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ISSN: 2155-6180
Journal of Biometrics & Biostatistics
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A Linear Regression Framework for the Receiver Operating Characteristic(ROC) Curve Analysis

Zheng Zhang1* and Ying Huang2

1Department of Biostatistics and Center for Statistical Sciences, Brown University, Providence, RI 02912, USA

2Fred Hutchinson Cancer Research Center, Seattle, WA 98109, USA

*Corresponding Author:
Zheng Zhang
Department of Biostatistics and Center for Statistical Sciences
Brown University, 121 South Main Street
Providence, RI 02912, USA
Tel: 401-863-2578
E-mail: [email protected]

Received Date: December 06, 2011; Accepted Date: March 22, 2012; Published Date: March 23, 2012

Citation: Zhang Z, Huang Y (2012) A Linear Regression Framework for the Receiver Operating Characteristic (ROC) Curve Analysis. J Biomet Biostat 3:137. doi:10.4172/2155-6180.1000137

Copyright: © 2012 Zhang Z, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Keywords

AUC; Covariate effect; Linear regression; ROC curve; Sensitivity

Introduction

New advances in medical technology have produced an array of potentially powerful tools to screen for and diagnose various medical conditions. Effective screening and accurate diagnosis can ensure optimal treatment and improve prognosis. Before a screening or diagnostic test can be applied in a clinical setting, rigorous statistical assessment of its performance in discriminating the diseased state from the non-diseased state is required. For tests measured on the continuous scale, the receiver operating characteristic (ROC) curve is a common statistical tool for describing the performance of such tests [1]. Let D be disease status (1 for disease and 0 otherwise) and Y the test result with positivity defined whenever Y ≥ c. Define the true positive fraction (TPF(c)) and the false positive fraction (FPF(c)) as P[Y ≥ c | D = 1] and P[Y ≥ c | D = 0], where TPF and 1− FPF are also called test’s sensitivity and specificity. The ROC curve is a plot of TPF(c) versus FPF(c) when the threshold c ranges from - ∞ to ∞. Alternatively, the ROC curve can be written as a function of t, by defining SD(c) = P[Y ≥ c | D = 1] = TPF(c), Equation, where D and Equation indicate the diseased and non-diseased population, then

Equation                 (1)

Different approaches have been proposed to estimate an ROC curve. Nonparametrically, we can obtain the empirical ROC curve based on Equation and Equation ,where nD and Equationare number of diseased and non-diseased observations respectively.

Alternatively, we can estimate the ROC curve parametrically by assuming a distributional form for SD and Equation and then calculating the induced ROC curve with equation (1). The derived ROC curve, however, is not invariant to transformation of the test results. Pepe [2] notes that ROC curve describes only the relationship between the distributions of SD and Equation not the distributions themselves. Semiparametric estimators that directly model the ROC curve as a parametric function without specifying the underlying distributions of SD and Equation provides a desirable alternative.

Semiparametric estimations of the ROC curve starts by first specifying a parametric model of the ROC curve, where the binormal model is most popular [3], and is defined as the following:

ROC(t) = Φ(α0 + α1Φ−1(t)),                           (2)

with Φ being the cdf of the standard normal distribution. We call α0 the intercept and α1 the slope of the binormal ROC curve. The binormal ROC curve was originally derived from normally distributed test results, where Equation and Equation, the resulting ROC curve has Equationand Equation. However, since the ROC curve is invariant to strictly increasing transformations of Y, to say that the ROC curve is binormal simply means that there exists some strictly increasing transformation, which would simultaneously transform the raw data, Equationand YD, into normally distributed random variables. In addition, under the binormal model, the area under the ROC curve (AUC) can be written as Equation.Various semiparametric methods have been proposed to estimate the ROC curve under the binormal assumption. LABROC [4] is a maximum likelihood-based procedure for ordinal test results. Specifically, it categorizes continuous data and then applies the Dorfman and Alf algorithm [5] to the categorized data. ROC-GLM [6-8] is a binary regression based method. Pepe and Cai [9] and Cai [10] estimated the ROC curve using the concept of placement values (PV).

