# Lesson 3: Measures of Risk

## Section 1: Frequency Measures

Numerator = upper portion of a fraction

Denominator = lower portion of a fraction

A measure of central location provides a single value that summarizes an entire distribution of data. In contrast, a frequency measure characterizes only part of the distribution. Frequency measures compare one part of the distribution to another part of the distribution, or to the entire distribution. Common frequency measures are ratios, proportions, and rates. All three frequency measures have the same basic form:

numerator denominator
× 10 n

Recall that:

100 = 1 (anything raised to the 0 power equals 1)
101 = 10 (anything raised to the 1st power is the value itself)
102 = 10 × 10 = 100
103 = 10 × 10 × 10 = 1,000

So the fraction of (numerator/denominator) can be multiplied by 1, 10, 100, 1000, and so on. This multiplier varies by measure and will be addressed in each section.

### Ratio

#### Definition of ratio

A ratio is the relative magnitude of two quantities or a comparison of any two values. It is calculated by dividing one interval- or ratio-scale variable by the other. The numerator and denominator need not be related. Therefore, one could compare apples with oranges or apples with number of physician visits.

#### Method for calculating a ratio

Number or rate of events, items, persons,
etc. in one group Number or rate of events, items, persons,
etc. in another group

After the numerator is divided by the denominator, the result is often expressed as the result “to one” or written as the result “:1.”

Note that in certain ratios, the numerator and denominator are different categories of the same variable, such as males and females, or persons 20–29 years and 30–39 years of age. In other ratios, the numerator and denominator are completely different variables, such as the number of hospitals in a city and the size of the population living in that city.

#### EXAMPLE: Calculating a Ratio — Different Categories of Same Variable

Between 1971 and 1975, as part of the National Health and Nutrition Examination Survey (NHANES), 7,381 persons ages 40–77 years were enrolled in a follow-up study.(1) At the time of enrollment, each study participant was classified as having or not having diabetes. During 1982–1984, enrollees were documented either to have died or were still alive. The results are summarized as follows.

 Original Enrollment Dead at Follow-Up (1971–1975) (1982–1984) Participant 189 100 3,151 811 218 72 3,823 511

Of the men enrolled in the NHANES follow-up study, 3,151 were nondiabetic and 189 were diabetic. Calculate the ratio of non-diabetic to diabetic men.

Ratio = 3,151 ⁄ 189 × 1 = 16.7:1

#### Properties and uses of ratios

• Ratios are common descriptive measures, used in all fields. In epidemiology, ratios are used as both descriptive measures and as analytic tools. As a descriptive measure, ratios can describe the male-to-female ratio of participants in a study, or the ratio of controls to cases (e.g., two controls per case). As an analytic tool, ratios can be calculated for occurrence of illness, injury, or death between two groups. These ratio measures, including risk ratio (relative risk), rate ratio, and odds ratio, are described later in this lesson.
• As noted previously, the numerators and denominators of a ratio can be related or unrelated. In other words, you are free to use a ratio to compare the number of males in a population with the number of females, or to compare the number of residents in a population with the number of hospitals or dollars spent on over-the-counter medicines.
• Usually, the values of both the numerator and denominator of a ratio are divided by the value of one or the other so that either the numerator or the denominator equals 1.0. So the ratio of non-diabetics to diabetics cited in the previous example is more likely to be reported as 16.7:1 than 3,151:189.

#### EXAMPLES: Calculating Ratios for Different Variables

Example A: A city of 4,000,000 persons has 500 clinics. Calculate the ratio of clinics per person.

500 ⁄ 4,000,000 × 10n = 0.000125 clinics per person

To get a more easily understood result, you could set 10n = 104 = 10,000. Then the ratio becomes:

0.000125 × 10,000 = 1.25 clinics per 10,000 persons

You could also divide each value by 1.25, and express this ratio as 1 clinic for every 8,000 persons.

Example B: Delaware’s infant mortality rate in 2001 was 10.7 per 1,000 live births.(2) New Hampshire’s infant mortality rate in 2001 was 3.8 per 1,000 live births. Calculate the ratio of the infant mortality rate in Delaware to that in New Hampshire.

10.7 ⁄ 3.8 × 1 = 2.8:1

Thus, Delaware’s infant mortality rate was 2.8 times as high as New Hampshire’s infant mortality rate in 2001.

#### A commonly used epidemiologic ratio: death-to-case ratio

Death-to-case ratio is the number of deaths attributed to a particular disease during a specified period divided by the number of new cases of that disease identified during the same period. It is used as a measure of the severity of illness: the death-to-case ratio for rabies is close to 1 (that is, almost everyone who develops rabies dies from it), whereas the death-to-case ratio for the common cold is close to 0.

