Statistics II

Statistics & HSCT
ESH-EBMT 2014 Vienna
Myriam Labopin
[email protected]
Complexity of the story….
Death
<D100
Transplant
Acute GVHD
Relapse
Alive
at D100
Chronic
GVHD
Death
w/o relapse
Statistics & HSCT – outline
Endpoints following HSCT
Censored data
Competing events
Statistical methods for survival & competing events
Intermediate events
Endpoints following SCT
Endpoints after HSCT
Endpoints : Response variables chosen to assess treatment effects that are related to
safety and efficacy
Primary endpoint chosen on the basis of the principal objective of the study
Its choice determines the sample size of the trial
Standardization of the endpoints used in transplantation CT to ensure comparability
between trials.
Endpoints after HSCT
5 endpoints irrespective to the type of disease studied
1. Overall survival (OS)
2. Disease-Free survival or Progression-Free survival (DFS or PFS)
3. Relapse Incidence (RI)
4. Time to progression (TTP)
5. Non-Relapse-Mortality (NRM)
Endpoints after HSCT
 Relapse incidence (RI): Time to relapse.
 Time to progression (TTP): Time to an advanced phase of the disease in
comparison to the stage at the study beginning.
 Non Relapse Mortality (NRM): Time to death without relapse/progression.
Deaths from any cause without prior progression are events.
Endpoints after HSCT
 Disease-free survival (DFS): Time to relapse or death from any cause,
whichever comes first.
 Progression-Free Survival (PFS): Time to relapse , progression or death from
any cause, whichever comes first.
Overall survival (OS): Time to death, irrespective of the cause.
Endpoints after HSCT
 NRM, OS : all patients
 RI DFS : Patients previously in disease remission
 TTP PFS : Patients transplanted with active disease
Specify in the protocol:
–
The definition of relapse or progression according to the disease
–
If patients transplanted with active disease who never achieved remission after SCT
are considered in progression at time of engraftment, at day 1 or 28.
Endpoints after HSCT
Most appropriate primary endpoint: PFS or DFS
 all clinically relevant events are included
 little opportunity for bias
 likely to be statistically sensitive to real treatment beneﬁts
 observed earlier than overall survival
Always report OS as secondary
Endpoints after HSCT
Many other events after SCT: Engraftment, Chimerism, Response rate, Acute Graftvs-host disease, Chronic Graft-vs-host disease
Specific endpoints could be needed :
 For specific diseases such as aplastic anaemia or inborn errors
 Remission rate
 In studies with a very specific goal (infection, GVHD, VOD...)
 Event-Free survival: acute GVHD free survival / fungal free survival
Endpoints after HSCT
 Survival (OS)
 Disease free survival (DFS)
Survival-like endpoints
 Event-Free Survival (EFS)
 Non relapse mortality (NRM)
 Relapse (RI)
 Engraftment: PMN>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 (<=1)
-  (probabilities CI) tends to 1
Statistical methods – Competing risks
Cumulative incidence (software R): cuminc function (package cmprsk)
RI
NRM
Statistical methods – survival analysis
Relation between CI and DFS
CI of relapse
CI OF NRM
DFS
• at t=60: RI=19% NRM=8% DFS=73%
• same number of patients at risk
Total risk of failure at time t = Risk of event 1 + Risk of event 2
Hazard of DFS (t) = CSHrelapse(t) + CSHNRM(t)
Comparison of the
endpoints
Statistical methods
Comparison of endpoints
Univariate analysis: unadjusted comparison
Multivariate analysis (regression models)
 Effect of one particular covariate, corrected for other factors
 More precise estimate of the effect
 Prognostic models
Statistical methods
Survival endpoint
Cumulative hazard H(t)
Survival function S(t)
A
B
We would like to quantify how much A is better than B
Statistical methods – Survival endpoint
Univariate analysis: Log-rank test
 Compares estimates of the hazard functions of the two groups at each
observed event time
 Based on comparison of observed with expected number of events (formally
identical to the Chi-square test for a series of 2x2 tables)
 Test of the global difference between curves
 The log-rank test has optimal power in case the ratio of the hazards is
constant with time, has little or no power if hazards cross
 Use other methods if you are Interested in:
• early event -> 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<1 indicates that the risk is lower in group A
HR provides a quantification of the difference, in percent :
HR=1.2 means that in group A, the risk is 20% higher than in group B
HR= 2 means that in group A, the risk is double than in group B
Statistical methods – Survival endpoints
Cox model - Interpretation of the results
Example: Model with 2 risk factors:
- age (in years) :
- stage of the disease :
continuous variable
categorical variable
Results:
- Stage 2 vs 1:
- Age:
HR=4
HR=1.02
A 50-year-old patient has a probability of dying 1.2 times greater than the
probability of dying for a 40-year-old patient with the same stage of the disease
Statistical methods – competing risks
What do we want to compare?
