1. The life table LT statistics: rates, probabilities, life expectancy (waiting time to event) Period life table Cohort life table

2. Life table from observational data 22 respondents Namboodiri and Suchindran, 1987, Chapter 4

4. sort

5. 0.9524 = 1-1/21 =1 - failure/number at risk before failure 0.8995 = 0.9524*[1-1/18] Kaplan-Meier 1/21

6. Discrete time interval (30 days)

10. Standard error of survival function: Greenwood formula With S t = survival function R t = risk set q t = probability of event p t = survival probability (probability of NO event)

11. Assume 100 respondents (R = risk set) Probability of event (q): 0.10 => survival probability (p): 0.90 Var(p) = pq/R = 0.9*0.1/100 = 0.0009 SQRT(0.0009) = 0.03 p 2 q/(pR) = 0.81*0.1/(0.9*100) = 0.0009 SQRT(0.0009) = 0.03 15 survivors experience event. Prob of event is 15/90 = 0.1667 Risk set: 90 (= 100*0.9) Prob of surviving second interval: S2 = 0.9*0.833=0.75 Var(S2) = 0.75 2 * [0.1/(0.9*100) + 0.1667/(0.8333*90)] = 0.00188

12. Standard error of (hazard) rate: With r t = hazard rate

13. Life table with grouped data Remarriage of divorced women, aged 25 to 34, USA, 1975 Source: 1975 US Current Population Survey Namboodiri and Suchindran, 1978, pp. 63 ff

14. 0.2109 = 238/(0.5*(1298+959)) 0.2117 = -ln(0.8092) Conditional density varies with duration

16. Leaving parental home, The Netherlands, Birth cohort 1961

17. Data

22. Leaving home: occurrences and exposures

23. SPSS output

25. A method for the nonparametric estimation of the survival function (1958) Also called product-limit estimator The risk set is calculated at every point in time where at least one event occurred. Hence all episodes must be sorted according to their ending times. It is a staircase function with a. Location of drop is random (time at event) b. Size of drop depends on censoring The Kaplan-Meier estimator

26. Kaplan-Meier estimator -->

27. Kaplan-Meier estimator Exact time of failure and censoring are known Where Y(x i ) is the risk set (individuals at risk just before time t (time at event))and D i is the failure indicator (1 in case of failure)

28. Kaplan-Meier estimator Time from diagnosis to death Clayton and Hills, 1993, p. 37

29. References: Kaplan-Meier Good introduction: Clayton and Hills, 1993, pp. 35ff Technical: Andersen and Keiding, 1996, pp. 180ff (includes several references)

30. A method for the nonparametric estimation of the cumulative hazard function The risk set is calculated at every point in time where at least one event occurred. Hence all episodes must be sorted according to their ending times. It is a staircase function with a. Location of drop is random (time at event) b. Size of drop is 1/risk set (number at risk: count of persons alive before the death Easier to generalise to multistate situations The Nelson-Aalen estimator

31. Nelson (1969) and Aalen (1978) Clayton and Hills, 1993, p. 48 Andersen and Keiding, 1996, p. 181 The Nelson-Aalen estimator A(t) = -lnS(t)

32. Clayton and Hills, 1993, p. 50 Nelson-Aalen estimator (rates) Time from diagnosis to death

33. Duration of job episodes (Blossfeld and Rohwer, 1995) (sorted) 201 respondents 600 job episodes

34. Product-limit estimate of survival function (Kaplan-Meier) Pisa99/blossfeld/rrdat_sort.xls

35. Product-limit estimate of survival function (Kaplan-Meier) and 95% interval

36. Plot of survival function, generated by TDA Duration up to 428 months (shown up to 300 months) Blossfeld and Rohwer, 1995, p. 70 (ehc6_1.cf => ehc6_1.ps)

37. Blossfeld and Rohwer, 1995, p. 73 (ehc8.ps)

38. Product-limit estimate of survival function (Kaplan-Meier)

39. Product-limit estimate of survival function (Kaplan-Meier): output D:\S\TEACH\99PISA\BLOSSF\OEF>d:\s\software\tda\62b\tda\tda_nt cf=ehc5.cf Creating new single episode data. Max number of transitions: 100. Definition: org=0, des=DES, ts=0, tf=TFP Mean SN Org Des Episodes Weighted Duration TS Min TF Max Excl ---------------------------------------------------------------------------- 1 0 0 142 142.00 128.18 0.00 428.00 - 1 0 1 458 458.00 49.30 0.00 350.00 - Sum 600 600.00 Number of episodes: 600 ple=ehc5.ple Product-limit estimation. Current memory: 367814 bytes. Sorting episodes according to ending times. Product-limit estimation. 1 table(s) written to: ehc5.ple ---------------------------------------------------------------------------- Current memory: 311232 bytes. Max memory used: 387781 bytes. End of program. Mon Mar 27 23:49:40 2000

