Receiver operating characteristic curve generalization for non-monotone relationships.

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Article

Receiver operating characteristic curve generalization for non-monotone relationships

Statistical Methods in Medical Research 0(0) 1–11 ! The Author(s) 2014 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0962280214541095 smm.sagepub.com

Pablo Martıńez-Camblor,1,2 Norberto Corral,3 Corsino Rey,3,4 Julio Pascual3,5 and Eva Cernuda-Morolloń3,5

Abstract The receiver operating characteristic curve is a popular graphical method frequently used in order to study the diagnostic capacity of continuous markers. It represents in a plot true-positive rates against the false-positive ones. Both the practical and theoretical aspects of the receiver operating characteristic curve have been extensively studied. Conventionally, it is assumed that the considered marker has a monotone relationship with the studied characteristic; i.e., the upper (lower) values of the (bio)marker are associated with a higher probability of a positive result. However, there exist real situations where both the lower and the upper values of the marker are associated with higher probability of a positive result. We propose a receiver operating characteristic curve generalization, Rg , useful in this context. All pairs of possible cut-off points, one for the lower and another one for the upper marker values, are taken into account and the best of them are selected. The natural empirical estimator for the Rg curve is considered and its uniform consistency and asymptotic distribution are derived. Finally, two real-world applications are studied. Keywords area under the curve, asymptotic distribution, resampling methods, receiver operating characteristic curve

1 Introduction The receiver operating characteristic (ROC) curve is a popular and valuable tool commonly used in order to study the diagnostic performance of a particular marker. It displays in a plot the 1

Oficina de Investigacioń Biosanitaria de Asturias (OIB-FICYT) Universidad Autonoma de Chile, Chile 3 Universidad de Oviedo 4 UCI Pedia´trica, Departamento de Pediatrıá, Hospital Universitario Central de Asturias (HUCA) 5´ Area de Neurociencias, Servicio de Neurologıá, Hospital Universitario Central de Asturias (HUCA) 2

Corresponding author: Pablo Martıńez-Camblor, Oficina de Investigacioń Biosanitaria de Asturias, Calle Matema´tico Pedrayes, 25 Entresuelo, 33005 Oviedo, Asturias, Spain. Email: [email protected]

Downloaded from smm.sagepub.com at Kungl Tekniska Hogskolan / Royal Institute of Technology on November 14, 2015


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Statistical Methods in Medical Research 0(0)

false-positive rate (FPR) or 1-specificity (1 – SP), i.e. the inability of the test to recognize a negative subject (frequently healthy) as negative against the sensitivity (SE) or true-positive rate (TPR), i.e. the capacity of a diagnostic criterion/test to identify a positive subject (frequently diseased) as positive. As classifying individuals into groups is a common problem in a variety of contexts, the ROC curve has received great attention in the specialized literature. There exist a huge number of papers which deal with both the theoretical and practical aspects of the ROC curve and its related problems (see, for instance, Martı´ nez-Camblor1 for a recent review). Also, the book by Zhou et al.2 provides a deep study of different ROC curve issues. Conventionally, without loss of generality, it is assumed that larger values of the marker indicate the larger confidence that a given subject is positive (diseased). Therefore, let and be two continuous random variables representing the values of the diagnostic test for the negative (without the studied characteristic/normal) and the positive (with the characteristic/diseased) subjects, respectively, for a fixed point t (FPR), the ROC curve is defined as follows RðtÞ ¼ 1 F ðF1 ð1 tÞÞ ¼ Pf 4 F1 ð1 tÞg ¼ Pf1 F ðÞ tg ¼ F1F ðÞ ðtÞ

ð1Þ

where F and F denote the cumulative distribution functions (CDF) for the variables and , respectively. Although other different parametric,3 semi-parametric4 and non-parametric estimators5 have been proposed, the empirical estimator, based on replacing the involved unknown CDFs for their respective empirical cumulative distribution functions (ECDF) is the most popular one. Hence, let X and Y be two random samples drawn from and , respectively, for each t 2 ½0, 1 the empirical ROC curve estimator is defined by b ¼ 1 F^n ðX, F^1 ðY, ð1 tÞÞ RðtÞ m