SURVIVAL ENDPOINT
Survival function S(t) : Kaplan Meier
Probability that an individual survives
from the time origin to t
COMPETING RISKS
Cumulative incidence function: CI(t)
Probability of event j (ie relapse) before t
One CI function for each competing event
Hazard function h(t)
Cause-specific hazard (CSH)
Probability that an individual dies at
time t, conditional he or she have
survived until that time
Probability of event j (relapse) at time t,
given that the patient did not fail from
any cause before that time
Statistical methods – Competing risks
Example : Relapse (competing = NRM)
Cumulative incidence function (CI)
Probability of relapse before t
Univariate:
Multivariate :
Gray-test
Fine-Gray model
Cause-specific hazard (CSH)
‘Instantaneous’ risk of relapse at each
time t for survivors without relapse
Univariate :
Multivariate :
Log-rank
Cox model
Statistical methods – Competing risks
Comparison : Cumulative Incidence or CSH ?
• They could return different conclusions!
• Cumulative incidence of relapse depends on both CSHrelapse and CSHNRM
• To understand if X is a risk factor for relapse, we need to see the
combined effect of X on CSHrelapse and CSHNRM
Statistical methods – Competing risks
Example: A is a risk factor for both CSH of relapse and CSH of NRM
Cause-specific hazard for relapse
Cause-specific hazard for NRM
However, the increase of NRM is much stronger
Cumulative incidence of relapse
Cumulative incidence of NRM
A is associated with higher NRM, but might be not associated with relapse
Statistical methods – Competing risks
Cumulative Incidence or CSH?
Cumulative incidence : probability of having relapse within a certain period after
transplantation:
 Prediction from the start of the disease history
 Strategic decisions prior to transplant
Analysis of the effects of factors on the CSH of relapse :
 Factors that increase the instantaneous risk of relapse among the survivors
 Biological mechanisms
 Clinical decisions to be made after transplantation.
BUT no estimation of the effects of the same factor on mortality.
Statistical methods – Competing risks
The results must always be presented for the competing events and the
effect should be discussed accordingly
The final benefit of one treatment should be studied on a global endpoint
such as DFS or EFS.
Statistical methods & HSCT
Summary
Survival
endpoints
Object
Estimation
Survival function S(t)
KaplanMeier
Hazard function h(t)
Competing risks
Cumulative incidence
(CI)
Cause-specific hazard
function (CSH)
proper nonparametric
unadjusted Regression
comparison model
Log-rank
Wilcoxon
Cox
Gray test
Fine-Gray
Log-rank
Cox
Intermediate events
Intermediate events
SCT
Relapse
GVHD
 Does GVHD modify the risk of relapse?
 Does the timing of GVHD modify the risk of relapse?
Intermediate events
Diagnosis
Death
SCT
Does SCT modify the risk of death?
Intermediate events
Very common source of errors in the medical literature when the (future) value of the
covariate is taken to be fixed at time 0
Intermediate event unknown at time Ø
 To experience the IE, patients have to survive long enough
 Patients who die « soon » end up belonging to the no-IE group
=> 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…

Download Report