2. Intermediate methods in observational epidemiology 2008 Instructor: Moyses Szklo Measures of Disease Frequency

3. MEASURES OF RISK Absolute measures of event (including disease) frequency: –Incidence and Incidence Odds –Prevalence and Prevalence Odds

4. What is "incidence"? Two major ways to define incidence Cumulative incidence (probability) SURVIVAL ANALYSIS (Unit of analysis: individual) Rate or Density ANALYSIS BASED ON PERSON-TIME (Unit of analysis: time)

5. OBJECTIVE OF SURVIVAL ANALYSIS: To compare the “cumulative incidence” of an event (or the proportion surviving event- free) in exposed and unexposed (characteristic present or absent) while adjusting for time to event (follow-up time) BASIS FOR THE ANALYSIS NUMBER of EVENTS TIME of occurrence Time Survival 1.0

6. Need to precisely define: “EVENT” (failure): –Death –Disease (diagnosis, start of symptoms, relapse) –Quit smoking –Menopause “TIME” : –Time from recruitment into the study –Time from employment –Time from diagnosis (prognostic studies) –Time from infection –Calendar time –Age

7. –Example: Follow up of 6 patients (2 yrs) –3 Deaths –2 censored (lost) before 2 years –1 survived 2 years Question:What is the Cumulative Incidence (or the Cumulative Survival) up to 2 years?

8. Death Censored observation (lost to follow-up, withdrawal) ( ) Number of months to follow-up Jan 1999 Jan 2000 Jan 2001 1 3 2 4 5 6 (24) (6) (18) (15) (13) (3) Person ID Crude Survival: 3/6= 50%

9. Change time scale to “follow-up” time: Person ID 0 12 1 3 2 4 5 6 (24) (6) (18) (15) (13) (3) Follow-up time (years)

10. One solution: Actuarial life table Assume that censored observations over the period contribute one-half the persons at risk in the denominator (censored observations occur uniformly throughout follow-up interval). ID 0 12 1 3 2 4 5 6 (24) (6) (18) (15) (13) (3) Follow-up time (years) It can be also calculated for years 1 and 2 separately: Year 1: S(Y1)= [1 - {1 ÷ [6 – ½(1)]}= 0.82 Year 2: S(Y2)= [1 – {2 ÷ [4 – ½(1)]}= 0.43 S(2yrs)= 0.82 × 0.43= 0.35 100 82 43 Year 1Year 2 Cumulative Survival Follow-up time

11. KAPLAN-MEIER METHOD E.L. Kaplan and P. Meier, 1958* Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event time Step 1: Sort the survival times from shortest to longest *Kaplan EL, Meier P.Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457-81. Person ID 0 12 1 3 2 4 5 6 (24) (6) (18) (15) (3) Follow-up time (years) (13)

12. KAPLAN-MEIER METHOD E.L. Kaplan and P. Meier, 1958* Calculate the cumulative probability of event (and survival) based on conditional probabilities at each event time Step 1: Sort the survival times from shortest to longest Person ID 0 12 4 1 (24) 2 (6) 3 (18) (15) 5 (13) 6 (3) Follow-up time (years) *Kaplan EL, Meier P.Nonparametric estimation from incomplete observations. J Am Stat Assoc 1958;53:457-81.

13. Step 2:For each time of occurrence of an event, compute the conditional survival Person ID 0 12 4 1 (24) 2 (6) 3 (18) (15) 5 (13) 6 (3) Follow-up time (years) When the first event occurs (3 months after beginning of follow-up), there are 6 persons at risk. One dies at that point; 5 of the 6 survive beyond that point. Thus: Incidence of event at exact time 3 months: 1/6 Probability of survival beyond 3 months: 5/6

14. Person ID 0 12 4 1 (24) 2 (6) 3 (18) (15) 5 6 (3) Follow-up time (years) When the second event occurs (13 months), there are 4 persons at risk. One of them dies at that point; 3 of the 4 survive beyond that point. Thus: Incidence of event at exact time 13 months: 1/4 Probability of survival beyond 13 months: ¾ (13)

15. Person ID 0 12 4 1 (24) 2 (6) 3 (18) (15) 5 6 (3) Follow-up time (years) When the third event occurs (18 months), there are 2 persons at risk. One of them dies at that point; 1 of the 2 survive beyond that point. Thus: Incidence of event at exact time 18 months: 1/2 Probability of survival beyond 18 months: 1/2 (13)

16. Step 3:For each time of occurrence of an event, compute the cumulative survival (survival function), multiplying conditional probabilities of survival. 3 months:S(3)=5/6=0.833 12 months:S(13)=5/6  3/4=0.625 18 months:S(18)=5/6  3/4  1/2 =0.3125 CONDITIONAL PROBABILITY OF AN EVENT (or of survival) The probability of an event (or of survival) at time t (for the individuals at risk at time t), that is, conditioned on being at risk at exact time t.