For example, in the United States in 2002, a total of 15,075 new cases of tuberculosis were reported.(3) During the same year, 802 deaths were attributed to tuberculosis. The tuberculosis death-to-case ratio for 2002 can be calculated as 802 ⁄ 15,075. Dividing both numerator and denominator by the numerator yields 1 death per 18.8 new cases. Dividing both numerator and denominator by the denominator (and multiplying by 10n = 100) yields 5.3 deaths per 100 new cases. Both expressions are correct.

Note that, presumably, many of those who died had initially contracted tuberculosis years earlier. Thus many of the 802 in the numerator are not among the 15,075 in the denominator. Therefore, the death-to-case ratio is a ratio, but not a proportion.

### Proportion

#### Definition of proportion

A proportion is the comparison of a part to the whole. It is a type of ratio in which the numerator is included in the denominator. You might use a proportion to describe what fraction of clinic patients tested positive for HIV, or what percentage of the population is younger than 25 years of age. A proportion may be expressed as a decimal, a fraction, or a percentage.

#### Method for calculating a proportion

Number of persons or events with a
particular characteristic Total number of persons or events, of which
the numerator is a subset
× 10 n

For a proportion, 10n is usually 100 (or n = 2) and is often expressed as a percentage.

#### EXAMPLE: Calculating a Proportion

Example A: Calculate the proportion of men in the NHANES follow-up study who were diabetics.

Numerator = 189 diabetic men
Denominator = Total number of men = 189 + 3,151 = 3,340

Proportion = (189 ⁄ 3,340) × 100 = 5.66%

Example B: Calculate the proportion of deaths among men.

Numerator = deaths in men
= 100 deaths in diabetic men + 811 deaths in nondiabetic men
= 911 deaths in men

Notice that the numerator (911 deaths in men) is a subset of the denominator.

Denominator = all deaths
= 911 deaths in men + 72 deaths in diabetic women + 511 deaths in nondiabetic women
= 1,494 deaths

Proportion = 911 ⁄ 1,494 = 60.98% = 61%

Your Turn: What proportion of all study participants were men? (Answer = 45.25%)

#### Properties and uses of proportions

• Proportions are common descriptive measures used in all fields. In epidemiology, proportions are used most often as descriptive measures. For example, one could calculate the proportion of persons enrolled in a study among all those eligible (“participation rate”), the proportion of children in a village vaccinated against measles, or the proportion of persons who developed illness among all passengers of a cruise ship.
• Proportions are also used to describe the amount of disease that can be attributed to a particular exposure. For example, on the basis of studies of smoking and lung cancer, public health officials have estimated that greater than 90% of the lung cancer cases that occur are attributable to cigarette smoking.
• In a proportion, the numerator must be included in the denominator. Thus, the number of apples divided by the number of oranges is not a proportion, but the number of apples divided by the total number of fruits of all kinds is a proportion. Remember, the numerator is always a subset of the denominator.
• A proportion can be expressed as a fraction, a decimal, or a percentage. The statements “one fifth of the residents became ill” and “twenty percent of the residents became ill” are equivalent.
• Proportions can easily be converted to ratios. If the numerator is the number of women (179) who attended a clinic and the denominator is all the clinic attendees (341), the proportion of clinic attendees who are women is 179 ⁄ 341, or 52% (a little more than half). To convert to a ratio, subtract the numerator from the denominator to get the number of clinic patients who are not women, i.e., the number of men (341 − 179 = 162 men.)Thus, ratio of women to men could be calculated from the proportion as:

Ratio = 179 ⁄ (341 − 179) × 1
= 179 ⁄ 162
= 1.1 to 1 female-to-male ratio

Conversely, if a ratio’s numerator and denominator together make up a whole population, the ratio can be converted to a proportion. You would add the ratio’s numerator and denominator to form the denominator of the proportion, as illustrated in the NHANES follow-up study examples (provided earlier in this lesson).

#### A specific type of epidemiologic proportion: proportionate mortality

Proportionate mortality is the proportion of deaths in a specified population during a period of time that are attributable to different causes. Each cause is expressed as a percentage of all deaths, and the sum of the causes adds up to 100%. These proportions are not rates because the denominator is all deaths, not the size of the population in which the deaths occurred. Table 3.1 lists the primary causes of death in the United States in 2003 for persons of all ages and for persons aged 25–44 years, by number of deaths, proportionate mortality, and rank.

