or continue smoking) because the pleasures might outweigh the unlikely danger of lung cancer. In fact, the absolute risk of lung cancer is much greater than one in a million, and so any sizable increase in relative risk will be even more important. According to the CDC, in 2003 there were approximately 191,000 new cases of lung cancer in the United States. The overall absolute risk of being diagnosed with lung cancer (i.e., the incidence rate) is about 70 per 100,000 (or, equivalently, about one in 1,400; the risk is higher for men and lower for women). Approximately 90% of diagnosed males and 80% of diagnosed females are active smokers; some of the remainder are former smokers (CDC, 2007). To illustrate the effect an increased relative risk can have on absolute risk, consider that females who have never smoked have an incidence of lung cancer of roughly 15 per 100,000, or one in 6,600 (Wakelee et al., 2007). If we apply to these non-smoking women the calculated relative risk assessed to the heaviest and longest-term smokers, then a 20-fold increase in relative risk translates to an absolute incidence risk of 300 per 100,000, or one in 330. In the context of the absolute risk, the 20-fold increase in relative risk, which in this case is associated with a controllable behavior, now becomes far more compelling.

Consider a second example, which highlights how a denominator can make a perceived risk less threatening. A 50% increased risk of something that affects four out of every 10 people (a small denominator) would then affect six out of every 10, a substantial and worrisome increase. A 50% increased risk of something that affects four out of every 100,000 (a large denominator), however, means that the new risk is still only six out of every 100,000, or 0.006%.

* Activity

1. Begin class by asking your students the questions in the introduction of this article. Or, if you prefer, ask similar questions about some other risk that has attracted recent media attention, such as the risk of cancer posed by electrical power lines or the likelihood of getting food poisoning from produce. Ask the students to explain whether they personally worry about such things and what behavioral steps they take to increase their safety. For example, some students might say that they would not take a dangerous drug such as Accutane because the potential side effects are not worth the benefits (clearing up acre). Do not offer your own opinions.

2. Ask the students to explain how they decide whether an activity or product is too risky to engage in or purchase. What information do they rely on to make those decisions? Some students will probably mention media reports of deaths or illnesses. Ask the students whether the sources of information they mentioned provide complete information.

3. Return to the side effects of Accutane mentioned in the opening paragraph. Ask the students:

Is the total number of birth defects and a general description of side-effect symptoms all the information you need to evaluate the safety or danger of Accutane?

The students should now suspect that the information is incomplete, although they may not know what is missing. If the students question the validity of the data, assure them that the symptoms and numbers are correct. If the students are stuck, prompt them to the answer by asking:

a) What percentage of people taking the drug suffer the negative side effects? (This information has not been provided.)

b) What information is necessary to calculate a percentage? (The introduction states only that "scores of infants [were] born with severe birth defects.")

What they need, of course, is the total number of women taking the drug, the denominator in the calculation of percentage.

4. List on the blackboard the following data, obtained from a study on the safety of Accutane (Mitchell et al., 1995):

a) 124,216 women taking Accutane were enrolled in a four-year study.

b) 99% of the women indicated that they were warned by their physicians not to become pregnant during drug therapy (which lasts 15-20 weeks) because of the risk of birth defects.

c) Despite the warning program, 402 women became pregnant during drug therapy. Of those women, 32 delivered live infants. (The other women chose to abort or had spontaneous miscarriages.)

d) Approximately seven of those infants had birth defects typical of Accutane exposure.

5. Now ask the students the following questions and help as needed with the calculations. (Do not reveal which of the risks are absolute and relative at this time.)

a) Approximately 4% of all newborns possess some type of birth defect (absolute). What percentage of infants born during the study suffered birth defects due to Accutane?

7/32 or approximately 22% (absolute)

b) What is the increase in risk of birth defects posed by Accutane?

22%/4% or about 5.5-fold greater risk of birth defects (relative)

c) What percentage of women involved in the study became pregnant?

402/124,216 or 0.3% (absolute)

Now reveal to the students that the number of pregnancies in the general population is about 10 times higher (relative). (This figure is much higher among teenagers.)

d) Was the pregnancy-warning program effective?

A perfect warning program would yield cord pregnancies; nonetheless, students should recognize that the pregnancy rate was reduced 10-fold.

e) What percentage of women involved in the study gave birth to a child with Accutane-induced birth defects?

