Main Points to be Covered • Incidence versus Prevalence • The 3 elements of measures of incidence • Cumulative incidence vs. incidence rate • Calculating cumulative incidence by the Kaplan-Meier method • Calculating cumulative incidence by the life table method
Prevalence versus Incidence • Prevalence counts existing disease diagnoses, often at a single point in time • Incidence counts new disease diagnoses during a defined time period
Prevalence: 3 types • Point prevalence - number of existing cases at one point in time divided by population • Period prevalence - number of existing cases in a time interval (eg, one year) divided by population • Cumulative (lifetime) prevalence - proportion of people who have had the outcome at any time in the past
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
Disease Occurrence Measures: A Confusion of Terms • Terminology is not standardized and is used carelessly even by those who know better • Key to understanding measures is to pay attention to how the 3 elements of number of events, number of persons at risk, and time are used • Even the basic difference between prevalence and incidence is often ignored
HIV/AIDS infection rates drop in Uganda Kyodo News Service KAMPALA, Sept. 10 (Kyodo) - Infection rates of the HIV/AIDS epidemic among Ugandan men, women and children dropped to 6.1% at the end of 2000 from 6.8% a year earlier, an official report shows. The report, compiled by the Ministry of Health together with the World Health Organization and the Medical Research Council of Britain, says the results were obtained after testing the blood of women attending clinics in 15 hospitals around the country. The ministry deduced the figures for men and children from the blood tests on women, according to the report. The report says the average rate of infection for urban areas fell from 10.9% to 8.7%. In rural areas, the average was 4.2%, not much different from the 4.3% average a year earlier. The highest infection rate of 30% was last reported in western Uganda in 1992.
The word “rate” should be avoided when existing cases at one point in time are what was measured. Although you may encounter “prevalence rate,” rate should be reserved for measuring incidence.
Measures that are sometimes loosely called Incidence • Count of the number of events (E) • eg, there were 84 traffic fatalities during the holidays • Count of the number of events during some time period (E/T) • eg, traffic accidents have averaged 50 per week during the past year • Neither explicitly includes the number of persons (N) giving rise to the events
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)
E/N E/T E/NT E
Cumulative Incidence • Perhaps most intuitive measure of incidence since it is just proportion of those observed who got the disease • Basis for Survival Analysis • Two primary methods for calculating • Kaplan-Meier method • Life table method
Survival Analysis • Data analysis for which the outcome is time to an event • The time variable is usually called survival time • The outcome event is usually called a failure (normally an adverse event but it doesn’t have to be) • Cumulative incidence is the complement of cumulative survival (1 - cumulative survival)
Calculating Cumulative Incidence • With complete follow-up cumulative incidence is just number of events (E) divided by the number of persons (N) = E/N • Rarely have equal follow-up on everyone so need to account for different follow-up times • Can be due to losses to follow-up • Can be due beginning follow-up at different times
Cumulative Incidence in the Setting of a Cohort Study If number of events (E) for all 1000 is known, cumulative incidence is just E/1000. But 7 persons left the cohort.
