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Bayesian hierarchical model for the estimation of proper receiver operating characteristic curves using stochastic ordering
Communications for Statistical Applications and Methods 2019;26:205-216
Published online March 31, 2019
© 2019 Korean Statistical Society.

Eun Jin Janga, Dal Ho Kim1,b

aDepartment of Information Statistics, Andong National University, Korea;
bDepartment of Statistics, Kyungpook National University, Korea
Correspondence to: 1Department of Statistics, Kyungpook National University, 80 Daehakro, Bukgu, Daegu 41566, Korea. E-mail: dalkim@knu.ac.kr
Received November 23, 2018; Revised January 5, 2019; Accepted February 7, 2019.
 Abstract

Diagnostic tests in medical fields detect or diagnose a disease with results measured by continuous or discrete ordinal data. The performance of a diagnostic test is summarized using the receiver operating characteristic (ROC) curve and the area under the curve (AUC). The diagnostic test is considered clinically useful if the outcomes in actually-positive cases are higher than actually-negative cases and the ROC curve is concave. In this study, we apply the stochastic ordering method in a Bayesian hierarchical model to estimate the proper ROC curve and AUC when the diagnostic test results are measured in discrete ordinal data. We compare the conventional binormal model and binormal model under stochastic ordering. The simulation results and real data analysis for breast cancer indicate that the binormal model under stochastic ordering can be used to estimate the proper ROC curve with a small bias even though the sample sizes were small or the sample size of actually-negative cases varied from actually-positive cases. Therefore, it is appropriate to consider the binormal model under stochastic ordering in the presence of large differences for a sample size between actually-negative and actually-positive groups.

Keywords : area under the curve, Bayesian hierarchical model, discrete ordinal data, receiver operating characteristic curve, stochastic ordering
1. Introduction

Diagnostic tests in medical fields detect or diagnose a disease with results measured by continuous or discrete ordinal data. The receiver operating characteristic (ROC) analysis is widely used to assess the diagnostic test performance and often presented as the ROC curve. It graphically represents the relationship between false positive and true positive rates. The false positive rate represents “1 – specificity” which indicates the probability that a truly non-diseased individual displays a positive test result, and the true positive rate represents “sensitivity” indicating the probability that a diseased individual will show a positive test result. The area under the curve (AUC) is often used to measure the accuracy of a ROC curve.

Numerous ROC curve estimation methods for continuous or discrete ordinal data have been proposed using a parametric, semiparametric, and nonparametric approach based on the frequentist or Bayesian method (Gonçalves et al., 2014). The most common model is a conventional binormal model which the distributions of the decision-variables of actually-negative and actually-positive populations are assumed to be the normal distribution. The conventional binormal model for discrete ordinal data can be estimated using a maximum likelihood method based on frequentist methods (Dorfman and Alf, 1969; Swets, 1986; Hanley, 1988; Metz, 1989; Metz et al., 1998) or Bayesian methods (Peng and Hall, 1996; Ishwaran and Gatsonis, 2000; Johnson and Johnson, 2006; Wang et al., 2007).

If the diagnostic test is effective, the ROC curve should be concave (Dorfman et al., 1996). The concave ROC curve is located above the main diagonal, which implies that ROC(u) > u for 0 ≤ u ≤ 1. It is referred to as proper if these conditions are satisfied and the non-concave curve with a hook is termed improper (Bandos et al., 2017). Based on their experiences, Pesce et al. (2010) outlined the increased probability of a hook when the total number of sample cases is small, the ratio of actually-negative to actually-positive cases is far from 1, the population ROC curve is strongly skewed, or when the distribution of operating point along the population ROC curve is very uneven. Bandos et al. (2017) investigated the effect of severe improperness of the fitted binormal ROC curves on the AUC estimates. They generated simulated data from an actually proper ROC curve and fitted the binormal model using a maximum likelihood approach (Dorfman and Alf, 1969). However, they only considered cases where actually-negative and actually-positive groups were of the same sample size.