ð2Þ

where F^n ðX, Þ is the ECDF referred to the sample X (with size n) and F^1 m ðY, Þ ¼ inffs 2 R = F^m ðY, sÞ g (with size m). Both the uniform consistency and the asymptotic behaviour b have been investigated under weak and standard assumptions (see, for instance, Hsieh and of R Turnbull4). In particular, if (i) both F and F have continuous densities, f and f , f ðF ðtÞÞ=f ðF ðtÞÞ is bounded in any subinterval (a, b) of (0, 1) and (ii) n=m ! as minðn, mÞ ! 1, there exists a probability space on which one can define independent sequences of Brownian bridges, ðnÞ fBðmÞ 1 ðtÞgf0t1g and fB2 ðtÞgf0t1g , such that for each t 2 ½0, 1 pffiffiffi b RðtÞ ¼ 1=2 rðtÞ BðmÞ ð1 tÞ þ BðnÞ ð1 RðtÞÞ þ oð1Þ a:s n RðtÞ 1 2

ð3Þ

where rðtÞ ¼ @RðtÞ=@t. However, sometimes not only the larger (lower) marker values are associated with disease but both the lower and larger values are related to the presence of the studied feature. For instance, in haemodialysis population, both the high and low levels of serum iPTH, calcium and phosphate are associated with higher risk of mortality.6 Also, in the intensive care units, leukocyte counts greater than 20,000 (leukocytosis) or below 5000 (leukopenia) are associated with bad prognosis in critically ill patients. Obviously, this issue has already been pointed out. Hilden7 exposed the following theoretical example: the (bio)marker values (T) for half of the diseased subjects are less than 80 and the other half have values greater than 120 while the (bio)marker values for the healthy patients are always



Martıńez-Camblor et al. (a)

3 (b)

Figure 1. Density functions for the negative, f, and the positive, f, populations (left) and the resulting ROC curve (right).

between 80 and 120. Zhou et al.2 reduced this problem to its usual formulation by performing the transformation T0 ¼ jT 100j, which solves the problem. However, it is not always possible to find easily a suitable transformation, specially, when the marker distribution for the diseased or for the healthy subjects is skewed. Figure 1 depicts the density functions for the negative and the positive populations (left) and the resulting right-side ROC curve (right). Note that, in this case, the sole difference between the negative and the positive subjects is their respective variability. Although not so extreme, the depicted situation is similar to that described by Hilden.7 From the usual right-side approach, the classification based on the point B obtains a higher percentage of false-positives (97.7%) than truepositives (81.4%). However, if a subject with a marker value lower than 2 or greater than 2 is declared as positive and negative otherwise, the false-positive and the true-positive percentages are 4.6% and 37.1%, respectively. This criterion is equivalent to using a threshold of two on the transformed marker T 0 ¼ jTj. In this paper, the authors propose a ROC curve generalization useful in the above context. Since it considers all possible pairs of cut-off points. The first element of each pair is associated with the lower bound and the second to the upper bound; those obtaining the same specificity and which achieve the largest sensitivity are selected. The uniform consistency and the asymptotic distribution for its direct empirical estimator are derived in Section 2. Section 3 is devoted to studying the properties of the associated area under the curve, AUCg. In Section 4, this methodology is applied to two real data sets. On the one hand, we study the association between the leukocyte count and the mortality risk in critically ill children, and, on the other hand, we study the relationship between the Calcitonin gene-related peptide (CGRP) levels and the response to the OnabotulinumtoxinA treatment for chronic migraine in adults. The results which guarantee the uniform consistency and the asymptotic distribution for the empirical general ROC curve estimator and a R function (Rg) to calculate the proposed general ROC curve are provided as supplementary material.



