The distribution of health states and events is an important focus of epidemiology. Because people differ in their probability or risk of disease, a primary concern is the identification of how they differ. Today, epidemiologists use tools such as geographic information systems (GISs) to study health-related events to identify disease distribution patterns, similar to John Snow’s mapping of cholera cases in London in the nineteenth century. However, mapping cases is limited in what it can reveal. A higher number of cases may simply be the result of a larger population with more people who are potential cases, or of a longer period of observation. Any description of disease patterns should take into account the size of the population at risk for the disease. That is, we should look not only at the numerator (the number of cases), but also at the denominator (the number of people in the population at risk) and at the length of time the population was observed. For example, 50 cases of influenza in a month might be viewed as a serious epidemic in a population of 250 but would indicate a low rate in a population of 250,000. On the other hand, even in a small population, one might observe 50 cases over a period of several years. Using rates and proportions instead of simple counts of cases takes the size of the population at risk into account (Koepsell and Weiss, 2003). How time is considered differs according to the measure being used.

Epidemiologic studies rely on proportions and rates. A proportion is a type of ratio in which the denominator includes the numerator. For example, in 2006 there were 2,426,264 deaths recorded in the United States, of which 631,636 were reported as caused by heart disease; so the proportion of deaths attributable to heart disease in 2006 was 631,636/2,426,264 = 0.260, or 26.0%. Because the numerator must be included in the denominator, proportions can range from 0 to 1. Proportions are often multiplied by 100 and expressed as a percent, literally meaning per 100. In public health statistics, however, if the proportion is very small, we use a larger multiplier to avoid small fractions; thus the proportion may be expressed as a number per 1000 or per 100,000.

A rate is a measure of the frequency of a health event in “a defined population, usually in a specified period of time” (Porta, 2008, p 207). A rate is a ratio, but it is not a proportion because the denominator is a function of both the population size and the dimension of time, whereas the numerator is the number of events (Rothman, 2002; Koepsell and Weiss, 2003; Gordis, 2009). Furthermore, depending on the units of time and the frequency of events, a rate may exceed 1. As its name suggests, a rate is a measure of how rapidly something is happening: how rapidly a disease is developing in a population or how rapidly people are dying. Conceptually, a rate is the instantaneous change in a continuous process. Notice the use of the words event and happening. Rates deal with change over time, as individuals move from one state of being to another (e.g., from well to ill, alive to dead, or ill to cured). In observing a population over time to observe such changes in status, we typically exclude from the population being followed those persons who have already experienced the event.

Using the same health event of death from heart disease detailed above, we can also compute a rate. To calculate the death rate, the denominator will be the number of individuals in the population during the year in which the deaths occurred, rather than total number of deaths. For example, in 2006, the U.S. population estimate was 299,398,484. Since rates are commonly expressed per 100,000 or per 1000, we will divide this total population figure by 100,000, to get 2993.98484. The resulting calculation of the rate of death from heart disease in the U.S. in 2006 would be 631,636/2993.98484 = 211 per 100,000.

Risk refers to the probability that an event will occur within a specified time period. A population at risk is the population of persons for whom there is some finite probability (even if small) of that event. For example, although the risk of breast cancer in men is small, a few men do develop breast cancer and therefore could be considered part of the population at risk. There are some outcomes for which certain people would never be at risk (e.g., men cannot be at risk of ovarian cancer, nor can women be at risk of testicular cancer). A high-risk population, on the other hand, would include those persons who, because of exposure, lifestyle, family history, social or environmental context, or other factors, are at greater risk for disease than the population at large. Although anyone may be susceptible to HIV infection, the degree of susceptibility does vary. Everyone in the population is at risk for HIV and AIDS, but persons who have multiple sexual partners without adequate protection or who use intravenous drugs are in the high-risk population for HIV infection. However, others who do not fit these categories may unknowingly be at high risk. An example is women who consider themselves to be in monogamous relationships but are unaware that their partners have sexual relations with other women or men. As proportions, risk estimates have no dimensions, but they are a function of the length of time of observation. Given a continuous rate, increasing time will mean that a larger proportion of the population will eventually become ill.