The development of our method is motivated by noting that the binormal model in equation (2) essentially states that Φ−1(TPF) and Φ−1 (FPF) have a linear relationship. Hence we propose to fit a linear line through the points of Equation,Equation and obtain estimates of α0 and α1 by the least squares technique, and consequently, estimate the ROC curve itself. This method is intuitive, conceptually easy to understand, and very easy to implement. More importantly, it can be readily extended to allow for covariate (Z) effects on the ROC curve by writing

Φ−1(ROCZ (t)) = α0 + α1 Φ−1(t) + θ Z.                                         (3)

Our linear regression approach provides a simple and widely accessible algorithm for fitting such models.

Hsieh and Turnbull [11] described a weighted least squares approach to estimate the binormal ROC curve. For continuous data, their approach groups the data into a pre-determined number (independent of the number of observations) of categories, and the largest they chose is 12. Our framework is similar in spirits but allows significant improvement in efficiency by eliminating the need of grouping, as will be shown later in the next section. Moreover, our framework offers the flexibility to model additional covariate effects and/or to model a segment of the ROC curve. In many applications only a part of the ROC curve is of interest [12]. Restricting fitting of model (3) to a subrange [a,b] within (0,1) is likely to confer robustness over the region of interest compared with fitting over the entire (0,1) range.

Linear Regression Framework and Estimation

Estimation

Write the empirical ROC curve as Equation, where Equation As shown in Appendix I,

Equation             (4)

meaning that the process converges to a mean zero Gaussian process with variance-covariance function

Σ(s, t) = 1 /{f (α0+α1Φ-1(s))Φ(α0+ α1Φ-1(t))} [λlΣ1(s,t)+Σ2(s,t)],

where Σ1(s,t) = ROC(t)∧ROC(s)−ROC(s)ROC(t), Equation 0 < s < 1,0 < t < 1, and λ is the limit of the ratio Equationas Equationapproaches infinity.

We propose the following estimating procedure:

1. For a fixed boundary point (a,b), choose the set T = {tp} such that 0 < a < t1 <…< tp <…< b < 1. For each tp, find the smallest threshold value cp, such that Equation

2. Calculate Equation, exclude Equation if Equationis either 0 or 1;

3. Set up the linear regression model as:

Equationwhere the normalized error vector, Equation is distributed as multivariate normal with mean 0 and asymptotic covariance matrix Σ ;

4. Define the design matrix M asEquation and the vectorEquation

5. Calculate the ordinary least squares(OLS) estimator of α asEquation

6. Then the estimated ROC curve and its associated AUC are

Equation and Equation

Asymptotic theory

We develop the asymptotic distribution of the OLS estimator Equationunder the following assumptions: Equationand Equationare i.i.d. random variables with a survival functions SD and Equation and density functions fD and Equationrespectively; Equation, 0 < λ < ∞, as Equation→ ∞ ; the slope of the ROC curve, Equation, is bounded on the subinterval [a,b] of (0,1),0 < a < b < 1.

Theorem 1

Equation

where

Equation and

Equation

where EquationEquation

Equation Equation

Equation Equation

with J1(s) = (φ(Φ−1(ROC(s))))−1, J2(s) = (φ(Φ−1(s)))−1, K1(s) = Φ−1(s) J1 (s) and K2(s) = Φ−1(s) J2 (s).

Proof: The proof for Theorem 1 can be found in the Appendix II.

Asymptotic efficiency relative to HT method

Here we compare the asymptotic efficiency between the OLS estimator and the estimator derived using Hsieh & Turnbull’s method (HT) for estimating α0 , α1, and ROC(t) at t = 0.2, 0.4, 0.7, when λ (the limit of the ratio of D n and nD) varies from 0.5 to 2. We choose (α0 , α1) = (1.2, 0.45) so that the area under the ROC curve (AUC) is 0.863 and the ROC curve takes values of 0.794, 0.861 and 0.924 at t = 0.2, 0.4, 0.7, respectively. Note that forEquation let Σ denote the asymptotic variance of Equationfrom an ROC modeling method (OLS or HT), the asymptotic variance expression for the corresponding Equationare:

Equation

Table 1 shows the asymptotic efficiency of HT relative to OLS. For the OLS estimator, the boundary points (a, b) chosen are (0.0001, 0.9999). For the HT estimator, the number of categories chosen is eight. Note that asymptotically, the OLS estimator can lead to substantial efficiency gain compared to the HT estimator. The biggest improvement is seen when estimating α1 with efficiency gain above 50%. The efficiency gain in estimating points on the ROC curve varies from 3% to 20% depending on the point of interest (Table 1).

l α0 α1 R(0.2) R(0.4) R(0.7)
0.5 1.10 1.72 1.09 1.06 1.20
1 1.10 1.64 1.05 1.06 1.20
2 1.10 1.59 1.03 1.06 1.20

Table 1: Asymptotic efficiency of OLS relative to HT (with 8 categories).