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2 ROC generalization: Definition and properties For each fixed threshold, x 2 R, the right-side ROC curve is the pair f1 SP ðxÞ, SE ðxÞg ¼ f1 F ðxÞ, 1 F ðxÞg. If t ¼ 1 F ðxÞ, the well-known ROC curve expression ft, RðtÞg is obtained. When different thresholds lead to the same specificity, the one which maximizes the sensitivity is taken (in the usual right-side convention, the minimum of all possible thresholds). When both the larger and lower values of the marker are associated with a higher probability of disease, two thresholds xl , xu 2 R ðxl xu Þ are necessary in order to make a diagnosis. A subject is classified as positive when its marker value is below xl or greater than xu; therefore, the specificity is defined by SP ðxl , xu Þ ¼ F ðxu Þ F ðxl Þ. For arbitrary t 2 ½0, 1, let F t ¼ fðxl , xu Þ 2 R2 such that xl xu and F ðxl Þ þ 1 F ðxu Þ ¼ tg. The generalized ROC curve is defined as follows sup fF ðxl Þ þ 1 F ðxu Þg

Rg ðtÞ ¼

ðxl ,xu Þ2F t

Obviously, for each ðxl , xu Þ 2 F t there exists a 2 ½0, 1 such that F ðxl Þ ¼ , and then 1 xl ¼ F1 ðÞ. Due to ðxl , xu Þ 2 F t , then F ðxu Þ ¼ 1 ½1 and xu ¼ F ð1 ½1 Þ. Therefore n o 1 ðÞÞ þ 1 F ðF ð1 ½1 tÞÞ Rg ðtÞ ¼ sup F ðF1 2½0,1

or equivalently Rg ðtÞ ¼ sup 1 Rð1 tÞ þ Rð½1 tÞ

ð4Þ

2ð0,1Þ

If t ¼ arg sup f1 Rð1 tÞ þ Rð½1 tÞg (if it is not a unique value, we take the minimum 2½0,1

among all possibilities), the above expression is equivalent to Rg ðtÞ ¼ f1 Rð1 t tÞ þ Rð½1 t tÞg The t value determines the optimum proportion of false-positives, t . . . t, due to the negative subjects with a marker below xl (left tail), and ½1 t t because of the negative subjects with a marker larger than xu (right tail). Logically, under symmetry (Figure 1), the optimum value is 1/2 for all t 2 ½0, 1. Directly, Rg generalizes the usual right-side (left-side) ROC curve. When lower (upper) marker values are associated with normal subjects, for each t 2 ½0, 1, t ¼ 0 ðt ¼ 1Þ and Rg ¼ R ðRg ðtÞ ¼ 1 Rð1 tÞÞ. Note that, in the Hilden case enunciated above Rg ¼ 1 when t ¼ 1/2 for all t 2 ½0, 1. In addition, taking into account that Rg is mainly based on the usual ROC curves, it inherits most of its properties, for instance, the invariance for monotone increasing transformations.4 Figure 2 represents the Rg construction for the populations considered in Figure 1. In this case, RðtÞ ¼ 1 Rð1 tÞ and the optimum value for is 1/2 for all t 2 ½0, 1. The set of all possible trajectories of the curve, 1 Rð1 tÞ þ Rð½1 tÞ for 2 ½0, 1, defines the grey zone. In this case, this area coincides with the difference between the area under the right-side (left-side) ROC curve and the area under the Rg curve. Conventionally, let X and Y be two independent random samples drawn from the random variables and , respectively, then for each t 2 ½0, 1, the empirical general ROC curve estimator is n o b tÞ þ Rð½1 b tÞ b g ðtÞ ¼ sup 1 Rð1 ð5Þ R 2½0,1



Martıńez-Camblor et al. (a)

5 (b)

Figure 2. Rg construction for the populations considered in Figure 1 (left). In grey, the set of all possible trajectories of f1 Rð1 tÞ þ Rð½1 tÞg for 2 ½0,1 and the Rg curve, in black (right).

