Two-stage receiver operating-characteristic curve estimator for cohort studies

Susana Díaz-Coto; Norberto Octavio Corral-Blanco; Pablo Martínez-Camblor

doi:10.1515/ijb-2019-0097

Artikel

Two-stage receiver operating-characteristic curve estimator for cohort studies

Susana Díaz-Coto , Norberto Octavio Corral-Blanco und Pablo Martínez-Camblor

Veröffentlicht/Copyright: 21. August 2020

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Aus der Zeitschrift The International Journal of Biostatistics Band 17 Heft 1

Abstract

The receiver operating-characteristic (ROC) curve is a graphical statistical tool routinely used for studying the classification accuracy in both, diagnostic and prognosis problems. Given the different nature of these situations, ROC curve estimation has been separately considered for binary (diagnostic) and time-to-event (prognosis) outcomes, even for data coming from the same study design. In this work, the authors propose a two-stage ROC curve estimator which allows to link both contexts through a general prediction model (first-stage) and the empirical cumulative estimator of the distribution function (second-stage) of the considered test (marker) on the total population. The so-called two-stage Mixed-Subject (sMS) approach proves its behavior on both, large-samples (theoretically) and finite-samples (via Monte Carlo simulations). Besides, a useful asymptotic distribution for the concomitant area under the curve is also computed. Results show the ability of the proposed estimator to fit non-standard situations by considering flexible predictive models. Two real-world examples, one with binary and one with time-dependent outcomes, help us to a better understanding of the proposed methodology on usual practical circumstances. The R code used for the practical implementation of the proposed methodology and its documentation is provided as supplementary material.

Keywords: asymptotic distributions; receiver-operating characteristic (ROC) curve; time-dependent ROC curve; two-stage estimator

Corresponding author: Pablo Martínez-Camblor, Biomedical Data Science Department, Geisel school of Medicine at Dartmouth, Hanover, NH, USA, E-mail: pablo.martinez.camblor@dartmouth.edu

Funding source: Gobierno del Principado de Asturias

Award Identifier / Grant number: FC-GRUPIN-IDI/2018/000132

Funding source: Ministerio de Ciencia e Innovación

Award Identifier / Grant number: MTM2017-89422-P

Acknowledgment

The authors thank the reviewers and the Associated Editor for all their comments, suggestions and their careful reading of the paper, which have helped us to improve the quality of the manuscript.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: This paper is supported by the grant FC-GRUPIN-IDI/2018/000132 of the Asturias Government. Besides PM-C is supported by the Spanish Ministry of Economy, Industry and Competitiveness; State Research Agency; and FEDER funds - MTM2017-89422-P.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix

Theoretical results

First we highlight the difference in the asymptotic behavior of the ECDF estimator for the global distribution when we consider a case-control or a cohort design. Let H=π⋅F+(1−π)⋅G, with π=P(D)(probability of having the studied characteristic, D=1) be the real distribution of the population including both positive and negative subjects. Notice that, if H^N(N=n+m) stands for the pooled empirical cumulative distribution function estimator (ECDF), we have the uniform weak convergence (see van der Vaart [33]),

N⋅[HN^(x)−H(x)]→LB{H(x)} ,

where ℬ{u}{0≤u≤1} is a standard Brownian bridge. However, if F^n, G^m denote the ECDFs for F and G, respectively, even when n/(n+m)→π, we have that

N⋅[π⋅Fn^(x)+(1−π)⋅Gm^(x)−H(x)]→Lπ⋅B1{F(x)}+1−π⋅B2{G(x)} ,

where ℬ1{u}{0≤u≤1} and ℬ2{u}{0≤u≤1} are two independent standard Brownian bridges. Of course, in general ℬ{H(x)}≠π⋅ℬ1{F(x)}+1−π⋅ℬ2{G(x)}.

Now, we formalize here the proofs of the results anticipated in section 3. Next, we are proving one general and useful lemma.

Preliminary result

Lemma 1.

