Evaluation of the predictive capacity of a...
Transcript of Evaluation of the predictive capacity of a...
Evaluation of the predictive capacity of abiomarker
Bassirou Mboup(ISUP Universite Paris VI)
Paul Blanche(Universite Bretagne Sud)
Aurelien Latouche(Institut Curie & Cnam)
GDR STATISTIQUE ET SANTE, 10/05/20171 / 19
Goal and Definition
Goal :To assess the predictive capacity of a biomarker.
I The prognostic value of a marker corresponds to its ability topredict the clinical course of the disease in the absence oftreatment.
I The predictive value of a marker corresponds to its ability topredict the response to a given treatment.
I Genomic test that stratify patients to low/med/high risk ofrecurrence to guide the treatment with chemo
I Proliferation Marker (Ki67) to guide the trt with chemo
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Evaluating marker predictive performance
I In practice, only a significant interaction test between the treatmentassignment and the marker value is used to establish the predictivecapacity of the marker.
I A strong interaction is a necessary condition for a predictive marker.However, this not a sufficient condition (Janes et al., 2011).
I Janes et al. (2014) proposed a formal and comprehensive approachto the evaluation of predictive marker with binary endpoint.
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Some notations
I D is a binary indicator of the clinical outcome.
I Treat the treatment indicator.
I Treat = 1 if treatedI Treat = 0 if no treated
I We note that
I ρ0 = P(D = 1|Treat = 0)I ρ1 = P(D = 1|Treat = 1)
I Y a continuous marker.
I We define the risk as the conditional probability of having the eventgiven treatment and biomarker.
risk(Y ) = P(D = 1|Treat,Y )
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Standard decisions in the Janes et al. (2014) approach
The marginal effect of treatment is defined as
ρ0 − ρ1 = P(D = 1|Treat = 0)− P(D = 1|Treat = 1)
I if ρ0 − ρ1 > 0 then the standard strategy is to treat all patients.
I if ρ0 − ρ1 ≤ 0 then the standard strategy is to treat nobody.
Treatment rule
I Let ∆(Y ) = P(D = 1|Treat = 0,Y )− P(D = 1|Treat = 1,Y )denote the absolute treatment effect given marker value Y .
I The treatment assignment is : do not treat if ∆(Y ) < 0.
I We refer to subject with
I ∆(Y ) < 0 as ”marker-negatives”I ∆(Y ) > 0 as ”marker-positives”.
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Estimation and Inference
The observed data (Yi ,Di ,Treati ),i = 1, ..., n
I The risk model (glm)
g(P(D = 1|Treat,Y ) = β0 + β1 Treat + β2 Y + β3 Treat × Y
with g the logit link
I risk0(Y ) = P(D = 1|Treat = 0,Y ) = g−1(β0 + β2Y )
I risk1(Y ) = P(D = 1|Treat = 1,Y ) = g−1(β0 + β1 + β2Y + β3Y )
I The absolute treatment effect is estimated by risk0 and risk1 ,
∆(Y ) = risk0(Y )− risk1(Y )
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Marker-by-treatment Predictiveness curves withρ0 − ρ1 > 0
The proportion of subjects with negative treatment effect who can avoidtreatment, 46% for Y .
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Distribution of the treatment effect : ∆(Y )
Distribution of treatment effect : ∆(Y ) = risk0(Y )− risk1(Y )
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Proposed extension to right censored data (1/2)
Objective : estimating the absolute treatment effect at time t? in thepresence of censoring.
I The dynamic prediction of risk can defined by :
risk(t,Y ) = P(T ≤ t|Treat,Y )
with T be the observed failure time and D(t) = 1(T≤t) theresponse indicator.
I The time-dependent absolute treatment effect is :
∆(t,Y ) = P(D(t) = 1|Treat = 0,Y )− P(D(t) = 1|Treat = 1,Y )
+ A biomarker can be predictive up to a given time t? andnot after t?
