2 Measuring Things in Epidemiology

2.1 Types of Counting

Epidemiologists are interested in the numbers behind health issues, including:

• How many people/animals are affected?
• How long are people/animals affected?
• Do the numbers differ by other factors?
• How many deaths are there?

Being able to do the work of an epidemiologist all comes down to appropriately counting what happens. There are two general categories of counting that we use in epidemiology: counts and ratios. Counts are the simplest and most frequently performed quantitative measures in epidemiology. Counts refer to the number of cases of a disease or other health phenomenon being studied. Most of the numbers you frequently see in epidemiology are types of ratios (rates, proportions, and percentages). Ratios are the values obtained by dividing one quantity (count) by another (numerator over a denominator).

We can use ratios to contrast different levels of injury severity from the most severe (e.g., death) to least severe (e.g., injured but not needing treatment) (figure 2.2). As the size and breadth of the population affected grows, the strength of definitive evidence for the marker decreases. We must strike a balance between the two when optimizing the use of the ratio.

2.1.1 Proportions

Proportions are a measure that states a count relative to the size of the group. It is a ratio in which the denominator contains the numerator. It can be expressed (written, said, etc.) as a percentage. A proportion can be used to demonstrate the magnitude of a health problem.

Prevalence is the number of existing cases of a disease or health condition in a population at some designated time. Prevalence is a proportion. It can be expressed (written, said, etc.,) as follows:

• A number
• A percentage
• The number of cases per unit size of the population

If no time is specified, we usually are discussing point prevalence or prevalence at a specific point in time. Period prevalence is the prevalence over a specified period of time. How do we use prevalence? As previously stated, we use it to find the magnitude or burden of a health problem in a population. We also use it to estimate the frequency of an exposure or to determine the allocation of health resources such as facilities and personnel.

The numerator for period prevalence is the sum of the prevalence at the beginning of the time period in question plus the cases that occur during the time interval.

The denominator for period prevalence is the average population over the time period in question. Sometimes we know the exact size of the population and we can use that number. But when we’re considering things that are dynamic, like the exact number of patients in and out of a hospital, an average is our best method. We could calculate this average a number of different ways, but often we use the following method:

The period prevalence includes everyone (alive, dead, cured) who had the condition during the period in question.

2.1.2 Rates

A rate is a ratio that consists of a numerator and a denominator and in which time forms part of the denominator. It must contain:

• Disease frequency
• Unit size of the population
• Time period during which an event occurs

When we report rates, we often use multipliers. You may recognize from news stories or journal articles rates being reported per 100,000 population. When we look at issues related to children or maternity, we often use 1000 as the multiplier (e.g., per 1000 live births). But 100,000 is the most used standard and assists us when we want to compare rates across populations.

In epidemiology we use three different forms of the rate. First is the crude rate. This rate is the rawest version of a rate. We have not considered any other reasons why that relationship could happen or any other related factors for the situation. It is just a simple numerator and a simple denominator.

Examples:

• Crude birth rate
• Fertility rate
• Infant mortality rate
• Fetal death rate
• Postneonatal mortality rate
• Maternal mortality rate

An example formula:

Use crude rates with caution when comparing disease frequencies between populations. Observed differences in crude rates may be the result of systematic factors (e.g., sex or age distributions) within the populations rather than true variation in rates. If this is the case, we are comparing apples to oranges. We need to make the populations as similar as possible to compare apples to apples.

If we want to compare rates across populations or even get a more accurate rate for our single population, we should do our best to use an adjusted rate. An adjusted rate is a measure in which statistical procedures have been applied to remove the effect of differences in composition of various populations. We can adjust using tools such as the direct method, indirect method, or regression.

The third type of rate we use is called a specific rate. This type of rate refers to a particular subgroup of the population defined in terms of factors such as race, age, sex, or single cause of death or illness (e.g., an age-specific death rate).

Incidence is the number of new cases of a disease that occur in a group during a certain time period. We can use incidence to help us research the etiology of disease and to provide estimates of the risk of developing disease. One way to calculate incidence is as a rate. The incidence rate describes the rate of development of a disease in a group over a certain time period. It has to include a numerator (number of new cases), denominator (population at risk), and time (period during which cases occur).

2.2 Incidence Versus Prevalence

For both incidence and prevalence, we must have:

• A clear, discrete definition of the event (either the event happened or it did not)
• The time frame for the event

An event (outcome) could be the start or end of a biological process (e.g., menopause), death, remission, disease (diagnosis, start of symptoms, or relapse), or the start or end a behavior (e.g., smoking cessation).