Let(X,D)be the test values and the subjects status, respectively. LetR(⋅), Se(⋅)andSP(⋅)be the ROC, the sensitivity and the specificity curves associated with the classification problem. LetR^(⋅), S^e(⋅)andS^P(⋅)be their respective estimators based on the random sample(XN, DN). If

A₁.-R(⋅)has two continuous and bounded derivatives,
A₂.-supx∈ℝ|S^P(x)−SP(x)|→N 0 a.s. (almost surely).

Then, for eachu∈(0, 1), there exists a real random numbers sequence, {xN*}N∈ℕ, satisfying that xN*→N x* (a.s.)(x*=[1−SP]−1(u)) such that,

(11)[R^(u)−R(u)]=[S^e(xN*)−Se(xN*)] +r(1−S^P(xN*))⋅[S^P(xN*)−SP(xN*)]+O([S^P(xN*)−SP(xN*)]2),

where r(⋅) is the first derivative of R(⋅). Besides, O(xN*−x)=O(S^P(xN*)−SP(xN*)).

Proof

From A1, for any sequence {uN*}N∈ℕ∈(0, 1), it is hold the equality

[R^(u)−R(u)]=[R^(u)−R(uN*)]+r(u)⋅(uN*−u)+O([uN*−u]2).

Let be xN*=[1−S^P]−1(u) and uN*=1−SP(xN*), then

[R^(u)−R(u)]=[S^e(xN*)−Se(xN*)] +r(1−S^P(xN*))⋅[S^P(xN*)−SP(xN*)]+O([S^P(xN*)−SP(xN*)]2).

On the other hand, A1 implies that SP(⋅) has two bounded derivatives. Therefore, there exists ξN∈(min{xN*, x*}, max{xN*, x*}), such that

sP(ξN)⋅|xN*−x*|=|SP(xN*)−SP(x*)|=|SP(xN*)−S^P(xN*)|,

where sP(⋅) is the first derivative of SP(⋅). From A2, xN∗→N x (a.s) and O(xN*−x*)=O(S^P(xN*)−SP(xN*)).□

Proof of Theorem 1 (Strong uniform consistency)

Let be π=P(D) and π^N=∫P^N(D|x)dH^N(x). From M1 we have that π^N→N π. Besides, for any x*∈ℝ,

π^N⋅[S^e(x∗)−Se(x∗)] =∫I(x∗,∞)(x)⋅P^N(D|x)dH^N(x)−π^N⋅Se(x∗) =∫Δse(x∗, x)⋅P^N(D|x)d[H^N(x)−H(x)]+∫Δse(x∗, x)⋅[P^N(D|x)−P(D|x)]dH(x),

with Δse(x*, x)=[I(x*,∞)(x)−Se(x*)]. The Glivenko-Cantelli’s Theorem and M1 guarantee the uniform convergence of the first and the second summands, respectively. The triangular inequality and π^N→N π lead to

(12)supx*∈ℝ|S^e(x*)−Se(x*)|→N 0 a.s.

Arguing similarly, we have that

(1−π^N)⋅[S^P(x*)−SP(x*)] =∫Δsp(x*, x)⋅Q^N(D|x)d[H^N(x)−H(x)] +∫Δsp(x*, x)⋅[Q^N(D|x)−Q(D|x)]dH(x),

where Δsp(x*, x)=[I(−∞,x*](x)−SP(x*)], Q(D|x)=1−P(D|x) and Q^N(D|x)=1−P^N(D|x). Then,

(13)supx*∈ℝ|S^P(x*)−SP(x*)|→N0 a.s.

Applying Lemma 1, the triangular inequality, Eq. (12) and Eq. (13) we get Eq. (7).□

Proof of Theorem 2 (Weak uniform convergence)

Arguing as in Theorem 1, from the Hungarian embedding (see, for instance, [33]) and M2, we know that, in a suitable probability space, there exist a sequence of Brownian bridges, ℬN{u}{0≤u≤1}, and a standardized Gaussian variable, P, such that

N⋅[S^e(x*)−Se(x*)]=1π⋅{∫Δse(x*, x)⋅P(D|x)dℬN{H(x)}+∫Δse(x*, x)⋅σ(x)dH(x)⋅P}+oP(1),

and

N⋅[S^P(x*)−SP(x*)]=11−π⋅{∫Δsp(x*, x)⋅Q(D|x)dℬN{H(x)}−∫Δsp(x*, x)⋅σ(x)dH(x)⋅P}+oP(1).