Epidemiologists and other health professionals are interested in measures of morbidity, especially incidence proportions, incidence rates, and prevalence proportions (Gordis, 2009). These measures provide information about the risk of disease, the rate of disease development, and the levels of existing disease in a population, respectively.

Measures of incidence reflect the number of new cases or events in a population at risk during a specified time. An incidence rate quantifies the rate of development of new cases in a population at risk, whereas an incidence proportion indicates the proportion of the population at risk who experience the event over some period of time, for example, the proportion of the population who develop influenza during a given year (Rothman, 2002). The population at risk is considered to be persons without the event or outcome of interest but who are at risk of experiencing it. Note that existing (or prevalent) cases are excluded from the population at risk for this calculation, since they already have the condition and are no longer at risk of developing it. (Calculations of incidence measures for events that can recur are more complicated.) The incidence proportion is also referred to as the cumulative incidence rate (and erroneously simply as the incidence rate) because it reflects the cumulative effect of the incidence rate over the time period, whether it is a month, a year, or several years. A constant incidence rate operating over a period of time results in an increasing proportion of the population who is affected, that is, the incidence rate may stay constant while the cumulative incidence rate increases with time. An incidence proportion can be interpreted as an estimate of risk of disease in that population over that period—that is, as a probability with limits from 0 to 1. The risk of disease is a function of both the rate of new disease development and the length of time the population is at risk. The interpretation can be for an individual (i.e., the probability that the person will become ill) or for a population (i.e., the proportion of a population expected to become ill over the specified period). In epidemiology, we often calculate proportions on the basis of population frequencies. These frequencies are then translated into personal risk statements for people representative of the population on which the estimates are based.

As an example, suppose a local health department and community hospital have a joint comprehensive screening program in a specific geographic area characterized by overcrowded housing, low-income families, limited access to services, and underutilization of preventive health practices. The outreach program includes conducting health histories and physical examinations; tuberculin skin tests with follow-up chest radiography where indicated; cardiovascular, glaucoma, and diabetes screening; and mammography for women and prostate screening for men more than 45 years of age. Of 8000 women screened through the program, 35 were previously diagnosed with breast cancer; through the screening efforts, 20 with no prior breast cancer diagnosis are identified as having cancer of the breast. Subsequently, the remaining 7945 women in whom no breast cancer was detected could be followed over the next 5 years to identify the number of new cases of breast cancer detected. Assuming no losses to follow-up (e.g., moved away or died from other causes), if 44 women were diagnosed over the 5-year period, the 5-year incidence proportion of breast cancer in this population would be as follows:

447945=0.005538,or553.8per100,000

Note the multiplication by 100,000, so that the number of cases is expressed as per 100,000 women. An incidence proportion estimates the risk of developing the disease in that population during that time. Also, as a proportion, each event in the numerator must be represented in the denominator, and only those persons at risk of the event counted in the numerator may be included in the denominator.

We estimate incidence rates by counting events relative to the total amount of time that persons in a population are observed (person-time). This measure is called incidence density. For many calculations, it is generally assumed that rates are constant over the period of observation. The numerator consists of the number of new events. In calculating the person-time denominator, we count the amount of time each person contributes from the time that observation of that person begins until the person: (1) experiences the event, (2) is lost to follow-up, dies from some other cause, or otherwise is no longer at risk, or (3) reaches the end of observation. Incidence density is an estimate of the true instantaneous rate (or hazard) of the event. It is an indication of how rapidly a disease is developing in a population. Conceptually, a rate is the instantaneous change in a continuous process. Note that incidence density does have dimensions; it is not bounded by 1 (as risk is); its value depends on the units of time chosen. It may be interpreted as the reciprocal of the average time until disease onset, assuming there are no competing risks and there is a fixed or steady-state population with complete follow-up (Rothman, 2002).