(a)

(b)

b g construction for two samples of negative and positive populations (n ¼ 100, m ¼ 100) drawn from the Figure 3. R b tÞ þ Rð½1 b tÞg for populations considered in Figure 1 (left). In grey, all possible trajectories of f1 Rð1 b 2 ½0,1 and the Rg curve, in black (right).

b is the empirical ROC curve estimator defined in equation (2). The R b g estimator has good where R asymptotic properties. See supplementary material for the results that cater for both the uniform consistency and the asymptotic distribution. Figure 3 is similar to Figure 2, but empirical curves based on independent random samples of positive and negative populations (sharing the same sample size, 100) were used instead of the theoretical ones. b g value was Here and onwards, the values of t were computed numerically. For t ¼ 1/2, the observed R b t tÞ ¼ Rð0:759Þ b 0.89. This value was obtained when t ¼ 0:481 ðRð1 ¼ 0:60 and



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b b t tÞ ¼ Rð0:259Þ ¼ 0:49). In this case, the minimum observed value for Rð½1 b b tÞg was 0.42, obtained for ¼ 1 (usual left-side ROC curve). f1 Rð1 tÞ þ Rð½1 Additionally, empirical results based on simulations (results not shown) suggest that the b g to its target is similar to the convergence ratio for the empirical convergence ratio of the R ROC curve estimator. However, it is worth pointing out that, when the diagnostic capacity of the b g over-estimates the marker diagnostic capacity. marker is poor (close to the main diagonal), R

3 AUC The area under the ROC curve (AUC) is the most commonly used global index of diagnostic accuracy.8 It ranges between 0.5 and 1 and represents the overall performance of a diagnostic test, in terms of its accuracy, at all relevant diagnostic thresholds (cut-off points) used to discriminate subjects with the considered characteristic or without. Moreover, in the usual rightside (left-side) assumption, the AUC (labelled by A) is Z1 Z A¼ RðtÞdt ¼ 1 F ðuÞdF ðuÞ ¼ Pf 4 g 0

Therefore, in this case, the AUC has a direct probabilistic interpretation. In particular, the AUC is the probability that the value of the marker in a randomly chosen positive subject will be higher than the value of the marker in a randomly chosen negative subject. Of course, the area under the general ROC curve can also be directly computed and it can be read as the sensitivity average for the different specificities. Its value can still be used in order to summarize the accuracy of the studied test by a single number; but, unfortunately, it loses the probabilistic interpretation, even when t constant, due to the fact that the relationship between the marker and the studied characteristic is not monotone. The direct non-parametric estimator for the area under the generalized ROC curve, AUCg, is Z1 bg ¼ b g ðtÞdt A ð6Þ R 0

R1 pffiffiffi b Theorem 2 (see supplementary material) guarantees that n ðA g Ag Þ (Ag ¼ 0 Rg ðtÞdt) follows, asymptotically, a normal distribution with mean zero. The variance depends on the first derivative of the t function and, in general, it is not possible to derive expressions useful in practice; therefore, a resampling method must be used in order to perform inferences. By using the traditional naive bootstrap method,9 the accuracy of the results, obtained via the Monte Carlo simulations, was similar to that of the usual AUC estimator In the supplementary material, the pffiffiffi b (results not shown). 0 explicit expression for the variance of n ðA A Þ, when ¼ 0 8t 2 ð0, 1Þ is derived. Unfortunately, g g t the simulation results suggest that this expression is not useful for general situations. In particular, when R t is replaced by t dt and a plug-in method is employed for estimating all the unknown parameters, the observed estimations for the variances were really unstable (results not shown).

4 Real data applications 4.1 Association between leukocyte count and mortality risk in critically ill children Having available tools to determine the risk of mortality at admission to the Paediatric Intensive Care Unit (PICU), or during the first 24 h after admission, is a clinical necessity.10 Leukocyte count



Martıńez-Camblor et al.