Then, from Lemma 1 and the Lebesgue’s Theorem we have that, in this suitable probability space,

(14)N⋅[R^(u)−R(u)] =∫{π−1⋅Δse(x*, x)⋅P(D|x)+(1−π)−1⋅r(1−SP(x*))⋅Δsp(x*, x)⋅Q(D|x)}dℬN{H(x)} +∫{π−1⋅Δse(x*, x)−(1−π)−1⋅r(1−SP(x*))⋅Δsp(x*, x)}⋅σ(x)dH(x)⋅P+oP(1),

where u=1−SP(x*). Hence, making v=1−SP(x), we have that (in a suitable probability space),

(15)N⋅[R^(u)−R(u)] =∫{LPse(u, v)+r(u)⋅LPsp(u, v)}dℬN{π⋅R(v)+(1−π)⋅v} +∫{Lse(u, v)+r(u)⋅Lsp(u, v)}⋅σ(1−SP−1(v))d[π⋅R(v)+(1−π)⋅v]⋅P+oP(1),

and the proof is concluded.□

Proof of Theorem 3 (AUC’s weak convergence)

Directly, we have that

(16)[A^N−A]=∫[S^P(x*)−SP(x*)] dSe(x*)−∫[S^e(x*)−Se(x*)] dS^P(x*).

From definition (2) we have that dSe(x*)=−π−1⋅P(D|x*)dH(x*), hence, by using the Fubbini’s Theorem and arguing as in Theorem 2

(17)N⋅∫[S^P(x*)−SP(x*)] dSe(x*)=−1π⋅(1−π){∫Wsp(x)⋅Q(D|x)dℬN{H(x)}−∫Wsp(x)⋅σ(x)dH(x)⋅P+oP(1)}.

Anagously, dSP(x*)=(1−π)−1⋅Q(D|x*)dH(x*), and

(18)N⋅∫[S^e(x*)−Se(x*)] dSP(x*)=1π⋅(1−π){∫Wse(x)⋅P(D|x)dℬN{H(x)}−∫Wse(x)⋅σ(x)dH(x)⋅P+oP(1)}.

The Slutski’s Lemma guarantees that the asymptotic distribution of N⋅[A^−A) is the distribution of

1π⋅(1−π){∫[Wse(x)⋅P(D|x)−Wsp(x)⋅Q(D|x)]dℬN{H(x)}+∫[Wsp(x)−Wse(x)]⋅σ(x)dH(x)⋅P}.

From the Brownian bridge properties (see, for instance, [34]) and M3, it is Gaussian with mean zero and variance v2, where

[π⋅(1−π)⋅v]2=∫[(Wse(x)+Wsp(x))⋅P(D|x)−Wsp(x)]2dH(x)+{∫[Wsp(x)−Wse(x)]⋅σ(x)dH(x)}2,

and the proof is concluded.□

Additional Monte Carlo results

We report here (4) full results regarding the competitors considered in the time-dependent scenario (Table 2). We provide mean ± sd for ∫[R^(t)−R(t)]2dt, where R^(⋅) is the ROC curve estimation and R(⋅) the real ROC curve and for the respective AUC. We consider the empirical estimator computed from those subjects with complete information (E); the NNE-based estimator proposed in Heagerty et al. [10], computed with the R package survivalROC [35] with two different bandwidth parameters: 0.25⋅N−1/5 (H₁) and 0.5⋅N−1/5 (H₂) and the estimator proposed in Li et al. [12], computed with the package nsROC [6] with two different bandwidth parameters: 0.1⋅N−1/5 (L₁) and 0.5⋅N−1/5 (L₂) (see Table 4).

Table 4:

Time-dependent models. Supplement. Complementary results to Table 2. Same information regarding five alternative procedures.

