mbcann01
commented
1 year ago Add these sections back
Relation between Incidence and Prevalence
$${Prevalence} \approx {Incidence} * {Duration}$$
$$\frac{Prevalence\, Proportion}{1 - Prevalence\, Proportion} = {Incidence\, Rate} * {Average\, Duration\, of\, Disease}$$
Briefly, it is worth explaining how prevalence and incidence relate to each other, conceptually and mathematically. Conceptually, prevalence over a time period is a function of both incidence (how many new cases are showing up?) and duration of disease (how long does each case last?). This is because the longer the duration of disease, the more incident cases accumulate during a time period. Consider a population of 100 people, 10 of whom get an upper respiratory infection on the first day of each month. If the infection lasts a full month, then, on average, the prevalence of such infections over time will be 10%: 10/100 for the first month, then a new 10/100 for the second month. But if the infection lasts 2 months, then, after the first month, the prevalence of infection will be about 20%: 10 cases from the first month and 10 more from the second month.6 This suggests a rough equivalency between prevalence, incidence, and duration, of the form:
where the squiggly equals sign (≈) means “approximately equal to.” More precisely, the relationship is
where the left side of the equation is exactly prevalence odds. This relationship will hold when the population is in “steady-state”; that is, constant (or nearly so) in composition and size and with constant (or nearly so) incidence rate and disease duration (Rothman, 2012; Rothman, Greenland, & Lash, 2008). If prevalence of the disease is low (typically, <10%), then this calculated prevalence odds will approximately equal to the prevalence proportion.
$$Risk = 1 - e^{-{Incident\, Rate\, *\, Time}}$$
$$Risk \approx {Incident\, Rate\, *\, Time}$$
In our preceding example, we might not get 10 cases in the second month: after the first month, there are only 90 people left who are not infected. If the incidence were 10% per month, we would expect 9 new cathan 10, because the population at risk of infection declines over time. This points to the relationship between the incidence rate and the incidence proportion (sometimes in this context called the cumulative incidence). In a closed cohort with a constant incidence rate, these two measures are related by the exponential formula as follows:ses, rather than 10, because the population at risk of infection declines over time. This points to the relationship between the incidence rate and the incidence proportion (sometimes in this context called the cumulative incidence). In a closed cohort with a constant incidence rate, these two measures are related by the exponential formula as follows:
where and time is the amount of time over which we wish to estimate risk (Rothman et al., 2008). Note again that here we are assuming a closed cohort (such that the number of individuals is diminishing over time), whereas in the relationship between prevalence and incidence, we assumed a steady-state population (people who leave the cohort are replaced by people entering the cohort). However, in parallel to the preceding relationship between prevalence and incidence, when risk is <10%, this relationship can be estimated as
Incidence Rate vs. Incidence Proportion
- Two key differences between the incidence proportion and incidence rate are worth highlighting. First, consider a situation in which we have data on 10 people, each followed-up for a different length of time. In this case a straightforward risk calculation would likely be invalid: the proportion of those 10 people who experienced an outcome would not be in a fixed period of time and thus would not typically or simply be interpretable as an incidence proportion. On the other hand, we could calculate the total person-time accumulated by those individuals and use that as the denominator for an incidence rate without concern that each individual contributed a different amount of time over follow-up.
- Second, the incidence rates can account for recurrent events, whereas incidence proportions cannot. Recall that the incidence proportion is constrained to be between 0 and 1: thus, with N people, the incidence proportion may falter at fully describing the occurrence of 2N events (as might happen if we were studying the number of upper respiratory infections experienced by individuals over a period of 6 months). Where an incidence rate can accumulate both a count of events and an experience of cumulative person-time among study participants and calculate an incidence rate, the incidence proportion might be forced to study the risk of first upper respiratory infection (and then, among those who had a first upper respiratory infection, the risk of a second and then third). These are different scientific questions and might both be of interest to public health.
Odds and proportions
knitr::include_graphics("img/10_part_intro_epi/03_measures_of_occurrence/odds_and_proportions.png")”Low” is generally considered to mean less than 10%. You can see why in the far right panel here.
As shown in Figure 1.1, as the prevalence proportion (the probability, on the x-axis) ranges from 0 to 1, the prevalence odds (y-axis) ranges from 0 to infinity.
Prevalence odds will approximate the prevalence proportion when the prevalence proportion is low but may otherwise overstate prevalence proportion.
Overview
In the Fall of 2023, I moved over a bunch of stuff from PowerPoint slides (nearly) verbatim. I was in a rush, so I told myself to move it just move it over and improve it later.
Go back, reread, and improve. PowerPoint doesn't always translate perfectly to book format.
Tasks