Main Points to be Covered

1. Main Points to be Covered • Difference between cumulative incidence based on proportion of persons at risk and incidence rate based on person-time • Calculating person-time incidence rates • Uses of person-time incidence rates • Relation of prevalence to incidence • Odds versus probability

2. The Three Elements in Measures of Disease Incidence • E = an event = a disease diagnosis or death • N = number of persons in the population in which the events are observed • T = time period during which the events are observed

3. E/N E/T E/NT E

4. Two Measures Described as Incidence in the Text • The proportion of individuals who experience the event in a defined time period (E/N during some time T) = cumulative incidence • The number of events divided by the amount of person-time observed (E/NT) = incidencerate or density (not a proportion)

5. Person-Time Incidence Rates • The numerator is the same as incidence based on proportion of persons = events (E) • The denominator is the sum of the follow-up times for each individual • The resulting ratio of E/NT is not a proportion; may be greater than 1; value depends on unit of time used

6. c

7. rates: year 1 = 3/7.083 = 42.4/100 person-years year 2 = 3/2.50 = 120/100 person-years both yrs 6/9.583 = 62.6/100 person-years

8. We have been calculating average rates; rate is often instantaneous change in one measure with respect to a second measure as interval 0 death rate Population size time In disease, the occurrence rate is often called a hazard or the force of morbidity (mortality)

9. Rates • We are used to rates being change in a measure with respect to time but time does not have to be involved • Accidents per passenger-mile, for example, is often used in transportation • Economics often uses rates in which time is not an element (eg, energy use per unit of gross national product)

10. Comparison of cumulative incidence and incidence rate (density) • Kaplan-Meier cumulative incidence estimate for these data was (1 - 0.18) = 0.82 (ie, 82% of persons will experience event in a two-year period) • Two-year incidence density is 62.6 / 100 person-years or 0.626 per person-year • Not a proportion--if calculated per person-days, rate would be 0.17 / 100 person-days

11. Incidence rate (density) value depends on the time units used An incidence rate of 100 cases per 1 person-yr: • 100 cases/person-year • 10,000 cases/person-century • 8.33 cases/person-month • 1.92 cases/person-week • 0.27 cases/person-day • Note: time period during which rate is measured can differ from the units used

12. Relationship of cumulative incidence and incidence rate • Our example E is large compared to N and the cumulative incidence and the rate differ • If E very small in relation to N (ie, incidence and losses to follow-up are low), then they will be approximately the same • Can make them nearly the same by choosing a small enough time interval so that no more than 1/N leaves pop. at risk • This is true if E is a non-recurrent event

13. Person-time incidence based on grouped vs. individual data • Szklo and Nieto use rate when based on group data and density when based on individual data (not followed by most) • Total person-time for grouped data is based on the time interval x the average population at risk during the interval • Assumes uniform occurrence of events and of censoring during the interval (like life table)

14. Calculating person-time incidence using grouped data • Use average number of persons at risk • In the text example, start with 10 persons, 6 die and 3 are lost to follow-up • Subtract 0.5 x (6 + 3) from 10 = 5.5 • uniformity assumption as in life tables • Total person-time is 5.5 x 2-years = 11 person-years. 6 events, so rate 6/11= 0.545 = 54.5 per 100 person-years (compare to 62.6 when calculated using individual data)

15. Incidence from grouped data • Most commonly used for large secondary data sets where precise information on occurrence of events and on persons leaving and entering population are not available • eg, annual cancer mortality rates per 100,000 population ( = per 100,000 person-years) • If times of events and of censoring available, would normally use individual level data

16. Group data rates versus individual data rates • How much they differ depends on how close events and losses are to being uniform throughout the follow-up • Total person-time calculation for the denominator (and thus the rate) is the same whether based on average population size or individual follow-up if losses are perfectly uniform

17. Individual calculation: 2 deaths / 5 pers-yrs = 0.4 per pers-yr Group data: average population = (4 + 1) / 2 = 2.5 rate = 2 / 2.5 x 2 = 0.4 pers-yr

18. Rates based on group data • Uniformity of events and losses likely to be approximately true for large secondary data sets • Rates using secondary data sets on free-living populations assume new members and losses balance out (= approx. stable) • Important for the use of population reference rates (eg, expected mortality in U.S. population)

19. Assumption of Person-Time Incidence Estimation • T time units of follow-up on N persons is the same as N time units on T persons • Observing 2 deaths in 2 persons followed for 50 years gives the same incidence rate as 2 deaths in 100 persons followed 1 year • Assumption is not reasonable if sample sizes and follow-up times differ greatly (as example above)

20. Assumption of Person-Time Incidence Estimation • If looking at relationship between exposure and outcome rate, one rate for a follow-up period implies exposure does not have cumulative effect on probability of event over time • Clearly false for exposures with cumulative effects like length of time smoking

21. Why use person-time rather than cumulative incidence? • Rates using group data can be calculated in open populations from a variety of data sources where population sizes are estimated • Incidence rates from a cohort study can be compared to standardized rates from the general population to obtain ratio measures called standardized mortality ratio (SMR) or standardized incidence ratio (SIR)

22. Why use person-time rather than cumulative incidence? • If E is a recurrent event, rate may seem more natural. • For example, cumulative incidence of episodes of the common cold, would have to be done separately for each (ie, proportion with 1st cold, proportion with 2nd cold given that you have had 1, etc.).

23. Calculating stratified person-time incidence rates in cohorts • For persons followed in a cohort some potential risk factors may be fixed but some may be variable • eg, ethnicity is fixed; smoking is a behavior that can change over time • Total person-time in an exposure category is one way to deal with risk factors that change over time

26. Relation of Prevalence and Incidence • Prevalence is a function of incidence and duration of disease by the equation: point prevalence = incidence x duration x (1 - point prevalence) [P = I x D (1 - P)] • For many typically low prevalence diseases prevalence becomes approximately I x D since (1 - P) is close to 1 if P is very low

27. Prevalence and Etiology • Because prevalence depends both on incidence and duration of disease, it is not a good measure for etiological studies • Cannot examine the determinants of occurrence alone when you have to account for determinants of duration (Rx, etc.) • Etiologic study designs should avoid sampling prevalent cases of disease

28. Odds versus Probability • Odds based on probability; expresses probability (p) as ratio: odds = p / (1 - p) • odds is always > p because divided by < 1 • For example, if probability of dying = 1/5, then odds of dying = 1/5 / 4/5 = 1/4 • Thinking of odds as 2 outcomes, the numerator is the # of times of one outcome and the denominator the # of times of the other • P = odds / (1 + odds), so 1/4 / 1 + 1/4 = 1/5

29. Odds versus Probability • Less intuitive than probability (probably wouldn’t say “my odds of dying are 1/4”) • No less legitimate mathematically, just not so easily understood • Used in epidemiology primarily because the ratio of two odds is given by the coefficients in logistic regression equations

30. Summary Points • Person-time incidence rate or density is not equivalent to cumulative incidence and is not a proportion • Person-time incidence rate can be calculated with group or individual data • Allows comparison with population reference rates from other data sources • Allows accumulation of time at risk for different strata

32. Calculating stratified person-time incidence rates in cohorts • Assumption of no temporal trends may not be true and therefore need to stratify the follow-up by calendar time • Person-time can be accumulated in strata defined by calendar periods or by age or by both • Graphically represented by Lexis diagram

33. Lexis Diagram