Statistics & HSCT ESH-EBMT 2014 Vienna
Myriam Labopin
[email protected]
Complexity of the story…. Death 500 Acute and chronic GVHD
Competing risks
Censored data
Censored data Time-to event variables
Time-to-event variables
We are interested in T, the time elapsed between a well defined starting point (SCT) and the endpoint. T is called « survival time ».
Censored data Typical type of data
Event Lost to follow-up Study termination
Time since start of study
Time since entry in study
Censored data
Survival time = T HSCT t1
Patient 1
t2
Still alive
Patient 2
Alive at time t:
Dead
T>t1
Dies at time t: T=t2
ti : observation time for patient i
Censored observation : we only observe that survival time T is greater than observation time t. T is unknown: we do not know if the event will happen and when Censored patients are by definition only those still at risk of failure Censored data = incomplete information missing
Censored data Censoring must be uninformative - independent of the event studied Assumption reasonable End of study
Loss to follow-up when unrelated to the disease Sometimes questionable Loss to follow-up • •
Healthy participants feel less need for medical services (underestimation of survival) Subjects with advanced disease progression are more likely to leave the study (overestimation of survival)
Another event has occurred, which prevents occurrence of the event of interest: competing risks
Censored data Question to have in mind when censoring: ‘Has a censored patient the same risk of an event as a patient still on follow-up?”
Do not censor: - Death without relapse when estimating relapse (competing risk) - Death from other causes when estimating a specific cause of death (competing risk) - Patients when they go ‘off-treatment’ for toxicity in prospective clinical trials (intent-to-treat) - Second transplant (competing or intermediate event)
Competing risks
Competing risks “Competing” : the occurrence of any event may modify or avoid the risk of the others
Dead in CR
Patients can experience only one of the events
HSCT
Relapsed
Competing risks
Relapse HSCT
Relapse at time t: T=t
T Death in CR
Competing event ≠ censoring
Alive relapse_free
Dies w/o relapse after time t: T>t
Event-free at time t: T>t
Failure from competing event is not an incomplete information Patients who relapse are no longer at risk for death in CR censor
Patients censored : only patients alive relapse-free
Competing risks Which competing events after HSCT? When analysing: RI : death without relapse (NRM) NRM: relapse Engraftment: second transplant/death GVHD: relapse / death / (± graft failure)
Statistical methods
Description of the endpoints
Statistical methods – survival analysis Survival endpoint -> (time, status=0/1) Survival time = T HSCT
Still alive
Patient 1
Dead
Patient 2 t : observation time
Death (complete data) -> time=t, status=1
Alive (censored) -> time=t, status=0
Basic concept describing survival
Probability that an individual survives from the time origin to t
Probability that an individual dies at time t, conditional he or she have survived until that time
Statistical methods – survival analysis Non parametric estimation of the survival function Example: survival time in 20 patients: 9, 50, 60, 114+, 153+, 272, 300, 364, 365+, 392, 400+, 450, 455, 530+, 687+, 722, 757+, 788+, 800+, 1316+ (+: censored)
Without censoring:
Nr patients surviving beyo nd t ˆ S (t ) N 17 Sˆ (100) 20
Problem: observation 114+
Statistical methods – survival analysis Kaplan Meier estimator : “product-limit” method Probably well-known to all of you Let t0 < t1 < t2 < … < tn ordered distinct event-times For each ti: ri : number at risk at ti di: number of events at ti di t 1 Probability of surviving ti surviving i 1 ri
Kaplan Meier estimate: product of the probabilities for each time This assumes that individuals at risk at ti are representative for all patients alive at ti : independence of censoring distribution
Statistical methods – survival analysis
Probability
Kaplan Meier curve -Median survival time : time t where estimated survival is equal to 0.5 (50% ) - Nr of patients still at risk at each time: Too few patients at risk at the end of follow-up : estimates unreliable. a ‘plateau’ is not the probability of being ‘cured’
Median survival time (half of the patients die within the median time)
Statistical methods – Competing risks Competing risk (time, status=0/1/2) Relapse HSCT
Relapse at time t: T=t
T Death in CR
Alive relapse_free
Relapse -> (time=t: status=1) Death in CR -> (time=t; status=2) Alive relapse-free (censored) -> (time=t;status=0)
Dies w/o relapse after time t: T>t Event-free at time t: T>t
Statistical methods – Competing risks
Do not use Kaplan Meier for competing risks! Example : estimation of RI (competing risk = NRM) If we use Kaplan Meier, patients who die from NRM are censored : 1. Patient who died from NRM are not remaining at risk of relapse 2. Censoring is not independent 3. The number of patients at risk of relapse decreases and the probability of
relapse tends to 1 => Overestimation of the probability of relapse. The Kaplan-Meier estimator is biased in the presence of competing risks
Statistical methods – Competing risks
Cumulative incidence functions Crude probability of each event = probability of a specific event in the presence of all other risks acting on the population Defined as the probability of failing from cause k before time t Depends on the hazard of both competing events
Statistical methods – Competing risks Kaplan Meier and cumulative incidence curves KM curves - Censoring at the occurrence of a competing event - Final estimation of each event tends to 1
CI curves - Increase from 0 up to the total rate ( Wilcoxon test … • late outcome • difference at a specific point => Klein et al, Logan et al
Statistical methods – Survival endpoint Multivariate analysis – Cox model Proportional hazards Hazard rate of A: hA(t) Hazard rate of B: hB(t) Hazard ratio = Ratio of these hazards
HR(t )
h A (t ) hB (t )
Both hA(t) and hB(t) depend on time => in principle, HR(t) would also depend on time
Proportional hazards assumption : HR(t) is constant and does not depend on time
Statistical methods – Survival endpoints Cox model - Interpretation of the results HR=1 indicates that the risk is the same
HR>1 indicates that the risk is higher in group A HR auto selection of a bad prognostic group.
Intermediate event CANNOT be included as a fixed-time covariate
Intermediate events
How do we deal with this situation?
- Landmark analysis - Time-dependent covariates
Intermediate events Landmark analysis Popularized by Anderson et al (1983) Fix a point in time t*: the landmark point Define covariates based on observations known before t* Study with the usual methods
Selection of landmark : Before analysis Based on some natural time of clinical significance
Intermediate events
Landmark analysis
Estimation of the OS after transplantation Survival analysis with another time origin (landmark ) = SCT Patients who die before the SCT do not contribute to the analysis Estimation of prognostic factors on patients who have received the transplant
Intermediate events Cox with intermediate event •
IE included as a time dependent variable in the model => modification of its value during time (ex SCT=“no” until SCT=“yes”) comparison in time of patients at each time series of landmark analysis On Study
SCT = 0
SCT = 1
Dead
SCT
-> evaluation of the effect of the IE on the estimation of survival
Intermediate events
Cox with time dependent variable : limitations Assumption : effect of the IE immediately after IE Sometimes not reasonable in practice: risk of death increase due to SCT
and then begin to decrease steadily
IE could reflect an auto selection of patients with low hazard independently on the real effect of the IE -> avoid the statement : « SCT reduces the risk of death of 20 % »
Cgvhd slide
P=0.003
Landmark at day 100
P=0.81
Statistics & HSCT – Conclusion
Endpoints are well defined after HSCT: they are most often censored
data, some of them are competing events Analysis:
Univariate: -> Kaplan Meier & Log rank for survival-like endpoints -> Cumulative incidence and Gray test for competing events Multivariate : Cox and Fine-Gray models Specific methods for intermediate events after transplant: landmark,
time dependent variables…