17. 0.833 0.625 0.3125 Time (mo) 3 13 18 Plotting the survival function: 0.60 0.40 0.20 0.80 Survival 25 20 15 10 5 0 Month of follow-up 1.00 The cumulative incidence (up to 24 months): 1-0.3125 = 0.6875 (or  69%) SiSi 0.833 0.625 0.3125

18. 0.833 0.625 0.3125 Time (mo) 3 13 18 Plotting the survival function: 25 20 15 10 5 0 Month of follow-up 0.60 0.40 0.20 0.80 Cumulative Survival 1.00 0.8 0.6 0.3

19. CEE Placebo CEE Placebo Cumulative Hazards for Coronary Heart Disease and Stroke in the Women’s Health Initiative Randomized Controlled Trial (The WHI Steering Committee. JAMA 2004;291:1701-1712) EXPERIMENTAL STUDY

20. 0.833 0.625 0.3125 Time (mo) 3 13 18 Plotting the survival function: 25 20 15 10 5 0 Month of follow-up 0.60 0.40 0.20 0.80 Cumulative Survival 1.00 0.8 0.6 0.3 Cumulative Hazard 0.20 0.80 1.00 0.60 0.40 0.2 0.4 0.7 The cumulative incidence (hazard) at the end of 24 months: 1-0.3 = 0.7 (or  70%)

21. ACTUARIAL LIFE TABLE VS KAPLAN-MEIER If N is large and/or if life-table intervals are small, results are similar Survival after diagnosis of Ewing’s sarcoma

22. ASSUMPTIONS IN KAPLAN-MEIER SURVIVAL ESTIMATES (If individuals are recruited over a long period of time) No secular trends Calendar time Follow-up time

23. ASSUMPTIONS IN SURVIVAL ESTIMATES (Cont’d) Censoring is independent of survival (uninformative censoring): Those censored at time t have the same prognosis as those remaining. Types of censoring: Lost to follow-up –Migration –Refusal Death (from another cause) Administrative withdrawal (study finished)

24. Calculation of incidence Strategy #2 ANALYSIS BASED ON PERSON-TIME CALCULATION OF PERSON-TIME AND INCIDENCE RATES (Unit of analysis: time) Example 1Observe 1 st graders, total 500 hours Observe 12 accidents Accident rate: IT IS NOT KNOWN WHETHER 500 CHILDREN WERE OBSERVED FOR 1 HOUR, OR 250 CHILDREN OBSERVED FOR 2 HOURS, OR 100 CHILDREN OBSERVED FOR 5 HOURS… ETC.

25. Person ID 0 12 4 1 (24) 2 (6) 3 (18) (15) 5 (13) 6 (3) Follow-up time (years) CALCULATION OF PERSON-TIME AND INCIDENCE RATES Example 2 Person ID No. of person-years in Total FU 1 st FU year2 nd FU year 625431625431 3/12=0.25 6/12=0.50 12/12=1.00 0 1/12=0.08 3/12=0.25 6/12=0.50 12/12=1.00 0.25 1.00 1.25 1.50 2.00 Total4.751.836.58 Step 1: Calculate denominator, i.e. units of time (years) contributed by each individual, and total: Step 2: Calculate rate per person-year for the total follow-up period: It is also possible to calculate the incidence rates per person-year separately for shorter periods during the follow-up: For year 1: For year 2:

26. Notes: Rates have units (time -1). Proportions (e.g., cumulative incidence) are unitless. As velocity, rate is an instantaneous concept. The choice of time unit used to express it is totally arbitrary. E.g.: 0.024 per person-hour = 0.576 per person-day = 210.2 per person-year 0.46 per person-year = 4.6 per person-decade

27. Person No.Year 1Year 2Total 11/12= 0.08 (D)00.08 22/12= 0.17 (C)00.17 33/12= 0.25 (C)00.25 44/12= 0.33 (C)00.33 55/12= 0.42 (C)00.42 66/12= 0.50 (D)00.50 77/12= 0.58 (C)00.58 88/12= 0.67 (C)00.67 99/12= 0.75 (C)00.75 1010/12= 0.83 (C)00.83 1111/12= 0.92 (C)00.92 1212/12= 1.00 (D)01.00 1312/12= 1.00 (C)1/12= 0.08 (C)1.08 1412/12 = 1.00 (C)2/12= 0.17 (C)1.17 1512/12 = 1.00 (C)3/12= 0.25 (D)1.25 1612/12 = 1.004/12= 0.33 (C)1.33 1712/12 = 1.005/12= 0.42 (C)1.42 1812/12 = 1.006/12= 0.50 (C)1.50 1912/12 = 1.007/12= 0.58 (C)1.58 2012/12 = 1.008/12= 0.67 (C)1.67 2112/12 = 1.009/12= 0.75 (D)1.75 2212/12 = 1.0010/12= 0.83 (C)1.83 2312/12 = 1.0011/12= 0.92 (C)1.92 2412/12 = 1.0012/12= 1.00 (C)2.0 Total18.56.525.0 Death rate per person-time (person-year) 5 deaths/25.0 person-years= 0.20 or 20 deaths per 100 person-years Death rate per average population, estimated at mid- point of follow-up Mid-point (median) population (When calculating yearly rate in Vital Statistics) = 12.5 Death rate= 5/12.5 per 2 years= 0.40 Average annual death rate= 0.40/2= 0.20 or 20/100 population No. of person-years of follow-up D, deaths C, censored