To estimate the proper ROC curve for discrete ordinal data, Metz and Pan (1999) proposed a proper binormal model and a new algorithm using the monotonic transformation of the likelihood ratio. In the frequentist method, the bi-gamma (Dorfman et al., 1996; Hughes and Bhattacharya, 2013) and the bi-beta model (Mossman and Peng, 2016) were proposed. Recently, Nandram and Peiris (2018) developed a robust Bayesian model using stochastic ordering to obtain proper ROC curves. Additionally, Hwang and Chen (2015) applied nonparametric Bayesian methods to estimate ROC curve/AUC under stochastic ordering (Gelfand and Kottas, 2001).

Pisano et al. (2005) analyzed the diagnostic performance of breast cancer screening mammography using the ordinal discrete scale including the 7-point malignancy score as well as breast imaging reporting and data system (BIRADS) score. The study detected 355 cases of breast cancer from 42,745 subjects through a film mammography screening test. The estimated ROC curves of film mammography using a conventional binormal model were improper and had a hook. In the case of low-prevalence disease, the number of people who actually have the disease, based on screening diagnostic tests is generally lower compared to those without a disease. Therefore, it is important to assess the performance of the conventional binormal model and compare with the model to estimate the proper ROC curve in cases of large differences in sample size.

In this study, we apply the stochastic ordering method in a Bayesian hierarchical model to estimate of the proper ROC curve and AUC when diagnostic test results are measured with discrete ordinal data. We describe the conventional binormal model and binormal model under stochastic ordering in Section 2, and compare these two models for various sample sizes using simulations in Section 3. In Section 4, we apply two models on breast cancer data. Finally, the conclusions are discussed in Section 5.

2. Bayesian hierarchical model for ROC curve

We consider K discrete ordinal categories in actually-negative and actually-positive population. Let n1j and n2j denote the observed frequency in the jth category, j = 1, …, K, in actually-negative cases and actually-positive cases, respectively. Then, n1 = (n11, …, n1K)T indicates the vector of the observed frequencies in actually-negative cases and n1=j=1Kn1j is the total number of cases in actually-negative cases. Similarly, n2 = (n21, …, n2K)T and n2=j=1Kn2j are defined in the actually-positive cases.

Suppose that the probability of a response in category j in the actually-negative population is p1j and the probability of a response in category j in the actually-positive population is p2j. Then the categorical data in the actually-negative population follow multinomial distributions with probabilities p1 = (p11, …, p1K)T, and the categorical data in the actually-positive population follow multinomial distribution with probabilities p2 = (p21, …, p2K)T (Bhattacharya and Nandram, 1996). Therefore, the joint likelihood function is expressed as:

p(n1,n2p1,p2)=n1!n2!j=1Kn1j!n2j!j=1Kp1jn1jp2jn2j.

Suppose that the latent decision-variable axis is partitioned into K categories by K − 1 boundaries c1, c2, …, cK−1 in the ROC model with K ordinal categories and the vector of boundaries is c = (c1, …, cK−1). The beginning and ending of the decision-variable axis are assumed as c0 ≡ −∞ and cK ≡ ∞, respectively (Metz et al., 1998; Metz and Pan, 1999).

Let F1 be the cumulative distribution function (CDF) in the actually-negative population and F2 be the CDF in the actually-positive population. Therefore, the probabilities of a response in category j in the actually-negative population and the actually-positive population are


p1j=F1(cj)-F1(cj-1), 듼 듼 듼p2j=F2(cj)-F2(cj-1).

Assume the decision-variables are distributed N(0, 1) in the actually-negative population and N(μ,σ2) in the actually-positive population (Nandram and Peiris, 2018). The hyperprior for μ is assumed to be Cauchy prior (Gelman et al., 2008) and the hyperprior for σ2 is assumed as shrinkage prior to avoid difficulties associated with improper priors of the form π(σ2) ∝ 1/σ2 (Gelman, 2006; Nandram et al., 2013) as:

π(μ)=1π(1+μ2), 듼 듼 듼π(σ2)=1(1+σ2)2.

Finally, we assume the standard logistic distribution with location 0 and scale 1 as a prior for the boundaries c (Nandram and Peiris, 2018),

c1,c2,,cK-1iidlogistic(0,1), 듼 듼 듼-<c1<c2<<cK-1<,

and the joint prior distribution for c is represented as

π(c)=(K-1)!j=1K-1rcj(1+ecj)2, 듼 듼 듼-<c1<c2<<cK-1<.