Time, as discussed in section 1.2, can vary widely. It is query dependent and could be calendar time, age, time from study recruitment, time from an exposure (e.g., time from employment), or time from diagnosis.

Figure 2.3 displays a comparison between incidence and prevalence and examples of both. The bathtub of prevention^[5] is a common graphical representation of the relationship between the two. Incidence is displayed as the water as it enters the tub. If the drain of the tub is closed, no water exits and it fills. This is similar to the relationship between incidence and prevalence. In reality, often some cases of disease do not survive or are cured; this part of the water escapes the tub. Sometimes those who are cured or are in remission have recurrences of disease and those cases are re-added to the mix. We can use them in the calculation of new cases (they are new) but also separately calculate the incidence of recurrence.

	Definition	Formula	Units	Example	Graphic representation
Incidence	All new cases in a given time (example: last year) in a population	# of new cases / # of people at risk	per unit of time	Of the 100 children at the sports camp yesterday, 20 got a sunburn. Incidence = 20/100
Prevalence	All cases (new + old) in a population in a given time	# of all cases / total # of people	at a point in time or during a period in time	Of the 1000 children in Town A during the summer of 2015, 50 children got a sunburn. Prevalence = 50/1000

Figure 2.3: The bathtub of prevention.

Prevalence includes all cases in a given time. If you are studying a patient’s survival time, mortality, or recovery time, you are interested in prevalence. Prevalence is approximately equal to the incidence of the disease times the duration (length) of the illness. What does this mean? If the duration of disease is short and incidence is high, prevalence becomes similar to incidence. Conditions that have short duration mean that cases recover rapidly or are fatal, so prevalence does not have time to build up. This is typical of certain infectious diseases such as the common cold. Conditions that have a long duration and a low incidence result in a prevalence that increases at a pace faster than incidence. This is very typical of chronic diseases. We have also come to recognize this with some infectious diseases such as HIV due to the wide availability of drug therapies and prevention strategies. If immunization programs work as intended for vaccine preventable diseases such as measles, the incidence should be low, as should prevalence. Figure 2.4 provides three scenarios in which we might calculate prevalence and what the expected outcome would be.

Figure 2.4: Scenarios and outcomes for prevalence.

Incidence includes new cases only in a given time. This, as well as prevalence, are impacted by prevention strategies and their acceptance in the population. When we see incidence and prevalence change, we need to understand why they are changing and appropriately adjust our numbers so we can best evaluate what our next steps should be (figure 2.5).

Figure 2.5: Scenarios and outcomes for incidence and prevalence.

Figure 2.6 summarizes various common types of prevalence and incidence and how we should interpret them. As you can see, there are at least four types of incidence that we can calculate. The difference is in how we capture the denominator and the population used. More details on when to use the different types of incidence can be found in section 2.3.

Figure 2.6: Prevalence and incidence summary table.

2.3 More Details on Calculating Incidence

There are two general types of incidence: cumulative incidence and the incidence rate. The difference is how the denominator is calculated.

2.3.1 Cumulative Incidence

If we have a constrained population (we know everyone involved and we have been tracking them; e.g., a clinical trial or prospective cohort study [covered in chapter 3]), we usually calculate cumulative incidence because we can focus on person-time at risk. Cumulative incidence is a proportion and can only range from 0 to 1. It is an extremely precise version of incidence and requires that we have a well-defined and closed population (i.e., a fixed population).

If we have completed follow-up (meaning we have details on every subject):

However, the number of times you will have complete follow-up information is rare. Study subjects are often censored, meaning they did not complete follow-up (e.g., disappeared, moved out of the study area, left because of other health problems, died, or were recruited late in the study), meaning you will need to use either the Kaplan-Meier method or the classic life table method to calculate cumulative incidence. Cumulative incidence is also known as the hazard of having an outcome. It is the complement of cumulative survival. If we can calculate the chance of having the event, we can also calculate the chance of surviving without it. This is one place where the Kaplan-Meier method (K-M) and the classic life table (CLT) shine. They allow us to have different follow-up time for study participants, including those who started later and those who are censored.

2.3.1.1 Classic Life Table

Example: Classic life table

For example, we follow 10 postoperative orthopedic patients for one year. We are interested in finding out how long it took before the patients were cleared from rehabilitation to resume normal activities. We found that five patients completed rehabilitation before the year was over, four were censored, and one was still in rehabilitation at the end of the year.