7/124,216 or 0.006 percent (absolute)

6. How does knowing the various denominators affect your initial reaction to the dangers of Accutane? Students' answers should reflect the contrast between the high relative risk of birth defects among those women who became pregnant while taking Accutane despite warnings (5.5-fold) and the low overall risk of birth defects among all women taking the drug (0.006%).

7. After the students have completed the exercise, formally introduce the terms relative risk and absolute risk. Revisit the Accutane example to clarify your explanation. Now ask the students to classify each of the following as relative or absolute and to explain why:

a) Smoking increases the risk of lung cancer approximately 15 to 20-fold.

Relative because ,to baseline comparison is provided

b) Nonsmoking males have a one in 1,400 chance of getting lung cancer.

Absolute because the denominator specifies the population size

c) One percent of passenger-car occupants who wear seat belts are ejected during serious accidents.

Absolute; in this case the statistic has been normalized to a population size of 100

d) Passenger-car occupants who do not wear seat belts are 20 times more likely to be ejected than belted occupants.

Relative because the number derives meaning only from the likelihood of being ejected when belted

e) Seventy-three percent of ejected occupants are killed.

Absolute because the statistic has been normalized to a population size of 100

f) Last year six children died as a result of the polio (or measles or whatever) vaccine.

A bit of a trick: This is neither absolute nor relative. The media often report numerical information without context, which is completely unhelpful because there is no denominator at all.

* Evaluation

Choose any article from the lay press or scientific literature that addresses risk and ask the students to analyze the article (individually or in teams), determining whether:

* there are sufficient data on which to base a risk determination and, if so, identify those data

* the stated risks are absolute or relative

* the conclusions in the article are justified on the basis of the data presented.

Current examples might include articles about:

* the adverse effects of vaccines (e.g., risk of autism) or cell phones (e.g., risk of neurological damage)

* increased genetic risk for common diseases based on the results of direct-to-consumer genetic testing offered by commercial firms such as 23andMe and Navigenics. For example, ask students to explain the meaning of an odds ratio of 1.2.

* Summary

Ultimately, risk analysis is related to values, but citizens can make the best decisions only when their personal and social perspectives are informed by accurate data and thoughtful analysis. Society should care about rational risk evaluation because many public health and safety initiatives are prompted by public perception and outcry. Some of these are unnecessary and ineffective. Conversely, potential initiatives that might yield tangible benefits may be overlooked, leading to needless suffering and expense.


Alberg, A.J. & Samet, J.M. (2003). Epidemiology of lung cancer. Chest, 123, 21S-49S.

Centers for Disease Control and Prevention. Available online at: cancer/lung/basic_info/risk_factors.htm. Date last reviewed March 23, 2007, Accessed May 7, 2007.

Ekwo, E.E., Kim, J.O. & Gosselink, C.A. (1987). Parental perceptions of the burden of genetic disease. American Journal of Medical Genetics, 28(4), 955-963,

Meryash, D.L. (1989). Perception of burden among at-risk women of raising a child with fragile X syndrome. Clinical Genetics, 36(1), 15-24.

Meryash, D.L, (1992). Characteristics of fragile X relatives with different attitudes toward terminating an affected pregnancy. American Journal on Mental Retardation, 96(5), 528-535.

Mitchell, A.A., Van Bennekom, C.M. & Louik, C, (1995). A pregnancy-prevention program in women of childbearing age receiving Isotretinoin. New England Journal of Medicine, 333, 101-106.

Morgan, G.M. (1993). Risk analysis and management. Scientific American, July, 32-41.

NRC (National Research Council). (1996). National Science Education Standards, Washington, DC: National Academy Press

Wakelee, H.A., Chang, E.T., Gomez, S.L. et al. (2007). Lung cancer incidence in never smokers. Journal of Clinical Oncology, 25, 472-478.


MICHAEL J. DOUGHERTY is Director of Education, American Society of Human Genetics, Bethesda, MD 20814; e-mail: JOSEPH D. MCINERNEY is Executive Director, National Coalition of Health Professionals Education in Genetics, Lutherville, MD 21093.
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Author:Dougherty, Michael J.; McInerney, Joseph D.
Publication:The American Biology Teacher
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2009
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