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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