Auc silver spring

Discrimination measures for survival outcomes:
Connection between the AUC and the predictiveness
curve

V. Viallon, A. Latouche

University Lyon 1
University Versailles Saint Quentin

October 13, 2011

Viallon (Univ. Lyon 1) Connecting the pred. curve and the AUC 1 / 22

1 The binary outcome setting
The AUC
The predictiveness curve
Connecting the AUC and the predictiveness curve

2 The survival outcome setting
Time-dependent outcomes deﬁnitions
Connecting time-dependent AUC and time-dependent predictiveness
curve
Estimates of the time-dependent AUC
Some synthetic examples


AUC for binary outcomes

Let D be a (0, 1)-variable representing the status regarding a given
disease:
D = 1 for diseased individuals;
D = 0 for non-diseased individuals.

Further let X be a continuous marker. For any c ∈ I
R
X > c: the test is positive;
X ≤ c: the test is negative.
TPR(c) = I P(X > c|D = 1) and FPR(c) = I
P(X > c|D = 0).

ROC curve: FPR(c), TPR(c) , c ∈ I
R

AUC: TPR(c)dFRP(c).


Binary outcomes: a toy example

FPR(c) = I
P(X > c|D = 0)
TPR(c) = I
P(X > c|D = 1)

6
1.0

4
0.8
0.6

2
TPR

0.4

0
0.2

−2
0.0

−4

0.0 0.2 0.4 0.6 0.8 1.0
0 1
FPR



FPR(c) = I
P(X > c|D = 0)
TPR(c) = I
P(X > c|D = 1)

6
1.0

4
0.8
0.6

2
TPR

0.4

0
0.2

−2
0.0

−4

0.0 0.2 0.4 0.6 0.8 1.0
0 1
FPR


Predictiveness curve for binary outcomes

Many alternative criteria have been proposed for evaluating
discrimination
proportion of explained variation,
standardized total gain
risk reclassiﬁcation measures (Pencina et al., SiM, 2006)
Most of them express as simple functions of the predictiveness curve
(Gu and Pepe, Int. J. Biostatistics, 2009).
Denote by G −1 the quantile function of X . For any q ∈ [0, 1], let

R(q) = P D = 1|X = G −1 (q)

be the risk associated to the qth quantile of X .

The predictiveness curve plots R(q) versus q.


Predictiveness curves and their corresponding AUC values
1
With p = I
P(D = 1) = 0 R(q)dq=0.5

1.0
R1 (AUC=0.500)
R2 (AUC=0.700)
R3 (AUC=0.833)
0.8

R4 (AUC=0.928)
R5 (AUC=1.000)
Predictiveness Curve

0.6
0.4
0.2
0.0

0.0 0.2 0.4 0.6 0.8 1.0

Quantiles

The relation in the binary outcome setting

Still denote by R the predictiveness curve of marker X ,
R(q) = P D = 1|X = G −1 (q) .

1
Then, denoting by p = IP(D = 1) = 0 R(q)dq the disease
prevalence, the AUC of marker X is given by
1
0 qR(q)dq − p 2 /2
AUC =
p(1 − p)

We can check that
AUC = 0.5 when R(q) = p;
AUC = 1 when R(q) = 1 [1−p,1] (q).
I


Extensions to survival outcomes

In prospective cohort study, the outcome (e.g., the disease status) can
change over time
⇒ we consider time-dependent outcomes, TPR, FPR, ROC curves,
AUC and predictiveness curve.

Notations:
Ti and Ci : survival and censoring times for subject i
(Zi , δi )1≤i≤n with Zi = min(Ti , Ci ) and δi = 1 i ≤ Ci )
I(T
Di (t): time-dependent outcome status for subject i at time t.


Heagerty and Zheng’s Taxonomy

Today’s talk focus on Cumulative cases & Dynamic controls:

cumulative cases: Di (t) = 1 if Ti ≤ t;

dynamic controls Di (t) = 0 if Ti > t;

so that Di (t) = 1 i ≤ t}.
I{T

⇒discrimination between subjects who had the event prior to time t
and those who were still event-free at time t.


Cumulative cases and Dynamic controls

For a given evaluation time t0

Cumulative true positive rates are

TPRC (c, t0 ) = I
P(X > c|D(t0 ) = 1)
P(X > c|T ≤ t0 );
= I

Dynamic false positive rates are

FPRD (c, t0 ) = I
P(X > c|D(t0 ) = 0)
= I
P(X > c|T > t0 );
∞
AUCC,D (t0 ) = C
−∞ TPR (c, t0 )d FPRD (c, t0 ) .

But 1 i ≤ t0 ) is not observed for all i due to censoring!!
I(T


Workaround for AUCC,D

Using Bayes’s theorem (see, e.g., Chambless & Diao)
∞ ∞
F (t0 ; X = x)[1 − F (t0 ; X = c)]
AUCC,D (t0 ) = g (x)g (c)dxdc
−∞ c [1 − F (t0 )]F (t0 )

with

P(T ≤ t) be the risk function at time t;
F (t) = I
P(T ≤ t|X = x) be the conditional risk function at
F (t; X = x) = I
time t;
g the density function of marker X .