Regression Model with Discrete Covariates

Suppose there are K categories that potentially could overlap with each other. For k = 1,…,K, let Z(k) be a vector of length K− 1, with value 1 for the k−1th element and zero else where (so Z(1) is a vector of zeros), and let Equation be the number of non-diseased observations in category k. Suppose the ROC curve within category k is characterized by

Φ−1(ROCk(t)) = β1 + γΦ−1(t) + θTZ(k)

where θ = (β2,…,βk)T.

That is, for subset 1(the reference subset)

Φ−1(ROC1(t)) = β1 + γΦ−1(t)                                                               (5)

and for subset k,k = 2,…,K,

Φ−1(ROCk(t)) = β1 + γΦ−1(t) + βk.                                                      (6)

Hence βk is the difference in the intercept parameter of the ROC curves between subset k and the reference subset. The parameter of interest here is θ * = (β1,…,βk, γ)T. The underlying assumptions for equation (5) is that there exists an unknown monotone increasing function h1 , such that Equationand h1(YD,1) : N(β1 / γ,1 /γ2). Similarly, for subsequent subset k, k = 2, 3,…,K, (6) implies there exists an unknown monotone increasing function hk, such that Equationand hk(YD,k) : N((β1 + βk / γ,1 / γ)2. Notice that hk s are not required to be the same for different k.

Let Equationbe the empirical estimate of ROCk(t) based on data from category k. Like in the case of equation (4), Equationconverge to a Gaussian process, therefore Φ−1 ROCk(t) can be approximated by Equation, which motivates the following estimation procedure:

1. Calculate pairs of Equationfor each subset separately;

2. Let design matrix D be

Equation                                            (7)

whereEquation, M1 is a vector of length p with constant value of one, and O is a zero value matrix.

3. Let Equation and Equation

4. Our linear model is: for the reference subset, Equation

where Equationis normally distributed with mean 0 and asymptotic covariance matrix Σr,1; and for subset k,k = 2,3,..,K

Equation

where Equationis normally distributed with mean 0 and asymptotic covariance matrix Σr,k;

5. Our OLS estimator for Equation

The above method assumes covariate effects can be explained adequately by the difference in the intercept parameter (α0). If in addition we allow the slope of a binormal ROC curve to be different across covariate categories by assuming

Φ−1(ROCk(t)) = β1 + γ1Φ−1(t) + (1 Φ−1(t)) θTZ(k), (8)

where

Equation

then the parameter of interest is θ* = (β1, γ1,…, βk, γk)T. The estimating procedure is similar to the case where only intercept parameters vary, but with M1 replaced by M in (7). Inference for the significance of θ* in both settings can be achieved by estimating the variance of θ* with the bootstrap resampling method.

Simulation

Estimation of the ROC curve

The estimation procedure specified in the previous section starts with the choice of the false positive set T. Although in theory, any chosen set of T would yield estimators with the same asymptotic property, their small sample properties need to be investigated. An obvious starting point is to choose the collection of the observed false positive fractions that fall into the interval [a,b], we call this observed FP (OFP) method. In the case where there are no ties in the test results of non-diseased subjects, this is equivalent to the method selecting the subset of Equationwithin [a,b], which we call the equal fraction (EF) method. Another possible choice is to divide interval [a,b] into Equationequally spaced sub-intervals and choose the midpoints of those subintervals to be the set T (midpoint(MP) method). For the last two methods, the number of points in T always equal to Equationregardless of the length of [a,b].

We compare the performance of the OLS estimators between those three different methods of selecting T (OFP, EF and MP). We simulate dataset both with and without ties and vary the values of [a,b] to estimate either a full curve or a partial ROC curve.