In figure 2.7, we can see a row for each of our 10 patients across the 12-month study period. We can see that some patients were enrolled in January, while others were enrolled later. Boxes that are shaded are the months the subject was in rehabilitation. The letter R denotes that the patient was released to normal activity. The letter C denotes that the patient was censored. In this example, we would calculate cumulative incidence for the patients that were released (the outcome of interest) and the cumulative survival (chance of remaining in the study without the outcome) for patients that were not released. Five patients were released and five patients were not released, so the cumulative survival is:

 

If we change the time scale to be months contributed to the study, we get figure 2.8.

Note that the x-axis has changed from calendar time to follow-up time. Cumulative incidence can now be calculated for each individual time point in the study. For example, if we wanted to know the incidence of being released by the end of six months (time point 6), we see that, including that time point, three patients have been released and four have been censored.

Cumulative incidence:

Cumulative incidence_{time point t}

Cumulative incidence_{time point 6}

Cumulative survival:

Cumulative survival_{time point t}

Cumulative survival_{time point 6}

There is a problem: we lost all of the data for the people that censored at or before time point 6! Our rules tell us that we must always take into account those that have censored. If we use the CLT approach, we choose to assume that everyone that censors during the time period contributes one-half the risk in the denominator. This method assumes that censoring happens uniformly throughout the period under study.

In action, this looks like:

Cumulative incidence_{time point t}

Cumulative incidence_{time point 6}

By the end of time point 6, 37.5 percent of patients had been released from rehabilitation to normal activity.

Cumulative survival_{time point t} = 1-cumulative incidence_{time point t}

Cumulative survival_{time point 6} = 1-0.375 = 0.625 or 62.5 percent

Patients remaining in the study at the end of time point 6 had a 62.5 percent chance to remain in rehabilitation past this time point.

As we can see from these results and figures 2.7 and 2.8, the probability that a patient stays in rehabilitation changes over the study. Because the chance of finishing rehabilitation or staying in rehabilitation changes over the study period (e.g., because of weather changes, changes in rehabilitation site, or changes in other medical conditions), we may need to examine this problem over multiple intervals of time. With an infectious disease such as the flu or a cold, we expect to need to calculate the cumulative incidence based on seasons of the year.

2.3.1.2 Kaplan-Meier Method

The primary difference between the classic life table and K-M methods is that in the latter we calculate the incidence and survival every time there is an event (outcome). This allows us to give all participants full credit for their time in the study.

Looking at figure 2.8, we see that patients were released from rehabilitation in month 3, month 5, month 6, and month 7. We will use this information to calculate the conditional probability of the event/survival during the study observation period. In figure 2.9, we have six columns: the time points when events happen (A); the number of patients under observation in the study at that time point (B); the number of patients with an event at that time point (C); the probability of the event occurring (conditional on it being at that time point) (D); the probability of surviving (conditional on it being at that time point) (E); and the cumulative probability of surviving to that time point (F). Note that participants are included in the denominator until after the time point when the event occurred. As an example, at time point 3, patient 5 is included in the numerator and in the denominator because that person was in the study until that point; at time point 4, the patient is no longer included in the denominator.

Figure 2.9: Kaplan-Meier table.

The K-M Curve (figure 2.10) is used often in clinical studies to show the cumulative survival or the cumulative incidence in graphic form. Known for its “stair step,” the graph shows how the probability (y-axis) changes as time passes (x-axis). As we know from our previous figure/table as the study starts, the probability of being in rehabilitation is 100 percent. Every time a patient completes rehabilitation, the probability of being in rehabilitation (“step down” in red [top left to bottom right]) and the probability of finishing rehabilitation (“step up” in blue [bottom left to top right]) change. These two probabilities mirror each other, so display the graph that best depicts what you are trying to convey.

When using this method, it is important to recognize that if your study period is long, you do not see what are called secular trends, meaning changes in risk over time of your study that could be attributed to something else. You also need to make sure that censoring is independent of survival (i.e., those who censor have the same prognosis as those who remain in the study). If those who censor are for some reason different from those who remain (e.g., older or sicker), the results of your study will be biased. If censored observations have a worse prognosis than those in the study, the observed survival will be greater than the real answer. If censored observations have a better prognosis than those in the study, the observed survival will be lower than the real answer. See chapter 5 for more on bias.

