Analysis of Survival Data.ppt

1、Analysis of Survival Data,Time to Event outcomes Censoring Survival Function Point estimation Kaplan-Meier,Introduction to survival analysis,What makes it different? Three main variable types Continuous Categorical Time-to-event Examples of each,Example: Death Times of Psychiatric Patients (K&M 1.15

2、),Dataset reported on by Woolson (1981) 26 inpatient psychiatric patients admitted to U of Iowa between 1935-1948. Part of larger study Variables included: Age at first admission to hospital Gender Time from first admission to death (years),Data summary,gender age deathtime death 1 51 1 1 1 58 1 1 1

3、 55 2 1 1 28 22 1 0 21 30 0 0 19 28 1 1 25 32 1 1 48 11 1 1 47 14 1 1 25 36 0 1 31 31 0 0 24 33 0 0 25 33 0 1 30 37 0 1 33 35 0 0 36 25 1 0 30 31 0 0 41 22 1 1 43 26 1 1 45 24 1 1 35 35 0 0 29 34 0 0 35 30 0 0 32 35 1 1 36 40 1 0 32 39 0,. tab gendergender | Freq. Percent Cum. -+-0 | 11 42.31 42.311

4、 | 15 57.69 100.00 -+-Total | 26 100.00,. sum ageVariable | Obs Mean Std. Dev. Min Max -+-age | 26 35.15385 10.47928 19 58,Death time?,. sum deathtimeVariable | Obs Mean Std. Dev. Min Max -+-deathtime | 26 26.42308 11.55915 1 40,Does that make sense?,Only 14 patients died The rest were still alive a

5、t the end of the study Does it make sense to estimate mean? Median? How can we interpret the histogram? What if all had died? What if none had died?,. tab deathdeath | Freq. Percent Cum. -+-0 | 12 46.15 46.151 | 14 53.85 100.00 -+-Total | 26 100.00,CENSORING,Different types Right Left Interval Each

6、leads to a different likelihood function Most common is right censored,Right censored data,“Type I censoring” Event is observed if it occurs before some prespecified time Mouse study Clock starts: at first day of treatment Clock ends: at death Always be thinking about the clock,Simple example: Type

7、I censoring,Time 0,Introduce “administrative” censoring,Time 0,STUDY END,Introduce “administrative” censoring,Time 0,STUDY END,More realistic: clinical trial,Time 0,STUDY END,“Generalized Type I censoring”,More realistic: clinical trial,Time 0,STUDY END,“Generalized Type I censoring”,Additional issu

8、es,Patient drop-out Loss to follow-up,Drop-out or LTFU,Time 0,STUDY END,How do we treat” the data?,Time of enrollment,Shift everything so each patient time represents time on study,Another type of censoring: Competing Risks,Patient can have either event of interest or another event prior to it Event

9、 types compete with one another Example of competers: Death from lung cancer Death from heart disease Common issue not commonly addressed, but gaining more recognition,Left Censoring,The event has occurred prior to the start of the study OR the true survival time is less than the persons observed su

10、rvival time We know the event occurred, but unsure when prior to observation In this kind of study, exact time would be known if it occurred after the study started Example: Survey question: when did you first smoke? Alzheimers disease: onset generally hard to determine HPV: infection time,Interval

11、censoring,Due to discrete observation times, actual times not observed Example: progression-free survival Progression of cancer defined by change in tumor size Measure in 3-6 month intervals If increase occurs, it is known to be within interval, but not exactly when. Times are biased to longer value

12、s Challenging issue when intervals are long,Key components,Event: must have clear definition of what constitutes the event Death Disease Recurrence Response Need to know when the clock starts Age at event? Time from study initiation? Time from randomization? time since response? Can event occur more

13、 than once?,Time to event outcomes,Modeled using “survival analysis” Define T = time to event T is a random variable Realizations of T are denoted t T 0 Key characterizing functions: Survival function Hazard rate (or function),Survival Function,S(t) = The probability of an individual surviving to ti

14、me t Basic properties Monotonic non-increasing S(0)=1 S()=0*,* debatable: cure-rate distributions allow plateau at some other value,Example: exponential,Weibull example,Applied example,Van Spall, H. G. C., A. Chong, et al. (2007). “Inpatient smoking-cessation counseling and all-cause mortality in pa

15、tients with acute myocardial infarction.“ American Heart Journal 154(2): 213-220. Background Smoking cessation is associated with improved health outcomes, but the prevalence, predictors, and mortality benefit of inpatient smoking-cessation counseling after acute myocardial infarction (AMI) have not