2.1. Conventional binormal model

In the conventional binormal model without considering the stochastic ordering, the two CDFs are

F1(cj)=Φ(cj), 듼 듼 듼F2(cj)=Φ(cj-μσ),

and the probabilities of a response in category j in actually-negative and actually-positive population in (2.1) are

p1j=Φ(cj)-Φ(cj-1), 듼 듼 듼p2j=Φ(cj-μσ)-Φ(cj-1-μσ).

Thus, the joint posterior density of ν, σ2, c using Bayes’ theorem is given by

π(μ,σ2,cn1,n2)j=1K[Φ(cj)-Φ(cj-1)]n1j[Φ(cj-μσ)-Φ(cj-1-μσ)]n2j×1π(1+μ2)1(1+σ2)2j=1K-1ecj(1+ecj)2.

The conditional posterior distributions in the conventional binormal model are calculated similar to the binormal model under stochastic ordering and we use the grid method for the Markov chain Monte Carlo (MCMC) computations.

2.2. Binormal model under stochastic ordering

The stochastic order of two populations is defined as: F2 is stochastically larger than F1 if F1(u) ≥ F2(u) for all u (Gelfand and Kottas, 2001; Hwang and Chen, 2015).

The ROC curve is proper when ROC(u) > u for 0 ≤ u ≤ 1 if and only if F1(u) ≥ F2(u) (Hanson et al., 2008). Therefore, two CDFs can be defined to estimate the proper ROC curve using the stochastic ordering method (Nandram and Peiris, 2018),

F1(cj)=Φ(cj), 듼 듼 듼F2(cj)=Φ(cj)Φ(cj-μσ),

where Φ(z) is the standard normal CDF and the probabilities of a response in category j in actually-negative and actually-positive populations in (2.1) and can be rewritten as:

p1j=Φ(cj)-Φ(cj-1), 듼 듼 듼p2j=Φ(cj)Φ(cj-μσ)-Φ(cj-1)Φ(cj-1-μσ).

Thus, the joint posterior density of μ,σ2, c using Bayes’ theorem is given by

π(μ,σ2,cn1,n2)j=1K[Φ(cj)-Φ(cj-1)]n1j[Φ(cj)Φ(cj-μσ)-Φ(cj-1)Φ(cj-1-μσ)]n2j×1π(1+μ2)1(1+σ2)2j=1K-1ecj(1+ecj)2.

We use the Griddy Gibbs sampler to generate samples from the posterior density because the conditional posterior distributions are not standard form (Lee et al., 2017; Molina et al., 2014; Nandram et al., 2011; Nandram and Yin, 2016). For the grid method, we divide 100 intervals between 0 and 1, and calculate 100 mid-points for each sub-interval. We then calculate the values according to the mid-points in the conditional posterior density, and divide by the sum of the calculated values to convert it to an approximate probability mass function. To draw a random sample, we choose one of the sub-intervals and draw a uniform random variable within its sub-interval. If the mode of the density lies in the interior of (0, 1), then the grid method procedure works well.

The conditional posterior distribution of μ is

π(μσ2,c,n1,n2)j=1K[Φ(cj)Φ(cj-μσ)-Φ(cj-1)Φ(cj-1-μσ)]n2j11+μ2,

and the parameter μ in (−∞, ∞) is transformed to the parameter ν in (0, 1) using ν = eμ/(1 + eμ). The transformed conditional posterior distribution for ν is

π(νσ2,c,n1,n2)j=1K[Φ(cj)Φ(cj-logit(ν)σ)-Φ(cj-1)Φ(cj-1-logit(ν)σ)]n2j11+{logit(ν)}21ν(1-ν),

where logit(ν) = log(ν/(1 − ν)) and we generate samples using the grid method.

The conditional posterior distribution of σ2 is

π(σ2μ,c,n1,n2)j=1K[Φ(cj)Φ(cj-μσ)-Φ(cj-1)Φ(cj-1-μσ)]n2j1(1+σ2)2,

and the parameter σ2 is generated using the grid method on (0, 1) after transformation to τ = σ2/(1 + σ2), 0 < τ < 1. The transformed conditional posterior distribution for τ is

π(τμ,c,n1,n2)j=1K[Φ(cj)Φ(cj-μτ/(1-τ))-Φ(cj-1)Φ(cj-1-μτ/(1-τ))]n2j.