2.3.2 Incidence Rate

In our previous examples discussing cumulative incidence, our population was fixed. More often, however, we have an open or dynamic population, such as the patients in a hospital, the population of a state, or the population of students at a university. Dynamic means that people can come in and out of the population (e.g., births, deaths, migration). In this case we typically use incidence rate (also known as incidence density). The incidence rate is a ratio, but it is not a proportion. The value can be from zero to infinity.

If you have a better-defined population, like a cohort of patients from a hospital where all have a specific diagnosis coming through the emergency department and each one of them participates a different amount of time, calculating incidence using person-time as the denominator is the best option. If you are looking at geographical-type areas, use the average population as the denominator. In this instance, the formula is:

The average population can be calculated one of two ways.

•   Method 1

Using our previous example (figure 2.9), we started with 10 people in rehabilitation and finished with 1 in rehabilitation.

Average population = (10+1) / 2 = 11 / 2 = 5.5

In a different example, if our town had a ski resort and the population changed several times over the year to include seasonal workers or seasonal residents, we might have more population numbers to consider. If the population during ski season was 10,000, immediately after the season was 6,000, and during the summer was 8,000, we would calculate the denominator as follows:

Average population = (10,000 + 6,000 + 8,000) / 3 = 24,000 / 3 = 8,000

•   Method 2

Average population = population at the beginning of the period – events- censored

Using our original example (figure 2.9), we started with 10 people in rehabilitation, 5 people had the event of interest (getting out of rehabilitation), and 4 people censored.

Average population = 10 – (5) – (4) = 10 – 2.5 – 2 = 10 – 4.5 = 5.5

In our ski example, if 1,500 people were injured and 2,200 censored, we would calculate the denominator as follows:

Average population = 10,000 – (1,500) – (2,200) = 10,000 – 750 – 1,100 = 10,000 – 1,850 = 8,150

2.3.2.1 Person Time

Using person-time, or incidence based on how much time was contributed by participants, requires precise data from a very defined population. The denominator is calculated as how much time each person contributed to the study. For example, if we studied 5 people for 5 years, the 5 people would have contributed a total of 25 person-years to our study. If we studied 25 people for 1 year, the 25 people would have contributed a total of 25 person-years to our study. We then take this information to calculate our incidence rate (density):

We use person-time when we cannot determine the incidence of the event for individuals like we would with cumulative incidence but need a similar answer. We use the time unit (e.g., years, months, days) that is most relevant to the situation. For the time at risk, we assume the participant with the event provided half of the relevant time period before their outcome (e.g., if the time period is one year, the participant is assumed to have had the event at six months).

If we look at figure 2.11, we see a graphical representation of a study cohort over a four-month study period. At the beginning of the study, there are eight volleyball players on a team. We want to observe the incidence of shoulder injuries during the four-month period but only have team-level information. In the first month, one player gets a shoulder injury. In the second month, no player is injured. In the third month, four players get a shoulder injury. In the fourth month, no player is injured.

At the end of month 1 (example January 1 to January 31), the person-time contributed by the eight volleyball players is 7.5 months.

7 players uninjured → person-time = 7 x 1 = 7

1 player injured → person time = 1 x 0.5 = 0.5

The total person-time after 1 month = 7+0.5 = 7.5

The amount of person time contributed in month 2 (example February 1 to February 28) by the seven remaining uninjured players is seven months.

7 players uninjured → person-time = 7 x 1 = 7

The total person-time to this point is 14.5 months (7.5 + 7).

The amount of person time contributed in month 3 (example March 1 to March 31) by the seven remaining uninjured players is five months.

3 players uninjured → person-time = 3 x 1 = 3

4 players injured → person-time = 4 x 0.5 = 2

Total person-time added in month 3 = 3+2 = 5

The total person-time to this point is 19.5 months (14.5 + 5).

The amount of person-time contributed in month 4 (example April 1 to April 30) by the three remaining uninjured players is three months.

3 players uninjured → person-time = 3 x 1 = 3

The total person-time at the end of the study period (example January 1 to April 30) is 22.5 months (19.5 + 3).

There were a total of five injuries during the study period.

Incidence of shoulder injuries = 5 / 22.5 person-months = 0.22 injuries per person-month

or 2.67 injuries per person-year or 0.88 injuries per season

Figure 2.12: Game and practice injury rates, 15 Sports, National Collegiate Athletic Association (1988–1989 through 2003–2004).