To continue the previous example, suppose the 7945 women we follow for 5 years accumulate a total of 39,615 person-years of observation. (Person-time, such as person-years, is the sum of the time observed for all the people under observation.) Remember, the assumption was that there was no loss to follow-up or deaths from other causes. Note that the total person-time is not equivalent to 5 × 7945 (or 39,725), because we stop counting time for the 44 women who developed breast cancer at the time of diagnosis. (Note that determining the exact time of disease onset is often a problem in epidemiologic studies. The use of time of diagnosis may be biased because diagnosis occurs at different stages of disease in different populations.) Consequently, the incidence rate for breast cancer diagnosis in this population, after 5 years of observation, would be estimated as 44 newly diagnosed cases per 39,615 person-years of observation, or 0.0011107; we could express this as 11.1 cases per 10,000 person-years.

We are often concerned about the risk—that is, the probability of disease occurring over some defined period of time, such as a year or several years. Earlier we estimated the risk directly as the number of new cases over a 5-year period in a population at risk. That was straightforward because we had no losses, and observation began at the same time for all the women. A more common situation is that people come under observation at different times and there are losses attributable to attrition or competing risks (i.e., other events or deaths). We can handle those easily by counting the amount of time that is observed and calculating the incidence density as an estimate of the incidence rate. By calculating the incidence rate of an event (e.g., diagnosis of breast cancer) in a population, we then can estimate the risk of the event in that population, both in terms of the expected proportion of the population who would experience the event and in terms of the risk to a representative member of the population. When certain assumptions are met, primarily a constant rate, the average incidence density over a period of time is related to the cumulative risk for that period by the following equation:

Risk=1−e(−1×T)

where 1 is the mean per-person-time incidence rate, T is the period of observation in the same units of time, and e is the base of the natural logarithm. Again, to return to the example, suppose we observed the 44 new cases in 39,615 person-years of follow-up, but there were losses to follow-up, and women entered and left the population at different times. Assuming that the rate of new breast cancer is fairly constant over that 5-year period, the cumulative risk over 5 years is estimated as follows:

Risk=1−e(−0.0011107×5)=0.005538,

which is the same as the risk we calculated for the simpler situation.

Note that the rate used is not the incidence proportion but the mean incidence densities for intervals of time comprising the total period, and the formula assumes that the rates do not vary over the period. Also, risk is a probability whose value depends on both the incidence rate and the period of observation but whose range is restricted to between 0 and 1. For example, the longer a person is at risk for an event, the greater the probability (or risk) the event will occur at some point. If the risk of being hit crossing the street is 0.01, then the risk that I will be hit at some point if I cross the street 100 times is very much greater than if I cross the street only once. The value of incidence density, on the other hand, does depend on the units of time and, because it is not a probability, is not restricted to the 0 to 1 range. Furthermore, when the incidence rate is low or the period of observation short—so that, relative to the size of the population, few people are removed from the population at risk by disease—the product of the per-person-time rate and the period of observation approximates the risk for the period (Rothman, 2002; Szklo and Nieto, 2007).

A ratio can be used as an approximation of a risk. For example, the infant mortality rate (IMR) is the number of deaths in infants less than 1 year of age that occur in a given year divided by the number of live births in that same year. The IMR approximates the risk of death in the first year of life for live-born infants in a specific year. Some of the infants who die that year were born in the previous calendar year, and some of the infants born that year may die in the following calendar year before their first birthday. However, because about two thirds of infant deaths occur within the first 28 days of life, the number of infants in the numerator (deaths in a given year) but not in the denominator (live births in that same year) will be small. It can be assumed that current year deaths from the previous year’s cohort approximately equal the deaths from the current year’s cohort occurring in the following year. Although technically a ratio, this is an approximation to the true proportion and, therefore, an estimate of the risk. In some publications, readers may notice that the rate of infant death is not equivalent to the number of deaths divided by the number of live births. These rates take into account the changes in the rate of death over the first year of life—that is, they use a person-time denominator.

The prevalence proportion measures existing cases of disease. The prevalence odds (P/[1 − P]) are roughly proportional to the incidence rate multiplied by the average duration of disease (Rothman, 2002). The prevalence proportion is, therefore, affected by factors that influence risk (incidence) and by factors that influence survival or recovery (duration). For that reason, prevalence measures are less useful when looking for factors related to disease etiology. Because prevalence proportions reflect duration in addition to the risk of getting the disease, it is difficult to sort out what factors are related to risk and what factors are related to survival or recovery. In mathematical notation:

P/(1−P)≈I×D,or,when P is small(<0.1),P≈I×D,

where P = prevalence, I = incidence rate, and D = average duration.

