Survival Analysis Rick Chappell, Ph.D. Professor, Department of Biostatistics and Medical Informatics University of Wisconsin School of Medicine & Public Health chappell@stat.wisc.edu BMI 542 – Week 6, Lecture 1 (slides adapted in part from D. DeMets’) 2

Survival Analysis Terminology Concerned about time to some event Event is often death Event may also be, for example 1. Cause-specific death 2. Recurrence of tumor or death, whichever comes first 3. Death or some non-fatal event 4. Release from hospital; marriage; divorce; job tenure; job acquisition …

Survival Rates at Yearly Intervals YEARS At 5 years, survival rates the same Survival experience in Group A appears more favorable, considering 1 year, 2 year, 3 year and 4 year rates together

Beta-Blocker Heart Attack Trial LIFE-TABLE CUMULATIVE MORTALITY CURVE

Survival Analysis 1. Estimation of survival curves 2. Comparison of survival curves Estimation Simple Case All patients entered at the same “time” and followed for the same length of time Survival curve is estimated at various time points by (number of deaths)/(number of patients) As intervals become smaller and number of patients larger, a smoother survival curve may be plotted Typical Clinical Trial Setting

Staggered Entry T years 1 T years 2 Subject T years 3 T years 4 T 2T Time Since Start of Trial (T years) Each patient has T years of follow-up Time for follow-up taking place may be different for each patient

• * * Subject o Administrative 1 Right Censoring 2 Failure 3 Loss to Follow-up * 4 Failure T 2T Time Since Start of Trial (T years) Failure time is time from entry until the time of the event Right Censoring means vital status of patient is not known beyond that point

* • * Subject Administrative Censoring 1 o 2 Failure 3 Censoring Loss to Follow-up 4 * Failure T Follow-up Time (T years)

Clinical Trial with Common Termination Date Subject 1 o 2 * 3 • 4 • o 5 • * • • 6 7 • • 8 • * 9 • o o 10 • o * 11 • o o T 2T Follow-up Time (T years) Trial Terminated

Kaplan-Meier Estimate (JASA, 1958) – see “KM.mechanics” pdf Assumptions 1. "Exact" time of event is known Failure = uncensored event Loss = censored event Failures are independent of Losses! See KMBias.ppt 2. For a "tie", failure always before loss (minor) 3. Divide follow-up time into intervals such that a. Each event defines left side of an interval b. No interval has both deaths & losses

2. Comparison of Two Survival Curves Assume that we now have a treatment group and a control group and we wish to make a comparison between their survival experience How do we do so in a way which builds upon the Kaplan-Meier estimator? Point-wise at a “landmark” time using Greenwood. Medians? Mantel-Haenszel (log rank) test, with equal weighting; Peto-Prentice (Gehan) test, with earlier events upweighted & later ones downweighted.