In SRI research, person-time is often referred to as an athlete-exposure. Figure 2.12 is an example of the use of person-time in the NCAA Injury Surveillance Study to calculate the rates of game and practice injuries from the 1988–1989 academic year through the 2003–2004 academic year.^[6] Because of the number of players and the lack of the ability to count precisely how much time each individual athlete is present for a game or a practice and how much time each spends at said event actually playing instead of not being active, being present at the event and on the roster that day counts as an exposure. Athletic-training and clinician records are often used to count the number of injuries that occurred and when they occurred for the numerator. In figure 2.12, for example, we see that for Division III sports, there were 115,725 athlete-exposures and 562 injuries during games in the preseason. Using our formula for calculating person-time and 1000 as a multiplier, we find the following:

Game injury rate = x 1000 = 0.00486 x 1000 = 4.86 game injuries per 1000 athlete-exposures in Division III sports

2.4 Dynamics of Disease

2.4.1 Demographic Transition

The demographic transition refers to the change in population makeup due to births, deaths, and migration. Populations move from high births and high death rates from a time before the Industrial Revolution to low birth rates and low death rates due to factors such as improved economic conditions, sanitation, and better health care. This change occurs as a population (e.g., a country) moves from being agrarian to postindustrial. The demographic transition is important in clinical medicine because as we become more dependent on industry and richer as a population, our birth rate decreases, people live longer so the death rate decreases, and the population as a whole starts to decrease in size. Figure 2.13 displays the transition using population pyramids.^[7]

Figure 2.14 shows how this transition has occurred in five countries. As we can see, from 1820 where the x axis starts through 2010 where it ends, the population size has increased for all countries, but each has a slightly different pattern of how births and deaths occurred. However, in all cases, the births and deaths eventually fell, and yet the overall population size remained high because people lived longer.

Despite overall population growth, the demographic transition also means that populations will stop growing and eventually start falling. This can be seen in figure 2.15. There are lines for the least developed countries, less developed regions, and more developed regions. We can see that from 1950 to the present day, the population growth rate for all three countries has dropped and then slightly leveled out. It is predicted that the rates will continue to drop through the end of the century.

2.4.2 Epidemiologic Transition

The epidemiologic transition is an extension of the demographic transition and refers to how as countries transition from being more agrarian to more industrial, the causes of their deaths tend to change as well. In more agricultural societies and when populations have high rates of births and deaths, the causes of death tend to be infectious diseases and complications from reproduction. As populations become more industrial and even postindustrial, the advent of better sanitation, health care, transportation, and so on allows for improvements in care to reduce the burden of infectious diseases and reproductive outcomes. As people live longer, they become more susceptible to noncommunicable diseases and injury such as heart disease and falls. The originator of the epidemiologic transition theory,^[8] Dr. Abdel Omran, has published in length about this idea (see figures 2.16 and 2.17).

In figure 2.16, we can see all of the dynamics that feed into the epidemiologic transition and how it is built on top of the demographic transition.

In figure 2.17, we can see the movement of the preventable disease burden over time and how this impacts what we see in the clinical space.

2.4.3 Epidemic Curve

One mechanism used to examine disease in populations, particularly infectious diseases, is the epidemic curve (epi curve). An epi curve is a visual display of the onset of illness among cases associated with an outbreak.

You can learn a lot about an outbreak from an epi curve,^[9] such as:

• The outbreak’s time trend; that is, the distribution of cases over time
• Outliers; that is, cases that stand apart from the overall pattern
• General sense of the outbreak’s magnitude
• Inferences about the outbreak’s pattern of spread
• The most likely time period of exposure

In an epi curve, the x-axis represents the time frame of interest. Depending on the condition, this time frame might need to be minutes, hours, days, weeks, months, or even years. The y-axis is the incidence of cases of disease. This scale depends on how many cases exist. In figure 2.18, example B shows what we might see if we plot a disease that occurs sporadically, like Creutzfeldt-Jakob Disease. Example C shows what we might see with an endemic disease. In the United States, an example of an endemic disease is influenza. Example D shows us what an epidemic disease that is spread from a single source looks like if we were to plot the cases like foodborne illness from a potluck. This is different from epidemic disease with a propagating source (Example E). In these types of epidemics, the initial wave of disease propagates (i.e., is the source of) the following cases of disease. An example would be a measles outbreak in the United States.

Figure Descriptions

Figure References