Finally, the conditional posterior distributions of the boundaries cj are given by

π(cjμ,σ2,n1,n2)ecj(1+ecj)2[Φ(cj)-Φ(cj-1)]n1j[Φ(cj+1)-Φ(cj)]n1,j+1×[Φ(cj)Φ(cj-μσ)-Φ(cj-1)Φ(cj-1-μσ)]n2j[Φ(cj+1)Φ(cj+1-μσ)-Φ(cj)Φ(cj-μσ)]n2,j+1,

and we also transform the parameter cj to parameter t j in (0, 1) using t j = ecj/(1+ecj), j = 1, …, K−1. We have t1, t2, …, tc−1 ~ U(0, 1) such that 0 < t1 < t2 < … < tc−1 < 1. Therefore, the transformed conditional posterior distribution for t j is

π(tjμ,σ2,n1,n2)[Φ(logit(tj))-Φ(logit(tj-1))]n1j[Φ(logit(tj+1))-Φ(logit(tj))]n1,j+1×[Φ(logit(tj))Φ(logit(tj)-μσ)-Φ(logit(tj-1))Φ(logit(tj-1)-μσ)]n2j×[Φ(logit(tj+1))Φ(logit(tj+1)-μσ)-Φ(logit(tj))Φ(logit(tj)-μσ)]n2,j+1,

and the samples of t j, j = 1, …, K − 1 are drawn using the grid method on the conditions that t j−1 < t j < t j+1.

The convergence of the MCMC algorithm are assessed using the trace plots and autocorrelation plots. We use a single chain; therefore, convergence is checked using the Geweke test, which compares the mean of the initial 10% and the last 50% samples of the total iteration (Geweke, 1992).

The ROC curves are estimated by ROC(u)=1-F2(F1-1(1-u)), for 0 ≤ u ≤ 1, and the AUC is obtained by AUC=01ROC(u)du after estimating the parameters.

3. Simulation study

We conduct a simulation to compare the performance of the binormal model under stochastic ordering and the conventional binormal model. Metz and Pan (1999) discussed that the true population ROC curve does not have a hook, but the empirical ROC curve may have a hook in small sample size. The true ROC curve is a convex curve, but the estimated ROC curve from the simulated data with 5 categories (30 actually-negative and 40 actually-positive cases) has a hook in Figure 9 in Metz and Pan (1999). This population ROC curve has operating points (FPR, TPR) = (0.00025, 0.10), (0.000328, 0.25), (0.03040, 0.50), (0.28114, 0.85) (Dorfman and Berbaum, 1995) and the true AUC is 0.879 (Metz and Pan, 1999). The probabilities of 5 categories using the operating points are calculated as: p1 = (0.71886, 0.25074, 0.030072, 0.000078, 0.00025)’ for actually-negative cases and p2 = (0.15, 0.35, 0.25, 0.15, 0.1)’ for actually-positive cases.

We generate 100 simulated data from the multinomial distribution with probabilities p1 and p2 and sample sizes of (60, 30), (45, 45), (30, 60), (600, 300), (450, 450), (1000, 100), (10000, 100), (15000, 100). We then apply the binormal model under stochastic ordering and the conventional binormal model and estimate AUCs. The performance of the model are compared using the absolute bias (AB), relative absolute bias (RAB) and root posterior mean squared error (RPMSE) for AUCs (Lee et al., 2017). The RPMSE in rth iteration is defined by RPMSEr=PSDr2+ABr2, where PSDr is the posterior standard deviation (PSD) in rth simulated data. We also calculate the 95% credible interval (CI) and highest posterior density (HPD) credible interval for AUCs of each simulated data, and calculate the width of the credible interval and the coverage probability. Finally, we calculate the average of AB, RAB, and RPMSE, the width of credible interval and coverage probability to summarize the values for each simulated data (Hidiroglou and You, 2016).

