1.5.4 measures incidence+incidence density

b

1. Measures - incidence Incidence time Not sufficient to just record proportion of population affected by disease Necessary to account for the time elapsed before disease occurs and the period of time during which the disease events take place

2. Measures - incidence A P(D A ) = 0.6 B P(D B ) = 0.6

3. Measures - incidence Incidence time Incidence time is time from referent or zero time (e.g., birth, start of treatment or exposure, start of measurement period) until the time at which the outcome event occurs Also called event time, failure time, occurrence time

4. Measures - incidence Incidence time “Censoring” occurs if the time of event is not known because something happens before the outcome occurs – Examples: lost to follow-up, death, surgery to make outcome impossible like hysterectomy, end of measurement period ime = avera ge time until an event Average incidence t occurs

5. Measures – incidence density Incidence density (ID) - aka incidence rate (IR) The rate of occurrence of new cases of disease during person-time of observation in a population at risk of developing disease Numerator: number of new cases of disease Only count cases in the numerator that are contributing to person-time in the denominator Denominator: person-time of observation in population at risk Only count contributions to the denominator that could yield cases for the numerator A rate Units are “inverse time” (1/time, time -1 ) Range is 0-infinity

6. Measures – incidence density Incidence density What is “person-time”? Person-time at risk: length of time for each individual that they are in the population at risk – Sum over population is total person time at risk When a person is no longer “at risk” they cease contributing to person-time, this includes when they get the outcome of interest One person year could be 2 people x 6 months each, 1 person x 12 months, 3 people x 4 months, etc. Helps account for censoring and different observation periods

7. Measures – incidence density “Figure 2 suggests that ID may be viewed as the concentration or 'density' of new case occurrences in a sea of population time. The more dots per unit area under the curve, the greater is the ID.” Morgenstern et al. 1980

8. Measures – incidence density Person-time calculations for individual level data 1) If exact time contribution of each individual is known: – Sum the disease-free observation time

9. Measures – incidence density Person-time calculations for individual level data If data on each individual is collected at regular intervals: Estimate the disease-free observation time in each interval Note: variants of this formula also subtract I j /2 from N’ 0j

10. Measures – incidence density Person-time estimation from group level data If the population is in steady state can estimate based on population size (N’) and duration of follow-up (Δt) If the population is not in steady state can estimate based on mid-interval population (N’ 1/2 ) and duration of follow-up (Δt) – Note: mid-interval population size can be estimated as: (N t0 + N t1 )/2

11. Measures – incidence density Uses and limitations of incidence density Appropriate for fixed or dynamic populations; does not assume that everyone is followed for specified time period Does not distinguish between people who do not contribute to disease incidence because they were not in the study population long enough for disease to develop and those who do not contribute because they never got the disease (relates to next point)

12. Measures – incidence density Uses and limitations of incidence density 100 person-years could come from following 100 people for one year or two people for 50 years – no way to tell the difference without knowing the incidence time Have to consider whether study design allowed appropriate time to elapse to plausibly consider an exposure disease relation Disease process is important to consider in developing appropriate study design and disease measures Example: disease free cohort of 50 exposed and 50 unexposed followed for 1 year might not allow sufficient time to elapse for exposure to cause disease

13. Measures – incidence density In class exercise Study population observed monthly for 6 months What is the person-time contributed by this population? What is the incidence density?

14. Measures – incidence Hazard rate The instantaneous potential for change in disease status per unit of time at time t relative to the size of the candidate (i.e., disease-free) population at time t Instantaneous rate in contrast to incidence density which is an average rate Cannot be directly calculated because it is defined for an infinitely small time interval Hazard function over time can be estimated using modeling techniques (more in the analyzing epidemiologic data section)

15. Measures – incidence Hazard rate

16. Measures – incidence Survival function

17. Measures of disease outline Big picture Illustration/discussion of measuring disease in time Populations Time scales affecting disease in populations Epidemiologic measures Basic concepts Measuring diseases Prevalence Incidence density (incidence rate) Cumulative incidence (risk) Relations among measures Standardization Summary Appendix: specific measures of disease