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measurement constitutes a routinely determination when a patient is admitted to the PICU. Classically, low leukocyte or high leukocyte counts are described as one of the criteria for the diagnosis of sepsis, a severe systemic infection.11 Recently, low leukocyte count has been found to be an independent risk factor of mortality in critically ill children with sepsis12 as well as in adults with necrotizing pneumonia.13 Therefore, the objective of our study was to do research into the leukocyte count in a well-defined cohort of consecutive PICU patients to check whether low or high leukocyte counts are associated with a higher prediction of mortality risk scores in critically ill children. We designed a prospective observational study set in two PICUs at university hospitals (eight-bed PICU at Hospital Universitario Central de Asturias (HUCA) in Oviedo and 11-bed PICU at Hospital Universitario Gregorio Maranõń in Madrid). The study protocol was approved by the Hospital Ethics Committee of the HUCA. The study was conducted in a number of consecutive patients, all under 18 years of age, who were admitted to one of these PICUs. Patients were divided into two groups according to mortality risk scores. The higher score risk mortality group (high MR) included patients with a PIM 2 and PRISM III scores greater than percentile 75 (N ¼ 12); the lower score risk mortality group (low MR) included patients with a PIM 2 and/or PRISM score below percentile 75 (N ¼ 176). For further reference, see Rey et al.10 Leukocyte count routine determinations were performed during the first 12 h after admission. A total of 188 (61.7% women) patients were included in the study. Twelve of them were classified within the high mortality risk group (high MR) and 176 in the low mortality risk group (low MR). Patients in the high MR were younger (mean standard deviation: 31.67 42.38 months) than patients in the low MR, 67.32 59.88 (the Student–Welch test provided a p-value of 0.016). Table 1 shows some descriptive statistics for the leukocyte count in both the low and high mortality risk groups. Note that, in the high MR group the observed variability of the leukocyte count was really large. In addition, in the low MR group, P25 was larger and P75 lower than in the high MR group. When a usual right-side (left-side) ROC curve is plotted, it crosses the diagonal line and a low AUC value is obtained. In particular, the observed AUC for the left-side ROC curve was 0.520 with a 95% (naive) bootstrap (based on 5000 iterations) confidence interval of (0.275–0.693). The general ROC curve detects that both the lower and the higher values of the marker are associated with the high MR group and achieved an Ag of 0.740 with a 95% (naive) bootstrap (based on 5000 iterations) confidence interval of (0.658–0.884). If the children with leukocyte count lower than 24,900 (10.12 in logarithmic scale) and greater than 10,200 (9.23 in logarithmic scale) were classified within the low MR group (optimal thresholds in the Youden index sense) the observed true-positive rate (truepositive rate) and the TNR were 0.750 and 0.636, respectively. Figure 4 depicts the kernel density estimation for the leukocyte count in logarithmic scale for both the high (black) and low (grey) mortality risk groups as well as the right-side, the left-side and the general ROC curves. A moderate increase in the number of leukocytes is the normal stress response of the body when it suffers an injury, this being considered one of the main elements in the systemic response syndrome. Table 1. Descriptive statistics for the leukocyte count in the low and high mortality risk groups.

Low MR High MR

N

Mean SD

P25

P50

P75

176 12

14290.7 6588 18325.8 20,343

9725 7250

13,400 11,400

17,875 22,850

Sample size (N), mean and standard deviation (mean SD), percentile 25 (P25 ), 50 (P50 ) and 75 (P75 ) are provided. MR: mortality risk.



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Statistical Methods in Medical Research 0(0) (a)

(b)

Figure 4. At left, kernel density estimation for the leukocyte count in logarithmic scale for both the high (black) b left-side and low (grey) mortality risk groups. Vertical lines stand for the optimal thresholds. At right, right-side (R), b tÞ) and general ROC curves. In grey, TPR and FNR achieved by the optimal threshold (in the Youden index (1 Rð1 sense).