N	π	CP	A	ROC curve					AUC
N	π	CP	A	E	H₁	H₂	L₁	L₂	E	H₁	H₂	L₁	L₂
Model IV
200	1/2	10%	0.65	4.68 ± 4.9	3.62 ± 3.8	3.08 ± 3.6	4.37 ± 3.9	4.32 ± 3.9	0.657 ± 0.05	0.647 ± 0.05	0.636 ± 0.04	0.644 ± 0.05	0.649 ± 0.05
			0.85	2.43 ± 1.9	1.99 ± 2.1	2.87 ± 3.1	2.67 ± 2.3	2.44 ± 2.1	0.857 ± 0.03	0.842 ± 0.03	0.822 ± 0.03	0.844 ± 0.03	0.849 ± 0.03
200	1/2	25%	0.65	5.25 ± 5.3	4.03 ± 4.3	3.41 ± 4.1	4.81 ± 4.6	4.77 ± 4.6	0.658 ± 0.05	0.647 ± 0.05	0.636 ± 0.05	0.641 ± 0.05	0.647 ± 0.05
			0.85	2.99 ± 2.5	2.41 ± 2.5	3.21 ± 3.6	3.85 ± 3.5	3.16 ± 2.8	0.861 ± 0.03	0.841 ± 0.03	0.822 ± 0.03	0.836 ± 0.03	0.845 ± 0.03
300	1/3	10%	0.65	2.92 ± 2.7	2.43 ± 2.5	2.24 ± 2.6	2.93 ± 2.6	2.87 ± 2.6	0.654 ± 0.04	0.645 ± 0.04	0.636 ± 0.04	0.648 ± 0.04	0.650 ± 0.04
			0.85	1.81 ± 1.5	1.66 ± 1.8	3.09 ± 2.8	1.91 ± 1.7	1.82 ± 1.6	0.854 ± 0.02	0.842 ± 0.03	0.822 ± 0.03	0.847 ± 0.03	0.850 ± 0.03
300	1/3	25%	0.65	3.18 ± 2.9	2.64 ± 2.8	2.40 ± 2.8	3.24 ± 2.9	3.14 ± 2.9	0.654 ± 0.04	0.646 ± 0.04	0.636 ± 0.04	0.646 ± 0.04	0.649 ± 0.04
			0.85	2.09 ± 1.8	1.84 ± 1.9	3.22 ± 3.0	2.31 ± 2.1	2.09 ± 1.9	0.857 ± 0.03	0.842 ± 0.03	0.823 ± 0.03	0.844 ± 0.03	0.848 ± 0.03
Model V
200	1/2	10%	0.65	3.07 ± 2.7	3.05 ± 2.8	5.03 ± 2.9	3.11 ± 2.8	3.03 ± 2.8	0.658 ± 0.04	0.662 ± 0.04	0.658 ± 0.04	0.653 ± 0.04	0.653 ± 0.04
			0.85	2.09 ± 1.4	1.50 ± 1.3	2.29 ± 1.9	2.05 ± 1.5	2.01 ± 1.5	0.859 ± 0.03	0.854 ± 0.03	0.842 ± 0.03	0.853 ± 0.03	0.853 ± 0.03
200	1/2	25%	0.65	3.47 ± 3.1	3.32 ± 3.1	5.27 ± 3.3	3.49 ± 3.1	3.37 ± 3.1	0.662 ± 0.05	0.661 ± 0.05	0.657 ± 0.05	0.654 ± 0.05	0.653 ± 0.05
			0.85	2.54 ± 1.8	1.75 ± 1.6	2.52 ± 2.3	2.39 ± 1.9	2.31 ± 1.8	0.866 ± 0.03	0.854 ± 0.03	0.842 ± 0.03	0.853 ± 0.03	0.854 ± 0.03
300	1/3	10%	0.65	2.63 ± 2.6	2.98 ± 2.9	5.69 ± 3.0	2.63 ± 2.7	2.57 ± 2.6	0.653 ± 0.04	0.661 ± 0.04	0.652 ± 0.04	0.651 ± 0.04	0.650 ± 0.04
			0.85	1.80 ± 1.4	1.44 ± 1.4	3.59 ± 2.4	1.78 ± 1.5	1.77 ± 1.5	0.856 ± 0.03	0.855 ± 0.03	0.840 ± 0.03	0.851 ± 0.03	0.851 ± 0.03
300	1/3	25%	0.65	2.94 ± 3.0	3.18 ± 3.1	5.86 ± 3.2	2.90 ± 2.9	2.79 ± 2.8	0.657 ± 0.04	0.662 ± 0.04	0.653 ± 0.04	0.652 ± 0.04	0.651 ± 0.04
			0.85	2.06 ± 1.5	1.62 ± 1.5	3.74 ± 2.6	1.99 ± 1.7	1.96 ± 1.6	0.861 ± 0.03	0.8550.03	0.840 ± 0.03	0.852 ± 0.03	0.852 ± 0.03