18. Measures – cumulative incidence Cumulative incidence (CI) – aka risk, incidence proportion (IP – Rothman) The proportion of a closed population at risk that becomes diseased within a given period of time Numerator: number of new cases of a disease or a condition (Rothman calls this A) Denominator: number of persons in population at risk (Rothman calls this N) A proportion Range is 0-1 – dimensionless

19. Measures – cumulative incidence Cumulative incidence Calculated for a fixed time period Only interpretable with information on time period over which it was measured Population measure that translates most readily to individual Interpreted as capturing individual risk of disease Different methods for calculating Variations depending on how time at risk is handled Option for calculating from rate measure

20. Measures – cumulative incidence Different methods for calculating Simple cumulative Actuarial Kaplan-Meier Density

21. Measures – cumulative incidence Subscript notation R (t0,tj) – risk of disease over the time interval t0 (baseline) to tj (time j) R (tj-1,tj) – risk of disease over the time interval tj-1 (time before time j) to tj (time j)

22. Measures – cumulative incidence Subscript notation N’ – number at risk of disease at t0 (baseline) N’ 0j – number at risk of disease at the beginning of interval j

23. Measures – cumulative incidence Subscript notation I j – incident cases during the interval j W j – withdrawals during the interval j

24. Measures – cumulative incidence Simple cumulative method: R (t0,tj) = CI (t0,tj) = I N' Risk calculated across entire study period assuming all study participants followed for the entire study period, or until disease onset – Assumes no death from competing causes, no withdrawals Only appropriate for short time frame

25. Measures – cumulative incidence Simple cumulative method: Example: incidence of a foodborne illness if all those potentially exposed are identified

26. Measures – cumulative incidence Actuarial method: R (tj-1, tj) = CI (tj-1, tj) = I j N' 0j - W j /2 Risk calculated accounting for fact that some observations will be censored or will withdraw Assume withdrawals occur halfway through each observation period on average Can be calculated over an entire study period – R (t0,tj) = CI (t0, tj) = I/(N’ -W/2) Typically calculated over shorter time frames and risks accumulated

27. Measures – cumulative incidence Modification of Szklo Fig. 2-2 – participants observed every 2 months (vs 1)

28. Where to start – set up table with time intervals Fill incident disease cases and withdrawals into appropriate intervals Fill in population at risk Measures – cumulative incidence Actuarial Method

29. Calculate interval risk • R (tj-1, tj) = I j /(N’ 0j -(W j /2)) • R (0,2) =1/(10-(1/2)) = 0.11 Measures – cumulative incidence Actuarial Method

30. Calculate interval survival • S (tj-1,tj) = 1-R (tj-1,tj) Measures – cumulative incidence Actuarial Method

31. Calculate cumulative risk – example of time 0 to 10 • R (t0, tj) = 1 - Π (1 – R (tj-1,tj) ) = 1 - Π (S (tj-1,tj) ) • R (0, 10) = 1 – (0.89 x 0.88 x 1.0 x 1.0 x 0.85) = 0.34 Measures – cumulative incidence Actuarial Method

32. Calculate cumulative survival • S (t0,tj) = 1-R (t0,tj) Measures – cumulative incidence Actuarial Method

33. Intuition for why R (t0, tj) = 1 - Π (S j ) using conditional probabilities Example of 5 time intervals: – Π (S j ) = P(S1)*P(S2|S1)*P(S3|S2)*P(S4|S3)*P(S5|S4) = P(S5) Product first two terms: P(S2|S1)*P(S1) = P(S2) Multiplying conditional probabilities gives you unconditional probability of surviving up to any given time point the value (1 - survival) up to (or at) a given time point is then the probability of not surviving up to that time point Measures – cumulative incidence

34. Measures – cumulative incidence Exercise for home (discuss in lab) Study population observed monthly for 6 months Calculate the cumulative incidence of disease from month 0 to 6