This inflammatory response favours the recovery of the patient. However, when the response is very intense (large increase in leukocyte count) or very poor (decrease in leukocyte count) the recovery of the patient is compromised. Therefore, the mortality risk, especially in critically ill patients, increases.

4.2

Relationship between the response to the OnabotulinumtoxinA treatment for the chronic migraine headaches and the CGRP levels in women

The use of OnabotulinumtoxinA is the first and only FDA-approved (United State Food and Drug Administration) preventive treatment for chronic migraine in adults (see, for instance, Dodick et al.14). On the other hand, serum CGRP levels are high in chronic migraineurs which indicates a chronic activation of the trigeminovascular system; so CGRP has been proposed as the first biomarker for this entity.15 For this reason, we explored the relationship between basal levels of CGRP, determined as it was previously described, and the response to OnabotulinumtoxinA in 70 women (age range between 20 and 63 years, mean standard deviation: 46.4 10.3) meeting chronic migraine criteria. The study was approved by the Ethic review board of the HUCA and the patients gave written consent. Migraine patients are usually considered as responders when attack frequency decreases by 50%, so we adopted this criterion and we checked it by mean of using of monthly headache calendar in all patients. The observed response percentage was 78.6% (55/70). Table 2 shows the main descriptive statistics for the CGRP levels in the response and non-response groups. The values in the response group were, in general, higher than in the non-response group. However, the minimum value in the former group (11.44) was lower than the minimum value in the latter group (27.44).




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Table 2. Descriptive statistics for the CGRP levels in the response and non-response groups.

No response Response

N

Mean SD

P25

P50

P75

15 55

57.68 22.01 70.97 33.01

38.71 45.01

50.45 71.04

66.91 88.34

Sample size (N), mean and standard deviation (mean SD), percentile 25 (P25 ), 50 (P50 ) and 75 (P75 ) are provided.

(a)

(b)

Figure 5. On the left, kernel density estimation for the CGRP levels (logarithmic scale) for both the response (black) and non-response (grey) mortality risk groups. Vertical lines stand for the optimal thresholds. On the right, b the left-side (1 Rð1 b tÞ) and the general ROC curves. In grey, TPR and FNR achieved by the the right-side (R), Youden-threshold.

Figure 5 depicts in logarithmic scale (see left) the kernel density estimations for both the response b (black) and the non-response (grey) groups for the CGRP levels whereas the right-side (RðtÞ), b tÞ) and the general ROC curves (see right). The area under the right-side the left-side (1 Rð1 ROC curve was 0.619 (0.473–0.765) (between brackets a 95% (naive) bootstrap (based on 5000 bg was 0.731 (0.639–0.881). Optimal thresholds (maximizing iterations) confidence intervals) while A TPR-FPR) were (36.51, 66.97); i.e., women with CGRP levels below 36.51 (3.60 in logarithmic scale) and larger than 66.97 (4.20 in logarithmic scale) are the optimal group. Particularly, with these cutoff points, the FPR and the TPR were 0.267 and 0.673, respectively. The remarkable response in a subset of patients with high CGRP levels supports its value as a biomarker of chronic migraine and suggests that the inhibition of CGRP release could be a mechanism accounting for OnabotulinumtoxinA positive effect in chronic migraine. There are several potential explanations for the fact that a subset of patients with low CGRP levels seemed to respond to botulinum toxin A treatment. The first could simply be a placebo response, considering that about one-third of patients with frequent migraine respond to placebo in clinical trials and that this response increases with injections versus oral medications. Second, some patients with a chronic migraine phenotype and low CGRP levels could, in fact, suffer from a psychogenic headache instead of a true chronic migraine. Finally, it is also possible that for some patients, other



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pain-producing peptides, such as vasoactive intestinal peptide (VIP) or pituitary adenylate cyclaseactivating polypeptide, – not CGRP – are released by the activated trigeminovascular system.