Model VI
200	1/2	10%	0.65	3.47 ± 3.2	2.94 ± 2.9	2.64 ± 3.0	3.43 ± 3.1	3.41 ± 3.0	0.657 ± 0.04	0.647 ± 0.04	0.636 ± 0.04	0.652 ± 0.04	0.654 ± 0.04
			0.85	2.33 ± 1.9	2.13 ± 2.2	3.02 ± 3.3	2.40 ± 2.1	2.37 ± 2.1	0.859 ± 0.03	0.844 ± 0.03	0.824 ± 0.03	0.852 ± 0.03	0.854 ± 0.03
200	1/2	25%	0.65	4.30 ± 4.2	3.38 ± 3.5	2.96 ± 3.6	4.04 ± 3.5	4.01 ± 3.6	0.661 ± 0.05	0.648 ± 0.04	0.637 ± 0.04	0.650 ± 0.04	0.653 ± 0.04
			0.85	2.78 ± 2.3	2.43 ± 2.5	3.24 ± 3.5	3.26 ± 2.9	2.89 ± 2.5	0.865 ± 0.03	0.844 ± 0.03	0.824 ± 0.03	0.845 ± 0.03	0.851 ± 0.03
300	1/3	10%	0.65	2.51 ± 2.2	2.08 ± 2.1	1.86 ± 2.0	2.47 ± 2.1	2.46 ± 2.1	0.656 ± 0.03	0.647 ± 0.03	0.638 ± 0.03	0.653 ± 0.03	0.653 ± 0.03
			0.85	1.66 ± 1.4	1.51 ± 1.6	2.47 ± 2.4	1.68 ± 1.4	1.64 ± 1.4	0.858 ± 0.02	0.846 ± 0.02	0.828 ± 0.02	0.853 ± 0.02	0.855 ± 0.02
300	1/3	25%	0.65	3.10 ± 2.9	2.56 ± 2.8	2.29 ± 2.7	3.05 ± 2.8	3.01 ± 2.9	0.658 ± 0.04	0.647 ± 0.04	0.638 ± 0.04	0.650 ± 0.04	0.652 ± 0.04
			0.85	2.08 ± 1.8	1.78 ± 1.8	2.65 ± 2.7	2.16 ± 1.8	2.01 ± 1.8	0.863 ± 0.02	0.847 ± 0.03	0.829 ± 0.03	0.850 ± 0.03	0.854 ± 0.03

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Supplementary Material

As supplementary material we provide the file sMSApproach.R which contains two functions for implementing the sMS ROC curve estimator for the binary and time-dependent cases together with the code used for analyzing the real-world examples (data sets are freely available on the net). The file sMSApproach.pdf contains indications about the use of the functions. The files are available at https://doi.org/10.1515/ijb-2019-0097.

Received: 2019-08-27

Accepted: 2020-05-25

Published Online: 2020-08-21

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https://doi.org/10.1515/ijb-2019-0097

Schlagwörter für diesen Artikel

asymptotic distributions; receiver-operating characteristic (ROC) curve; time-dependent ROC curve; two-stage estimator

Two-stage receiver operating-characteristic curve estimator for cohort studies

Artikel

Abstract

Acknowledgment

Theoretical results

Preliminary result

Lemma 1.

Proof

Additional Monte Carlo results

References

Supplementary Material

Zusatzmaterial

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