Simulation results of AUCs using 100 simulated data with true AUC = 0.879 are summarized in Table 1. The means of AUC in the binormal model under stochastic ordering are closer to 0.879 than those in the conventional binormal model. All measures for AB, RAB, RPMSE, the widths of the credible interval, and the widths of the HPD credible interval in each sample size are smaller in the binormal model under stochastic ordering than in the conventional binormal model. The coverage probabilities in the binormal model under stochastic ordering are closer to 0.95 than those in the conventional binormal model. Figure 1 and Figure 2 represent the estimated ROC curves derived from the binormal model under stochastic ordering and the conventional binormal model for 100 simulated data with the true ROC curve. The estimated ROC curves gets closer to the true ROC curve and the variation decreases as the sample size in both models increases. Many estimated ROC curves are improper ROC curves with a hook in the case of smaller number of samples in addition, a few estimated ROC curves have small degree of hook when the ratio of actually-negative to actually-positive cases is larger. Therefore, this simulation study indicates that the binormal model under stochastic ordering could estimate the proper ROC curve with a small bias despite sample sizes being small or significant variations in the sample size for actually-negative cases from actually-positive cases.

4. Real data application

Pisano et al. (2005) analyzed the diagnostic performance of film mammography for breast-cancer screening using the seven-point malignancy scale: 1 (definitely not malignant), 2 (almost definitely not malignant), 3 (probably not malignant), 4 (may be malignant), 5 (probably malignant), 6 (almost definitely malignant), 7 (definitely malignant). The estimated ROC curve using the conventional binormal model presented in Figure 1 in Pisano et al. (2005) and all ROC curves of film mammography represent improper ROC curves with a hook. Pesce et al. (2010) measured the FPF and TPF values from the ROC curve in panel D of Figure 1 which displays the obvious hook and simplified them to 5-category data. The probability of 5 categories were: p1 = (0.74, 0.18, 0.06, 0.02, 0.00)’ for actually-negative cases and p2 = (0.43, 0.11, 0.09, 0.24, 0.13)’ for actually-positive cases. Pesce et al. (2010) considered 100 actually-positive and 16,000 actually-negative cases. This is highly unbalanced and often occurs in the diagnostic test for a disease with a low prevalence. We create artificial data according to Pesce et al. (2010) (Table 2).

We apply the binormal model under stochastic ordering and the conventional binormal model. We draw 60,000 samples from the conditional posterior distribution for each parameter and select every 5th iterate after discarding 10,000 samples to obtain 10,000 samples for inference. All p-values of the Geweke test are greater than 0.1 and the effective sizes are higher than 9,000 in both models.

Table 3 summarizes the posterior mean (PM), PSD, 95% CI and HPD CI of the AUC for the breast cancer data. The PM is 0.764, PSD is 0.026, 95% CI is (0.714, 0.815), and 95% HPD CI is (0.714, 0.814) in the binormal model under stochastic ordering. The PM is 0.695, PSD is 0.044, 95% CI is (0.607, 0.777), and 95% HPD CI is (0.612, 0.780) in the conventional binormal model. The PM from the binormal model under stochastic ordering is larger than the conventional binormal model because the fitted ROC curve obtained from the conventional binormal model has a hook. The PSD and the width of CI and HPD CI of the binormal model under stochastic ordering are smaller as compared to the conventional binormal model.

In Figure 3, we display the ROC curves and the posterior densities of AUCs of the binormal model under stochastic ordering and the conventional binormal model for breast cancer data. The circles represent empirical operating points. The fitted ROC curve obtained from the conventional binormal model are an improper ROC curve with a hook, but proper in the binormal model under stochastic ordering. The AUC distribution in the conventional binormal model highly vary and shift to the left of those in the binormal model under stochastic ordering.

5. Concluding remarks

In the case of low-prevalence diseases, the number of people who actually have a disease in screening diagnostic tests is generally lower compared to those without a disease. Pesce et al. (2010) mentioned that the probability of having a hook often increased with a small sample size, and the ratio of actually-negative to actually-positive cases is far from 1, based on experience. Therefore, it is important to assess the performance of the conventional binormal model and compare it with the model for estimating the proper ROC curve in sample size with large variations.

In this study, we apply the stochastic ordering method in a Bayesian hierarchical model to estimate the proper ROC curve and AUC when diagnostic test results are measured with discrete ordinal data and compare with the conventional binormal model and binormal model under stochastic ordering. The simulation study indicates that the binormal model under stochastic ordering can be used to estimate the proper ROC curve with a small bias despite the sample sizes being small or the large variation between actually-negative and actually-positive cases. In breast cancer data, the fitted ROC curve derived from the conventional binormal model represents an improper ROC curve with a hook, but proper in the binormal model under stochastic ordering. The distribution of AUC derived from the conventional binormal model varied greatly and shifted to the left in the binormal model under stochastic ordering. Therefore, it is appropriate to consider the binormal model under stochastic ordering in the case of a sample size large variation between actually-negative and actually-positive groups.