28. Person No.Year 1Year 2Total 11/12= 0.08 (D)00.08 22/12= 0.17 (C)00.17 33/12= 0.25 (C)00.25 44/12= 0.33 (C)00.33 55/12= 0.42 (C)00.42 66/12= 0.50 (D)00.50 77/12= 0.58 (C)00.58 88/12= 0.67 (C)00.67 99/12= 0.75 (C)00.75 1010/12= 0.83 (C)00.83 1111/12= 0.92 (C)00.92 1212/12= 1.00 (D)01.00 1312/12= 1.00 (C)1/12= 0.08 (C)1.08 1412/12 = 1.00 (C)2/12= 0.17 (C)1.17 1512/12 = 1.00 (C)3/12= 0.25 (D)1.25 1612/12 = 1.004/12= 0.33 (C)1.33 1712/12 = 1.005/12= 0.42 (C)1.42 1812/12 = 1.006/12= 0.50 (C)1.50 1912/12 = 1.007/12= 0.58 (C)1.58 2012/12 = 1.008/12= 0.67 (C)1.67 2112/12 = 1.009/12= 0.75 (D)1.75 2212/12 = 1.0010/12= 0.83 (C)1.83 2312/12 = 1.0011/12= 0.92 (C)1.92 2412/12 = 1.0012/12= 1.00 (C)2.0 Total18.56.525.0 Death rate per person-time (person-year) 5 deaths/25.0 person-years= 0.20 or 20 deaths/100 person-years Death rate per average population, estimated at mid- point of follow-up Mid-point (median) population (When calculating yearly rate in Vital Statistics) = 12.5 Death rate= 5/12.5 per 2 years= 0.40 Average annual death rate= 0.40/2= 0.20 or 20/100 population No. of person-years of follow-up D, deaths C, censored

29. Notes: Rates have an undesirable statistical property Rates can be more than 1.0 (100%): –1 person dies exactly after 6 months: No. of person-years: 1 x 0.5 years= 0.5 person-years

30. Use of person-time to account for changes in exposure status (Time-dependent exposures) Example: Adjusting for age, are women after menopause at a higher risk for myocardial infarction? ID1 2 3 4 5 6 7 8 9 10 No. PY PRE meno No. PY POST meno C C : Myocardial Infarction; C: censored observation. Rates per person-year: Pre-menopausal = 1/17 = 0.06 (6 per 100 py) Post-menopausal = 2/18 = 0.11 (11 per 100 py) Rate ratio = 0.11/0.06 = 1.85 3 4 0 5 6 0 0 1 5 5 3 3 17 18 Year of follow-up Note: Event is assigned to exposure status when it occurs

31. ASSUMPTIONS IN PERSON-TIME ESTIMATES Risk is constant within each interval for which person-time units are estimated (no cumulative effect): –N individuals followed for t time  t individuals followed for N time –However, are 10 smokers followed for 1 year comparable to 1 smoker followed for 10 years (both: 10 person-years) No secular trends (if individuals are recruited over a relatively long time interval) Losses are independent from survival Rate for 1 st Year= 0.21/PY Rate for 2 nd Year= 1.09/ PY Total for 2 years = 0.46/PY

32. ASSUMPTIONS IN PERSON-TIME ESTIMATES Risk is constant within each interval/period for which person-time units are estimated (no cumulative effect): –N individuals followed for t time  t individuals followed for N time –However, are 10 smokers followed for 1 year comparable to 1 smoker followed for 10 years (both: 10 person-years) No secular trends (if individuals are recruited over a relatively long time interval) Losses are independent of survival

33. MethodEstimateValue Life-table Kaplan-Meier q (2 years) q(Y1) × q(Y2) q (2 years) 0.60 0.65 0.64 Person-year Midpoint (median) population Rate (  yearly) 0.46/py 0.43 per year SUMMARY OF ESTIMATES

34. POINT REVALENCE

35. Point Prevalence “The number of affected persons present at the population at a specific time divided by the number of persons in the population at that time” Gordis, 2000, p.33 Relation with incidence --- Usual formula: Point Prevalence = Incidence x Duration* P = I x D * Average duration (survival) after disease onset. True formula:

36. ODDS

37. Odds The ratio of the probabilities of an event to that of the non-event. Example:The probability of an event (e.g., death, disease, recovery, etc.) is 0.20, and thus the odds is: That is, for every person with the event, there are 4 persons without the event.