Cumulative incidence with Kaplan-Meier estimate • Set in a cohort • subjects have different starting dates • subjects have different amounts of follow-up time • Requires date last observed or date outcome occurred on each individual (end of study can be the last date observed) • Analysis is performed by dividing the follow-up time into discrete pieces • calculate probability of survival at each event
3 Ways Censoring Occurs 1) Death (if death is not the study outcome) 2) Loss to follow-up (refuse, move, can’t be found) 3) End of study observation (if still alive and haven’t experienced outcome) • Each subject either experiences the outcome or is censored in a survival analysis
Calculating Cumulative Incidence • Probability of two independent events occurring is the product of the two probabilities for each occurring alone • eg, if event 1 occurs with probability 1/6 and event 2 with probability 1/2, then the probability of both event 1 and 2 occurring = 1/6 x 1/2 = 1/12 • Conditional probability of living to time 2 given that one has already lived to time 1 is independent of the probability of living to time 1
Cumulative calculated by multiplying probabilities for each prior failure time: e.g., 0.9 x 0.875 x 0.857 = 0.675 and 0.9 x 0.875 x 0.857 x 0.800 x 0.667 x 0.500 = 0.180
Graphical representation of K-M survival analysis (survival curve with discrete steps)
Kaplan-Meier Cumulative Incidence of the Outcome • Cannot calculate cumulative event probability directly by multiplying--it keeps getting smaller • (in our example, 0.100 x 0.125 x 0.143 x 0.200 x 0.333 x 0.500 = 0.0000595) • Cumulative probability of the event is obtained by subtracting the cumulative probability of surviving from 1; eg, (1 - 0.180) = 0.82 • Since it is a proportion, it has no time unit connected to it, so time period has to be added; e.g, 2-year cumulative incidence
Kaplan-Meier using STATA Need a data set with one observation per person. Each person either experiences event or is censored. Need a variable for the time from study entry to date of event or date of censoring (time variable=timevar). Need a variable indicating whether follow-up ended with the event or with censoring (failure variable=failvar) STATA syntax: (1) declare that you have survival data stset timevar, failure(failvar) (2) List Kaplan-Meier function: sts list (3) Graph Kaplan-Meier function: sts graph
Two assumptions in survival analysis • Censoring is unrelated to survival (unrelated to the probability of experiencing the outcome) • There are no temporal trends in the probability of the outcome
Life table method of estimating cumulative incidence • Key difference from Kaplan-Meier is that probabilities are calculated for fixed time intervals, not at the exact time of each event • Fixed time intervals can vary in length but are often uniform • Probability of surviving each fixed time interval is calculated • Cumulative survival is product of probabilities from each prior time period
Life table method of estimating cumulative incidence • Since exact event times not used, assumption required that events and censoring occur uniformly during the fixed time intervals • Calculations are based on assigning censored individuals follow-up for half of the time period (follows from the uniformity assumption) • Subtract one-half of subjects lost during interval from denominator at interval beginning
6 deaths in 2 years; 3 censored before 2 years of follow-up c
Example of Life Table Calculation from the Text Data Taking the full 2-year period as one interval: 6 deaths in the numerator and 10 (initial number) minus 0.5 x 3 censored persons in the denominator = 6 / 8.5 = 0.71, the cumulative probability of death (1 - 0.71) = 0.29, the cumulative probability of survival (NB - text gives this incorrectly as 0.39) Note this differs from the Kaplan-Meier cumulative survival estimate of 0.18
Example of Life Table Calculation Using Two One-Year Intervals Taking two 1-year time intervals: First year: Starts with 10 at risk, 3 deaths, 2 censored, probability of survival is 7 / 10 - 0.5 x 2 = 7/9 = 0.777 Second year: Starts with 5 at risk, 3 deaths, 1 censored, probability of survival is 2 / 5 - 0.5 x 1 = 2/4.5 = 0.444 Cumulative probability of survival = 0.777 x 0.444 = 0.345 This differs from both the Kaplan-Meier estimate and the life table using only one 2-year interval.
Life Table Method • Can see from inspecting the data used in the text that for these 10 observations the life table uniformity assumption doesn’t hold • Life table more commonly used on large secondary data sets where information on exact failure times are not available • With very large numbers the uniformity assumption is more likely to be valid
Calculating a 95% confidence interval for either a Kaplan-Meier or a life table estimate of cumulative survival SE (Si) = Si x Σ (dj/nj x (nj - dj)) 95% CI = Si + [1.96 x SE (Si)] eg, SE (S9) = 0.675 x 1/(10(10-1) + 1/8(8-1) + 1/7(7-1)) = 0.155 95% CI = 0.675 + (1.96 x 0.155) = 0.371 to 0.979
Summary Points • Prevalence counts existing cases and incidence counts new cases • Word “rate” is very loosely used • Two main types of incidence rate • incidence based on proportion of persons • incidence based on person-time • Kaplan-Meier or life table method of estimating cumulative incidence assume losses unrelated to outcome probability and that there are no temporal trends in outcome probability