EPI 5344: Survival Analysis in Epidemiology Left Truncation & Other Issues April 1, 2014

1. EPI 5344:Survival Analysis in EpidemiologyLeft Truncation & Other IssuesApril 1, 2014 Dr. N. Birkett, Department of Epidemiology & Community Medicine, University of Ottawa

2. Objectives • Left truncation, left censoring, unknown ‘0’ time • Imprecise outcome time • Interval censoring

3. Background (1) • Time ‘0’ is the starting point for follow-up • ‘time-on-study’ • Measured from the time of entry into study • Commonly used in RCTs • ‘age-at-follow-up’ • Time ‘0’ is date of birth • useful for • observational studies • when no clear point of initial exposure

4. Background (2) • Subject observed at time ‘t’ means: • If subject had an event at time ‘t’, then subject would have been recorded with an event in the study. • Point of initial observation may not be time ‘0’ • age-at-follow-up time scale • enter study at age 45 • first follow-up is at age 45, not 0 • subject must have survived to the time of first observation

5. Left Truncation (1) • Left truncation (delayed entry) • subject enters at time ‘0’ • Initial observation is at later time ‘t’ • Subject must be outcome free at time of initial observation • Distribution of survival times is ‘truncated’ or ‘cut-off’ in the left tail

6. Left Truncation (2) • Example #1 • 500 subjects with an acute MI enter hospital (time ‘0’) • 461 subjects were discharged alive • Only these subjects are in study • BUT, survival time is measured from the time of entry to hospital Truncation time

7. Left Truncation (3) • Left truncation (delayed entry) • What do to with the person-time between time ‘0’ and first observation? • include it bias of h(t) towards ‘0’ • exclude it can  bias away from ‘0’ • depends on the hazard during the interval • Analysis depends on assuming that the hazard starting at the first observation time is unaffected by the delayed entry

8. Left Truncation (4) • Example study (a bit artificial but shows the methods) • Residential drug-abuse treatment programme • 2 RCT’s at 2 sites • Site ‘A’ • 6 months vs. 12 months • Site ‘B’ • 3 months vs. 6 months • sites combined (for this example) • Short vs. long treatment • n=628 • follow-up from date of admission • up to 1,170 days of follow-up

9. Left Truncation (5) • In this example, we actually know subject history from time ‘0’ • Could analyze using time varying covariate • ‘0’ for time during treatment • ‘1’ for time after treatment. • Includes the entire sample (n=628) • Short vs. long Rx effect (HR): 1.02 (0.85 – 1.23) • On vs. off-Rx effect (HR): 13.1 (9.6 – 17.8) • Length of treatment has no effect but being ‘off-treatment’ has a strong adverse effect

10. Left Truncation (6) • Suppose that we know nothing about outcomes during treatment. • Study includes only people who complete treatment and remain drug-free at the end of treatment • n = 546 • Why is this an issue? • Using the standard analysis, ‘long-term’ Rx subjects are assured of having more months of drug-free follow-up time than those in ‘short-term’ Rx • We ignore events during these ‘immortal’ periods. • Introduces a serious bias in favor of ‘long term’ treatment.

11. Left Truncation (7) Truncation time End Prog: recruit Recid Enter Prog Minimum event time

12. Left Truncation (8) • No analytic method can recover information on events during the truncation period • If we can ignore this, solution is ‘easy’ • Set the ‘immortal person-time’ as outside the period of follow-up. • The assumption is that the hazard starting from the time of initial observation is unaffected by the delayed entry

13. Left Truncation (9) • Two approaches can be used in SAS. • Define a truncation time variable • Add a command to the Model statement giving entry time to risk set • Use the Counting process input style • Need to specify (for each subject): • Start of their follow-up time • End of their follow-up time • Outcome status (event vs.. censored) • No need to do any data re-structuring in this case

14. Left Truncation (10) • Continue Example #2 • Define a variable (LOS) which is the number of days from treatment start to entry into follow-up • Method 1: • add to model statement as ‘entrytime’ • Method 2: • No need to rearrange data • only one interval per subject • starts at ‘LOS’ • ends at study time

15. Left Truncation (11) PROC PHREG DATA=uis1; MODEL time*censor(0)= age becktota hercoc race treat site / TIES=EFRON; run; PROC PHREG DATA=uis1; MODEL time*censor(0)= age becktota hercoc race treat site / TIES=EFRON entrytime=los; run; PROC PHREG DATA=uis1; MODEL (los, time)*censor(0)= age becktota hercoc race treat site / TIES=EFRON; run;

16. Ignores late entry

17. Late entry, method #1

18. Late entry, method #2

19. Left Censoring (1) Left Censoring • Subject already has the outcome event when they enter the study • Differs from left truncation • In truncation, subject hasn’t had the event yet. • Truncation involves a subject selection process applied to all subjects • Left censoring is an individual subject event

20. Left Censoring (2) • Example • Outcome is ‘age of smoking uptake’ • Interview 12 year olds (time ‘0’ = age 12) • One child reports starting regular smoking before age 12

21. Left Censoring (3) • Approach • Inclusion/exclusion criteria • Exclude prevalent cases in cohort recruitment • Specialized regression methods • Parametric regression. • PH models • Too complex to explore further

22. Unknown ‘0’ time (1) Unknown ‘0’ time • Subject meets entry criteria but it is unclear when they met the criteria • Example • Study is to examine prognosis in people with AIDS • Some recruited subjects (diagnosed with AIDS) lack information on when they converted from HIV+ to AIDS  lack of a clear time ‘0’ • A common issue which is usually just ignored • Use date of recruitment instead • Models can be used but are complex

23. Imprecise measure of outcome date (1) • Some examples (share common features) • Tied data • Interval censoring • Sero-conversion to HIV+ status • Jan 1: OK • July 1: (+)ve only know conversion was between Jan 1 and July 1 • Discrete time for event • Event can only occur at fixed points in time • Examination failure can only occur when exam is administered

24. Interval censoring (1) • Issue is frequently ignored and regular Cox models are used. • Use the time when first known to have outcome as the time-to-event • can build more complex models • estimate distribution of event time within interval • Better approach converts to a discrete Cox model • Uses binary regression models to get the estimates, etc. • Logistic model • Complementary log-log model • Allows analysis using existing software

25. Interval censoring (2) • Divide the follow-up time into intervals • Equal size is easiest but is not required • Now, we assume that there is a continuous hazard underlying the data • Proportional hazards is assumed: • Now, define:

26. Interval censoring (3) • It can be shown that: • ‘αt’ depends on the baseline hazard • This is a regression model similar to: but with a different ‘link’ function: ln(-ln(1-pit)) rather than logit(pit) 1