Acknowledgements

This work was supported by a 2016 Research Funds of Andong National University grant.

Figures
Fig. 1. The estimated ROC curves (dash line) and 95% pointwise credible intervals (dotted line) derived from the conventional binormal model and the true ROC curve with AUC = 0.879 (solid line) from simulated data. ROC = receiver operating characteristic; AUC = curve and the area under the curve.
Fig. 2. The estimated ROC curves (dash line) and 95% pointwise credible intervals (dotted line) derived from the binormal model under stochastic ordering and the true ROC curve with AUC = 0.879 (solid line) from simulated data. ROC = receiver operating characteristic; AUC = curve and the area under the curve.
Fig. 3. The ROC curves and the posterior densities of AUCs of the binormal model under stochastic ordering (dotted line) and the conventional binormal model (dash line) for the breast cancer data. ROC = receiver operating characteristic; AUC = curve and the area under the curve.
TABLES

Table 1

Simulation results for AUCs using 100 simulated data with true AUC = 0.879

(1) Conventional binormal model

Sample size AUC AB RAB RPMSE C-CI W-CI C-HPD W-HPD
(60, 30) 0.818 0.063 0.072 0.091 0.82 0.230 0.88 0.225
(45, 45) 0.820 0.060 0.068 0.082 0.87 0.200 0.90 0.196
(30, 60) 0.770 0.109 0.124 0.131 0.69 0.277 0.75 0.268
(600, 300) 0.863 0.017 0.019 0.024 0.84 0.061 0.86 0.060
(450, 450) 0.850 0.029 0.033 0.033 0.51 0.060 0.53 0.059
(1000, 100) 0.867 0.017 0.020 0.033 0.97 0.100 0.99 0.098
(10000, 100) 0.861 0.023 0.026 0.036 0.89 0.098 0.90 0.097
(15000, 100) 0.861 0.024 0.028 0.037 0.90 0.097 0.92 0.096

Average 0.839 0.043 0.049 0.058 0.811 0.140 0.841 0.137
(2) Binormal model under stochastic ordering

Sample size AUC AB RAB RPMSE C-CI W-CI C-HPD W-HPD
(60,30) 0.830 0.052 0.059 0.074 0.82 0.186 0.830 0.183
(45,45) 0.827 0.053 0.060 0.073 0.88 0.180 0.890 0.177
(30,60) 0.800 0.079 0.090 0.097 0.71 0.209 0.780 0.206
(600,300) 0.871 0.011 0.013 0.019 0.96 0.054 0.980 0.054
(450,450) 0.862 0.017 0.019 0.022 0.85 0.053 0.870 0.052
(1000,100) 0.875 0.014 0.016 0.027 1.00 0.084 1.000 0.083
(10000,100) 0.873 0.017 0.019 0.028 0.94 0.079 0.940 0.079
(15000,100) 0.873 0.018 0.020 0.028 0.95 0.078 0.950 0.077

Average 0.851 0.033 0.037 0.046 0.889 0.115 0.905 0.114

AUC = area under the curve; AB = absolute bias; RAB = relative absolute bias; RPMSE = root posterior mean squared error; C-CI = coverage probability for 95% credible interval; W-CI = width of a credible interval; C-HPD = coverage probability HPD credible interval; W-HPD = width of the HPD credible interval; HPD = highest posterior density.


Table 2

Artificial breast cancer data of film mammography screening

1 2 3 4 5
Patients without breast cancer 11837 2839 1002 322 0
Patients with breast cancer 37 12 13 24 14

Table 3

Posterior mean, standard deviation, 95% credible intervals, and HPD CI of the area under the curve for the breast cancer data

PM PSD 95% CI 95% HPD CI
Binormal model under stochastic ordering 0.764 0.026 (0.714, 0.815) (0.714, 0.814)
Conventional binormal model 0.695 0.044 (0.607, 0.777) (0.612, 0.780)

PM = posterior mean; PSD = standard deviation; CI = 95% credible intervals; HPD = highest posterior density.


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