5 Conclusions In this paper, we have explored a ROC curve generalization for the case where the relationship between the marker and the studied feature is not monotone, i.e. extreme values of the marker are associated with high probability of having the considered characteristic. In spite of the proposed ROC curve generalization does not have a direct probabilistic interpretation; for a considered (bio)marker, the Rg ðtÞ is – can be interpreted as – the greatest reached sensitivity when the specificity is, at the most, 1 – t. The explored approach has the advantage of solving the non-monotonicity problem without transforming the original data. Note that a suitable transformation does not always exist and, in addition, when it does it is usually made taking into account information extracted from the sample and therefore the same sample is used in order to calculate the ROC curve. In the cases where such transformation exists, both methods result in the same ROC curve. In the analysed datasets, working with the transformed markers jT 0 j, jT m0 j and jT p0 j where 0, m0 and p0 stand for the mean, the median, and the middle point (0:5 ½max min) in the control group, respectively, and T is the original marker (leukocyte count and CGRP level, in the considered examples), the obtained curves b g . The best AUCs were 0.69 and 0.67 improved the original right-side ones but both were worse than R for the leukocyte and the CGRP data, respectively. Additionally, its direct non-parametric empirical estimator is studied. It is the result of replacing the unknown CDFs by their usual non-parametric estimators, that is to say their respective ECDFs. From the properties of the traditional right-side ROC curve, both the uniform convergence and the asymptotic distribution are derived. Our simulations (not shown here) suggest that the quality of b g is similar to the usual R; b although, logically, the new estimator estimations provided by R overestimates the diagnostic capacity when that is poor. This problem is shared with the traditional ROC curve analysis if the desired direction (left or right) is not previously defined. Probably, the main handicap of the proposed curve is that its associated area under the curve loses the direct probability interpretation. The AUCg can still be used in order to summarize the diagnostic capacity of the studied biomarker. It still ranges between 0.5 and 1 and higher values are associated with better diagnostic capacity. In addition, it can be read as the average of the achieved sensitivity for all possible specificities. The asymptotic normality of the area under the proposed bg , has also been derived. However, the explicit expression for the general ROC curve estimator, A variance depends on the first derivative of the function t and it is not useful in practice. Unfortunately, the result obtained in Theorem 3 (see supplementary material) is not robust against the unrealistic assumption that t0 ¼ 0 8t 2 ð0, 1Þ; hence, in order to make inference, the use of some resampling method is advisable. Finally, the proposed methodology is used in two real datasets. In the first one, we explored the use of the leukocyte count in order to predict the mortality risk in critically ill children. In the second one, the relationship between the response to OnabotulinumtoxinA treatment for chronic migraine headaches and the CGRP levels in 70 Spanish women has been studied. It has been found out that both the higher and lower values of CGRP are associated with a higher probability of response. The proposed general ROC curve allows to detect a potential placebo effect in those women with low CGRP levels. The customer software used for developing the computes involved in this work (in particular, the R function called Rg) is included as supplementary material. The authors expect to develop additional functions which perform ROC curve comparison, ROC meta-analysis and other




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proposed solutions for different problems related to the ROC curve and to upload it all together in an R package. Funding This work was supported by the Grant MTM2011-23204 of the Spanish Ministry of Science and Innovation (FEDER support included) and PI11/00889 FISSS grant (Fondos Feder, ISCIII, Ministry of Economy, Spain) and the Allergan grant to JP.

Conflicts of interest None declared.

Acknowledgements The authors are grateful to Susana Dı´ az-Coto and Beatriz Giudici-Fernańdez for the manuscript revision. The authors wish to thank the PICUs of the Hospital Universitario Central de Asturias (HUCA) and of the Hospital Universitario Gregorio Maranõń for granting permission to use their respective PICU data.