For example, the 5-year survival rate for breast cancer is about 85%, but the 5-year survival rate for lung cancer in women is only about 15%. Even if the incidence rates of breast and lung cancer were the same in women (and they are not), the prevalence proportions would differ because, on average, women live longer with breast cancer (i.e., it has a longer duration). Incidence rates and incidence proportions, on the other hand, are the measure of choice to study etiology because incidence is affected only by factors related to the risk of developing disease and not to survival or cure. Prevalence proportions are useful in planning health care services because they indicate the level of disease existing in the population and therefore the size of the population in need of services. At the level of a local health department, epidemiologists and nurses would rely on both incidence and prevalence data in planning services focused on the prevention and control of tuberculosis (TB). They would examine the existing level of TB within the community (prevalence) to plan services and direct prevention and control measures, and they would take into consideration the rate of new TB cases (incidence) to study risk factors and evaluate the effectiveness of prevention and control programs.

Mortality rates are key epidemiologic indicators of interest to nurses (Table 12-2). Although measures of mortality reflect serious health problems and changing patterns of disease, they are limited in their usefulness. Mortality rates are informative only for fatal diseases and do not provide direct information about either the level of existing disease in the population or the risk of contracting any particular disease. Also, it is not uncommon for a person who has one disease (e.g., prostate cancer) to die from a different cause (e.g., stroke).

TABLE 12-2

COMMON MORTALITY RATES

Rate/Ratio	Definition and Example
Crude mortality rate	Usually an annual rate that represents the proportion of a population who die from any cause during the period, using the midyear population as the denominator Example: In 2006 there were 2,426,264 deaths in a total population of 299,398,484, or 810.4 per 100,000: 2,426,264299,398,484=810.4per100,000
Age-specific rate	Number of deaths among persons of given age group per midyear population of that age group Example: 2006 age-specific mortality rate for 20- to 24-year-olds: 21,14821,111,240=100.2per100,000
Cause-specific rate	Number of deaths from a specific cause per midyear population Example: 2006 cause-specific rate for accidents: 121,599deaths from accidents299,398,484midyear population=40.6per100,000
Case-fatality rate	Number of deaths from a specific disease in a given period divided by number of persons diagnosed with that disease Example: If 87 of every 100 persons diagnosed with lung cancer die within 5 years, the 5-year case fatality rate is 87%. The 5-year survival rate is 13%.
Proportionate mortality ratio	Number of deaths from a specific disease per total number of deaths in the same period Example: In 2006 there were 631,636 deaths from diseases of the heart, and 2,426,264 deaths from all causes: 631,6362,426,264=0.26or26%of all deaths in2006were attributable to heart disease
Infant mortality rate	Number of infant deaths before 1 year of age in a year per number of live births in the same year Example: In 2006 there were 28,527 infant deaths and 4,265,555 live births: 28,5274,265,555=668.79per100,000live births or6.69per1000live births
Neonatal mortality rate	Number of infant deaths under 28 days of age in a year per number of live births in the same year Example: In 2006 there were 18,989 neonatal deaths and 4,265,555 live births: 18,9894,265,555=445.17per100,000or4.45per1000live births
Postneonatal mortality rate	Number of infant deaths from 28 days to 1 year in a year per number of live births in the same year Example: In 2006 there were 9538 postneonatal deaths and 4,265,555 live births: 95384,265,555=223.61per100,000or2.24per1000live births

From Heron MP, Hoyert DL, Murphy SL, et al: Deaths: Final data for 2006, Natl Vital Stat Rep 57(14), Hyattsville, MD, National Center for Health Statistics, 2009; Martin JA, Hamilton BE, Sutton PD, et al: Births: Final data for 2006, Natl Vital Stat Rep 57(7), Hyattsville, MD, National Center for Health Statistics, 2009.

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