Table 3.1 Number, Proportionate Mortality, and Ranking of Deaths for Leading Causes of Death, All Ages and 25–44 Year Age Group — United States, 2003

All_Ages Ages 25–44 Years Number Percentage Rank Number 2,443,930 100 128,924 100 684,462 28 1 16,283 12.6 3 554,643 22.7 2 19,041 14.8 2 157,803 6.5 3 3,004 2.3 8 126,128 5.2 4 401 0.3 * 105,695 4.3 5 27,844 21.6 1 73,965 3 6 2,662 2.1 9 64,847 2.6 7 1,337 1 10 63,343 2.6 8 0 0 * 33,615 1.4 9 305 0.2 * 34,243 1.4 10 328 0.2 * 30,642 1.3 11 11,251 8.7 4 27,201 1.1 12 3,288 2.6 7 17,096 0.7 13 7,367 5.7 5 13,544 0.5 * 6,879 5.3 6 456,703 18.7 29,480 22.9

* Not among top ranked causes

Data Sources: Centers for Disease Control and Prevention. Summary of notifiable diseases, United States, 2003. MMWR 2005;2(No. 54).
Hoyert DL, Kung HC, Smith BL. Deaths: Preliminary data for 2003. National Vital Statistics Reports; vol. 53 no 15. Hyattsville, MD: National Center for Health Statistics 2005: p. 15, 27.

As illustrated in Table 3.1, the proportionate mortality for HIV was 0.5% among all age groups, and 5.3% among those aged 25–44 years. In other words, HIV infection accounted for 0.5% of all deaths, and 5.3% of deaths among 25–44 year olds.

### Rate

#### Definition of rate

In epidemiology, a rate is a measure of the frequency with which an event occurs in a defined population over a specified period of time. Because rates put disease frequency in the perspective of the size of the population, rates are particularly useful for comparing disease frequency in different locations, at different times, or among different groups of persons with potentially different sized populations; that is, a rate is a measure of risk.

To a non-epidemiologist, rate means how fast something is happening or going. The speedometer of a car indicates the car’s speed or rate of travel in miles or kilometers per hour. This rate is always reported per some unit of time. Some epidemiologists restrict use of the term rate to similar measures that are expressed per unit of time. For these epidemiologists, a rate describes how quickly disease occurs in a population, for example, 70 new cases of breast cancer per 1,000 women per year. This measure conveys a sense of the speed with which disease occurs in a population, and seems to imply that this pattern has occurred and will continue to occur for the foreseeable future. This rate is an incidence rate, described in the next section, starting on page 3-13.

Other epidemiologists use the term rate more loosely, referring to proportions with case counts in the numerator and size of population in the denominator as rates. Thus, an attack rate is the proportion of the population that develops illness during an outbreak. For example, 20 of 130 persons developed diarrhea after attending a picnic. (An alternative and more accurate phrase for attack rate is incidence proportion.) A prevalence rate is the proportion of the population that has a health condition at a point in time. For example, 70 influenza case-patients in March 2005 reported in County A. A case-fatality rate is the proportion of persons with the disease who die from it. For example, one death due to meningitis among County A’s population. All of these measures are proportions, and none is expressed per units of time. Therefore, these measures are not considered “true” rates by some, although use of the terminology is widespread.

Table 3.2 summarizes some of the common epidemiologic measures as ratios, proportions, or rates.

Table 3.2 Epidemiologic Measures Categorized as Ratio, Proportion, or Rate

Condition
Ratio
Proportion
Rate
Morbidity
(Disease)
Risk ratio
(Relative risk)

Rate ratio
Odds ratio
Period prevalence

Attack rate
(Incidence proportion)

Secondary attack rate
Point prevalence
Attributable proportion

Person-time incidence rate
Mortality
(Death)
Death-to-case ratio
Proportionate mortality
Crude mortality rate
Case-fatality rate
Cause-specific mortality rate
Age-specific mortality rate
Maternal mortality rate
Infant mortality rate
Natality
(Birth)
Crude birth rate
Crude fertility rate

### Exercise 3.1

For each of the fractions shown below, indicate whether it is a ratio, a proportion, a rate, or none of the three.

1. Ratio
2. Proportion
3. Rate
4. None of the above

1. ____ 1.
number of women in State A who died from heart disease in 2004 number of women in State A who died in 2004
2. ____ 2.
number of women in State A who died from heart disease in 2004 estimated number of women living in State A on July 1, 2004
3. ____ 3.
number of women in State A who died from heart disease in 2004 number of women in State A who died from cancer in 2004
4. ____ 4.
number of women in State A who died from lung cancer in 2004 number of women in State A who died from cancer (all types) in 2004
5. ____ 5.
number of women in State A who died from lung cancer in 2004 estimated revenue (in dollars) in State A from cigarette sales in 2004

Check your answer.

## References (This Section)

1. Kleinman JC, Donahue RP, Harris MI, Finucane FF, Madans JH, Brock DB. Mortality among diabetics in a national sample. Am J Epidemiol 1988;128:389–401.
2. Arias E, Anderson RN, Kung H-C, Murphy SL, Kochanek KD. Deaths: final data for 2001. National vital statistics reports; vol. 52 no. 3. Hyattsville, Maryland: National Center for Health Statistics, 2003; 9:30–3.
3. Centers for Disease Control and Prevention. Reported tuberculosis in the United States, 2003. Atlanta, GA: U.S. Department of Health and Human Services, CDC, September 2004.

Page last reviewed: May 18, 2012