Predictiveness curve and AUCC,D

Introduce

P(D(t) = 1|X = G −1 (q))
R(t; q) := I
= P(T ≤ t|X = G −1 (q))
I

the time-dependent predictiveness curve

We established that
1 2
C,D 0 qR(t0 ; q)dq − F (t0 )
2
AUC (t0 ) =
F (t0 )[1 − F (t0 )]

Proper estimation of R(t0 ; q) (especially for q 1) should yield
proper estimation of AUCC,D (t0 )


Deriving estimates for AUCC,D (t)

G and g : cdf and pdf of X .
X(i) : i-th order statistic of (X1 , . . . , Xn ).
Fn (t0 ; x): estimator of the conditional risk F (t0 ; X = x).

Using the change of variable x = G −1 (q), we have
1 ∞
0 qR(t0 ; q)dq = −∞ G (x)F (t0 ; X = x)g (x)dx, so that

n
1 i
Fn (t0 ; X(i) ),
n n
i=1

1
is the empirical counterpart of the 0 qR(t0 ; q)dq.


Deriving estimates for AUCC,D (t)

To estimate the marginal risk function F
the KM estimator Fn,(1) (t0 ).

Since F (t0 ) = F (t0 ; x)g (x)dx, we can also use
n
1
Fn,(2) (t0 ) = Fn (t0 ; Xi ).
n
i=1

This yields two estimators for AUCC,D (t0 ), namely, for k = 1, 2,
1 n i 2
n i=1 n Fn (t0 ; X(i) ) − Fn,(k) (t0 )/2
AUCC,D (t0 ) =
n,(k) .
Fn,(k) (t0 ) 1 − Fn,(k) (t0 )

Experimental results (not shown) suggested better performances results
obtained with k = 2.


Simulation study

X : continuous marker

T : survival time generated under
a Cox model λ(t) = λ0 (t) exp(αX )
a TV coeff. Cox model λ(t) = λ0 (t) exp(α(t)X )

Various censoring schemes were considered.

Estimation of the cond. risk F (t; X = x)
a Cox model
an Aalen additive model
conditional KM estimator

Goal: assess the effect of model misspecification – when estimating
the conditional risk function– on the AUCC,D (t) estimation.


Simulation under a Cox model

1.0
True
HLP
KM cond.
0.9 Add. Aalen
Cox
0.8
AUC C/D

0.7
0.6
0.5

0 1 2 3 4 5

Time


Simulation under a Time-varying coeﬃcient Cox model

1.0
True
HLP
KM cond.
0.9 Add. Aalen
Cox
0.8
AUC C/D

0.7
0.6
0.5

0.2 0.4 0.6 0.8 1.0 1.2

Time


Conclusion

Relation between the predictiveness curve and the AUCC,D (t).

Enables to easily derive estimates of the AUCC,D (t) given estimates of
the cond. risk function.

Correctly specifying the model (when estimating the cond. risk
function) is crucial to get proper estimation of AUCC,D (t);

A similar relation can be obtained to connect the partial AUC (or
partial AUCC,D (t)) to the (time-dependent) predictiveness curve;

The conditional risk function, through the predictiveness curve, is the
key when assessing discrimination of prognostic tools


Sketch of the proof

∞ ∞
F (t; X = x)[1 − F (t; X = c)]g (x)g (c)dxdc
−∞ c
1 1
−1 −1
= F (t; X = G (u))[1 − F (t; X = G (v ))]dudv
0 v
1 1
−1 −1
= [1 − S(t; X = G (u))]S(t; X = G (v ))dudv
0 v
1 1 1
−1 −1 −1
= (1 − v )S(t; X = G (v ))dv − S(t; X = G (u))S(t; X = G (v ))dudv
0 0 v
1 1 1
−1 −1 −1
= (1 − v )S(t; X = G (v ))dv − S(t; X = G (u))S(t; X = G (v ))1I(u ≥ v )dudv .
0 0 0

Setting
−1 −1
L(u, v ) = S(t; X = G (u))S(t; X = G (v )),

we have L(u, v ) = L(v , u) so that

∞ ∞
F (t; X = x)[1 − F (t; X = c)]g (x)g (c)dxdc
−∞ c
1 1 1 1
−1 −1 −1
= (1 − v )S(t; X = G (v ))dv − S(t; X = G (u))S(t; X = G (v ))dudv
0 2 0 0
1 1 1 2
−1 −1
= (1 − v )S(t; X = G (v ))dv − S(t; X = G (v ))dv .
0 2 0

Simulation under a Time-varying coeﬃcient Cox model
1.0

1.0
True True
KM cond. KM cond.
0.8

0.8
Add. Aalen Add. Aalen
Cox Cox
Predictiveness curve

Predictiveness curve
0.6

0.6
0.4

0.4
0.2

0.2
0.0

0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

Quantile of marker Quantile of marker

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