16、 been described in detail. Methods The study was a retrospective, cohort analysis of a population-based clinical AMI database involving 9041 inpatients discharged from 83 hospital corporations in Ontario, Canada. The prevalence and predictors of inpatient smoking-cessation counseling were determined

17、. Results Conclusions Post-MI inpatient smoking-cessation counseling is an underused intervention, but is independently associated with a significant mortality benefit. Given the minimal cost and potential benefit of inpatient counseling, we recommend that it receive greater emphasis as a routine pa

18、rt of post-MI management.,Applied example,Adjusted 1-year survival curves of counseled smokers, noncounseled smokers, and never-smokers admitted with AMI (N = 3511). Survival curves have been adjusted for age, income quintile, Killip class, systolic blood pressure, heart rate, creatinine level, card

19、iac arrest, ST-segment deviation or elevated cardiac biomarkers, history of CHF; specialty of admitting physician; size of hospital of admission; hospital clustering; inhospital administration of aspirin and -blockers; reperfusion during index hospitalization; and discharge medications.,Hazard Funct

20、ion,A little harder to conceptualize Instantaneous failure rate or conditional failure rateInterpretation: approximate probability that a person at time t experiences the event in the next instant. Only constraint: h(t)0 For continuous time,Hazard Function,Useful for conceptualizing how chance of ev

21、ent changes over time That is, consider hazard relative over time Examples: Treatment related mortality Early on, high risk of death Later on, risk of death decreases Aging Early on, low risk of death Later on, higher risk of death,Shapes of hazard functions,Increasing Natural aging and wear Decreas

22、ing Early failures due to device or transplant failures Bathtub Populations followed from birth Hump-shaped Initial risk of event, followed by decreasing chance of event,Examples,Median,Very/most common way to express the center of the distribution Rarely see another quantile expressed Find t such t

23、hatComplication: in some applications, median is not reached empirically Reported median based on model seems like an extrapolation Often just state median not reached and give alternative point estimate.,X-year survival rate,Many applications have landmark times that historically used to quantify s

24、urvival Examples: Breast cancer: 5 year relapse-free survival Pancreatic cancer: 6 month survival Acute myeloid leukemia (AML): 12 month relapse-free survival Solve for S(t) given t,Competing Risks,Used to be somewhat ignored. Not so much anymore Idea: Each subject can fail due to one of K causes (K

25、1) Occurrence of one event precludes us from observing the other event. Usually, quantity of interest is the cause-specific hazard Overall hazard equals sum of each hazard:,Example,Myeloablative Allogeneic Bone Marrow Transplant Using T Cell Depleted Allografts Followed by Post-Transplant GM-CSF in

26、High Risk Myelodysplastic Syndromes Interest is in RELAPSE Need to account for treatment related mortality (TRM)? Should we censor TRM? No. that would make things look more optimistic Should we exclude them? No. That would also bias the results Solution: Treat it as a competing risk Estimate the inc

27、idence of both,Estimating the Survival Function,Most common approach abandons parametric assumptions Why? Not one catch-all distribution No central limit theorem for large samples,Censoring,Assumption: Potential censoring time is unrelated to the potential event time Reasonable? Estimation approache

28、s are biased when this is violated Violation examples Sick patients tend to miss clinical visits more often High school drop-out. Kids who move may be more likely to drop-out.,Terminology,D distinct event times t1 t2 t3 . tD ties allowed at time ti, there are di deaths Yi is the number of individual

29、s at risk at ti Yi is all the people who have event times ti di/Yi is an estimate of the conditional probability of an event at ti, given survival to ti,Kaplan-Meier estimation,AKA product-limit estimatorStep-function Size of steps depends on Number of events at t Pattern of censoring before t,Kapla

30、n-Meier estimation,Greenwoods formula Most common variance estimator Point-wise,Example:,Kim paper Event = time to relapse Data: 10, 20+, 35, 40+, 50+, 55, 70+, 71+, 80, 90+,Plot it:,Interpreting S(t),General philosophy: bad to extrapolate In survival: bad to put a lot of stock in estimates at late

31、time points,Fernandes et al: A Prospective Follow Up of Alcohol Septal Ablation For Symptomatic Hypertrophic Obstructive Cardiomyopathy The Ten-Year Baylor and MUSC Experience (1996-2007)”,R for KM,library(survival) library(help=survival)t - c(10,20,35,40,50,55,70,71,80,90) d - c(1,0,1,0,0,1,0,0,1,0) cbind(t,d)st - Surv(t,d) sthelp(survfit) fit.km - survfit(st) fit.km summary(fit.km) attributes(fit.km)plot(fit.km, conf.int=F, xlab=“time to relapse (months)“,ylab=“Survival Function“, lwd=2),Kaplan-Meier Curve,