Survival Analysis Introduction and Terminology Rick Chappell Ph

Survival Analysis – Introduction and Terminology 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 #1

Survival Analysis Introduction • 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 … #2

Fundamental Features of Survival Analysis - Loss to followup (right censoring) - Interest in conditional inference: given survival to time t, what is a patient’s risk? Fundamental Assumption of Survival Analysis - Censoring is independent of failure #3

Subject o 1 2 * Failure • 3 4 * Administrative Censoring Loss to Follow-up Failure T 0 Follow-up Time (T years)

Questions We May Want to Address • What is the probability of 5 -year survival? • What is the probability of “X”-year survival, for all X (within the range of the data)? • What is the median survival (if it exists)? • Is one group’s survival larger than another’s? • Can we create a regression to model survival as a function of predictors – not just treatment group but age, biomarkers. . . ? #5

Terminology 1. Origin (baseline): time from which events are measured. • Time of randomization, if applicable • Otherwise time of “registration” • In observational studies it may be birth 2. Time scale: time (t) since origin. 3. Right censoring: largest time at which patient is known to be “alive”. #6

Terminology 4. Left censoring: smallest time before which patient is known to be alive (not discussed). 5. Interval censoring: pair of times between which patient is known to fail (not discussed). 6. Left truncation: in observational studies, the time at which the patient comes under observation (not discussed). #7

Terminology 7. Survival Curve: S(t) = Probability of survival past t. 8. Hazard function – has units, eg. per year: h(t) = l(t) = slope of S(t)/S(t) = “conditional risk at t” = “instantaneous force of mortality at t”. Given that I have survived to time t, how much danger am I in? #8

Constant hazard: h(t) = a, S(t) = e-a. Exponential distribution; Useful in physics (radioactive decay); Not applicable in medicine except as a gross approximation. #9

Terminology 8. Positive Aging: h(t) increases with t; For human mortality, h(t) ~ t 4. 9. Negative Aging: h(t) decreases with t; 10. Nonmonotone hazard: h(t) goes up then down, or down then up, or. . . #10

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Annual hazard in U. S. , 2003 #12