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Proposed extension to right censored data (2/2)
I Zheng et al. (2006) proposed a time-dependent logisticregression model for prognostic accuracy with multiplebiomarkers
g(P(T ≤ t|Mi )) = α(t) + β1(t)M1 + · · ·+ βp(t)Mp
I We use this model to predict the overall risk :
g(P(T ≤ t|Treat,Y )) = α(t)+β1(t)Treat+β2(t)Y+β3(t)Treat×Y
I We consider a simplified model,
g(P(T ≤ t|Treat,Y )) = α(t)︸︷︷︸+β1Treat +β2Y +β3Treat ×Y
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Estimation of model parameters (1/2)
To estimate the β = (β1, β2, β3) and α under this model, we partition thetime axis into J non-overlapping intervals with cut-off points :
I t1...tj ...tJ−1 failure times.
I Dij = 1(tj−1 ≤ Ti < tj) response between tj−1 and tj
I θ = {α1 = α(t1), ..., αJ−1 = α(tJ−1), β}′,I xi = (1,Treati ,Yi ,Treati × Yi )
J−1∑j=1
n∑i=1
∂ log∇µij (θ)∇µi,j+1(θ)
∂θxi′{Dij(tj)− µi (αj , β)} = 0 (1)
where
µi (αj , β) = Λ(xiθ) = Λ(αj + β1Treat + β2Y + β3Treat × Y )
Λ(x) =exp(x)
1 + exp(x)
∇µi,j(θ) = E (Dij |Treati ,Yi , θ)
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Estimation of model parameters (2/2)
Suppose for the ith subject, we observe
I Ti = min(Ti ,Ci )
I δi = 1(Ti ≤ Ci ), where C is the censoring time
To account for censoring, we modify the previous equation and considerthe following inverse probability of censoring weighting (IPCW)estimating equation :
J−1∑j=1
n∑i=1
Vi (tj)
πi (tj)
∂log∇µi,j (θ)∇µi,j+1(θ)
∂θxi′{Di (tj)− µi (αj , β)} = 0 (2)
Vi (tj) =
{0 si Ti < tj and δi = 0
1 sinon
πi (tj) is a consistent estimator of the probability of censoring.
πi (tj) = P(Vi (tj) = 1|Treat,Y )
=
{P(Ci > Ti |Treat,Y , Ti ) if Ti ≤ tj and δi = 1
P(Ci > tj |Treat,Y ) if Ti > tj12 / 19
Time-dependent marker by treatment predictiveness curves
18% of relapses, 46% of survivors and 36% of censorings with alarge sample size.
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Replication with ”perfect” calibration of the predicted riskmodel
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Clinical interest
I The predictiveness curve makes it possible to obtain the cut-off ofthe marker.
I Determination of sub-group of patient who will benefit or not fromthe treatment.
I Improve and rationalize the performance of care, avoid unnecessarytreatment and improve the quality of life of patient.
As an illustration we consider the case of the proliferation marker Ki67
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Illustration : Ki67 as a predictive biomarker ?
Estimating the cut-off for Ki67
Observational study of 3310 women with a non metastatic breast cancerfollowed from 2000 to 2008 at Institut Curie. The Time-to-eventendpoint is Distant Metastasis.
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Perspectives
I Predictive biomarkers of response to treatment can potentiallyimprove clinical outcomes and lower medical costs.
I The proposed extension rely on a perfect calibration → A test ofcalibration would be of practical interest
I Hopefully, our request for the MINDACT data will be fulfilled ! ! !
I Extend the package R TreatmentSelection.
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THANK YOU FOR YOUR ATTENTION
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References
Janes, H., Brown, M. D., Huang, Y., and Pepe, M. S. (2014). Anapproach to evaluating and comparing biomarkers for patienttreatment selection. Int J Biostat, 10(1) :99–121.
Janes, H., Pepe, M. S., Bossuyt, P. M., and Barlow, W. E. (2011).Measuring the performance of markers for guiding treatmentdecisions. Ann. Intern. Med., 154(4) :253–259.
Zheng, Y., Cai, T., and Feng, Z. (2006). Application of thetime-dependent ROC curves for prognostic accuracy withmultiple biomarkers. Biometrics, 62(1) :279–287.
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