40. Product-limit estimate of survival function (Kaplan-Meier): output # SN 1. Transition: 0,1 - Product-Limit Estimation ehc5.ple # Number Number Exposed Survivor Std. Cum. # ID Index Time Events Censored to Risk Function Error Rate 0 0 0.00 0 0 600 1.00000 0.00000 0.00000 0 1 2.00 2 0 600 0.99667 0.00235 0.00334 0 2 3.00 5 1 597 0.98832 0.00439 0.01175 0 3 4.00 9 2 590 0.97324 0.00660 0.02712 0 4 5.00 3 0 581 0.96822 0.00717 0.03230 0 5 6.00 10 1 577 0.95144 0.00880 0.04978 0 6 7.00 9 0 567 0.93634 0.00999 0.06578 0 7 8.00 6 1 557 0.92625 0.01070 0.07661 0 8 9.00 7 3 548 0.91442 0.01146 0.08947 0 9 10.00 8 1 540 0.90087 0.01225 0.10439 0 10 11.00 4 4 528 0.89405 0.01262 0.11200 0 11 12.00 24 0 524 0.85310 0.01455 0.15888 0 12 13.00 8 1 499 0.83942 0.01510 0.17504 0 13 14.00 10 3 488 0.82222 0.01574 0.19575 0 14 15.00 6 1 477 0.81188 0.01610 0.20841 0 15 16.00 4 0 471 0.80498 0.01633 0.21694 0 16 17.00 9 0 467 0.78947 0.01681 0.23640 0 17 18.00 6 0 458 0.77913 0.01711 0.24958 0 18 19.00 8 0 452 0.76534 0.01749 0.26744 0 19 20.00 9 1 443 0.74979 0.01789 0.28797 Blossfeld and Rohwer, 1995, p. 69

41. Deaths in first years of life, Kerala. Source: NFHS, 1992-93

42. continued Deaths in first years of life, Kerala. Source: NFHS, 1992-93

45. The fetal life table 9564 pregnancies identified retrospectively from urine tests as well as first prenatal care visits at three Kaiser Permanente clinics in San Francisco Bay area during 10-month period in 1981-1982. Twin and triplet pregnancies, pregnancies with less than 2 days follow-up, and few other pregnancies were omitted => 9055 pregnancies. Of these, 103 withdrew during follow-up (pregnancy outcome not known), 6629 resulted in live births, 549 in spontaneous fetal loss (including 27 ectopic pregnancies), and 1774 induced abortion. 2-day lag was used to avoid bias arising when women selectively report for medical care because of threatened abortion. Many of these women miscarry within 2 days (selection!). Inclusion would overestimate the risk of abortion! Measurement issues: onset of pregnancy (date of last menstrual period) and pregnancy outcome. 459 women entered observation in week 5 of gestation (days 0-6 after last menstrual period = week 0; days 35-41 = week 5; days 308-314 = week 44) Goldhaber and Fireman (1991)

46. Fetal life table by gestational weeks (LMP), 1981-1982, California, USA

48. Application and confusion (discussion) Miller and Homan (1994) “Determining transition probabilities: confusion and suggestions”, Medical Decision Making, 14(1):??? (based on Kleinbaum et al.). Terminology used in this paper is confusing (and wrong!) a. Distinguish between rates and risk Rate (incidence rate): occurrences (incidences; new cases) over exposure. Exposure is measured by ‘summing each subject’s time exposed to the possibility of transiting’ (includes censored cases).. Instantaneous incidence rate (‘also known as the hazard function’). Average incidence rate (also known as the ‘incidence density’ [ID]) Density  rate

49. Application and confusion (discussion) Risk: Risk used to denote probability. Three methods for estimating risk: 1. Simple cumulative method: new cases / number of disease-free individuals at beginning of interval (no censoring or withdrawal): I/N 0 where I is number of new cases and N 0 is the number of disease-free individuals at t=0. 2. Actuarial (life-table) method: new cases / number of disease-free individuals at beginning of interval minus half of the number of withdrawals: I/[N 0 -W/2] where W is number of withdrawals 3. Density method: uses age-specific incidence densities (e.g. rates) to estimate the risk for given age or time interval: P(0,t) = 1 - exp[-ID*t] where ID is the average rate and t is the elapsed time. Rates are translated into probabilities. = risk set