References 1. Martı´ nez-Camblor P. Area under the ROC curve comparison in the presence of missing data. J Korean Stat Soc 2013; 42: 431–442. 2. Zhou XH, Obuchowski NA and McClish DK. Statistical methods in diagnostic medicine. New York: Wiley, 2002. 3. Hanley JA. The use of the binormal model for parametric ROC analysis of quantitative diagnostic tests. Stat Med 1996; 15: 1575–1585. 4. Hsieh F and Turnbull BW. Nonparametric and semiparametric estimation of the receiver operating characteristic curve. Ann Stat 1996; 24: 25–50. 5. Zou KH, Hall WJ and Shapiro DE. Smooth nonparametric receiver operating characteristic (ROC) curves for continuous diagnostic tests. Stat Med 1997; 16: 2143–2156. 6. Floege J, Kim J, Ireland E, et al. Serum iPTH, calcium and phosphate, and the risk of mortality in a European haemodialysis population. Nephrol Dial Transplant 2011; 26: 1948–1955. 7. Hilden J. The area under the ROC curve and its competitors. Med Decis Making 1991; 11: 95–101. 8. Fluss R, Faraggi D and Reiser B. Estimation of the Youden Index and it’s associated cutoff point. Biometrical J 2005; 47: 458–472. 9. Efron B and Tibshirani RJ. An introduction to the bootstrap. London, England: Chapman & Hall, 1993.

10. Rey C, Garcı´ a-Hernańdez I, Concha A, et al. Proadrenomedullin, pro-endothelin-1, procalcitonin, Creactive protein and mortality risk in critically ill children: a prospective study. Crit Care 2013; 17: R240. 11. Goldstein B, Giroir B and Randolph A. International paediatric sepsis consensus conference: definitions for sepsis and organ dysfunction in paediatrics. Pediatr Crit Care Med 2005; 6: 2–8. 12. Pe´rez DV, Jordan I, Esteban E, et al. Prognostic factors in paediatric sepsis study, from the Spanish society of paediatric intensive care. Pediatr Infect Dis J 2014; 33: 152–157. 13. Khanafer N, Sicot N, Vanhems P, et al. Severe leukopenia in Staphylococcus aureus-necrotizing, communityacquired pneumonia: risk factors and impact on survival. BMC Infect Dis 2013; 13: 359. 14. Dodick DW, Turkel CC, DeGryse RE, et al. OnabotulinumtoxinA for treatment of chronic migraine: pooled results from the double-blind, randomized, placebo-controlled phases of the PREEMPT clinical program. Headache: J Head Face Pain 2010; 50: 921–936. 15. Cernuda-Morolloń E, Larrosa D, Ramoń C, et al. Interictal increase of CGRP levels in peripheral blood as a biomarker for chronic migraine. Neurology 2013; 81: 1191–1196.


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CT.

Receiver operating characteristic curves: a basic understanding.

The average receiver operating characteristic curve in multireader multicase imaging studies.

RE: "RECEIVER OPERATING CHARACTERISTIC CURVE INFERENCE FROM A SAMPLE WITH A LIMIT OF DETECTION".

Evaluation of the predictive performance of nutritional indicators by receiver-operating characteristic curve analysis.

Jackknife variance of the partial area under the empirical receiver operating characteristic curve.

Weighted Area Under the Receiver Operating Characteristic Curve and Its Application to Gene Selection.

Effects of 'matched filter' smoothing as measured by receiver operating characteristic curve.

Comparison of semiparametric receiver operating characteristic models on observer data.

Understanding receiver-operating-characteristic curves: a graphic approach.

Study on diagnosis criteria of fire-heat syndrome based on receiver operating characteristic curve and principal component analysis.

Oxidative status in different settings and with different methodological approaches compared by Receiver Operating Characteristic curve analysis.

Receiver Operating Characteristic curve analysis determines association of individual potato foliage volatiles with onion thrips preference, cultivar and plant age.

Evidence Based Emergency Medicine; Part 5 Receiver Operating Curve and Area under the Curve.

Anthropometric measurements as a screening test for carpal tunnel syndrome: receiver operating characteristic curves and accuracy.

Evaluation of a digital workstation for interpreting neonatal examinations. A receiver operating characteristic study.