First, we generate Equation and Equation . The resulting ROC curve follows a binormal model: ROC(t) = Φ(α0 + α1Φ−1(t)). The parameters (α0, α1) are chosen to be (1.2, 0.45), corresponding to an area under the curve (AUC) value of 0.863 and a partial AUC (pAUC) value of 0.142 for t ∈ (0, 0.2). With sample size of Equation, we compare the biases and sampling standard errors in estimating AUC, ROC(0.2), ROC(0.4) and ROC(0.7) in the full curve estimation and pAUC(0.2) and ROC(0.1) in the partial curve estimation. To simulate data with ties, we chose 20% tied values within each population.

From Table 2, when the test results have no ties, midpoint(MP) method has the best performance. When data has ties, observed false positive fraction(OFP) method has the smallest bias when estimating the entire curve, but MP has the best performance for estimating the partial curve. We recommend using either method MP or OFP in practice and we choose MP method for the subsequent simulations (Table 2).

Parameter True value OFP EF MP
Percent of ties: 0%        
(a,b) = (0.0001, 0.9999)        
R(0.2) 0.794 0.791(0.039) 0.792(0.039) 0.795(0.038)
R(0.4) 0.861 0.860(0.031) 0.860(0.031) 0.862(0.031)
R(0.7) 0.924 0.923(0.024) 0.923(0.024) 0.924(0.024)
AUC 0.863 0.861(0.027) 0.861(0.027) 0.863(0.027)
(a,b)= 0.0001,0.2)        
R(0.1) 0.733 0.727(0.049) 0.729(0.048) 0.733(0.048)
pAUC(0.2) 0.142 0.141(0.010) 0.141(0.010) 0.142(0.010)
Percent of ties: 20%        
(a,b) = (0.0001, 0.9999)        
R(0.2) 0.794 0.794(0.043) 0.805(0.041) 0.798(0.042)
R(0.4) 0.861 0.862(0.034) 0.879(0.031) 0.864(0.034)
R(0.7) 0.924 0.924(0.026) 0.942(0.022) 0.926(0.025)
AUC 0.863 0.863(0.030) 0.875(0.027) 0.865(0.030)
(a,b) = 0.0001,0.2)        
R(0.1) 0.733 0.728(0.053) 0.739(0.051) 0.734(0.053)
pAUC(0.2) 0.142 0.141(0.011) 0.143(0.010) 0.143(0.011)

Table 2: Inference of the ROC curve by the semiparametric least squares based method (OLS) with different choices of the false positive sets (OFP: observed false positive fraction, EF: equal fraction or MP: midpoint method). Cases and controls are drawn from normal distributions. (α0, α1) = (1.2, 0.45). (nD, nD) = (100, 100). Sample means and sampling standard errors from 1000 simulations are shown.

Next we compare performance of the OLS estimator with two existing semiparametric ROC modeling approach (ROC-GLM and PV), which have been shown to have good performance among others. Table 3 summarizes relative biases and standard errors for the three estimators for estimating either a full or a partial ROC curve. We observe that when the full ROC curve is of interest, the OLS and the GLM estimators have comparable performances while the PV estimator has somewhat larger biases. When estimating a partial ROC curve, however, the OLS estimator can have substantially smaller bias compared to other estimators (Table 3).

Parameter True value OLS GLM PV
(a,b) = (0.0001, 0.9999)        
a0 1.2 1.09(0.67) 1.46(0.68) 2.40(0.72)
a1 0.45 2.42(0.37) 1.38(0.36) 9.44(0.45)
R(0.2) 0.794 -0.14(0.177) 0.16(0.172) -0.53(0.179)
R(0.4) 0.861 -0.02(0.139) 0.12(0.138) 0.15(0.143)
R(0.7) 0.924 -0.02(0.104) -0.002(0.105) 0.38(0.108)
 (a,b) = 0.0001,0.2)        
a0 1.2 0.18(1.10) -1.79(1.05) 7.45(1.19)
a1 0.45 -0.62(0.63) -6.82(0.58) 14.04(0.71)
R(0.05) 0.677 0.11(0.226) 1.27(0.221) -1.00(0.226)
R(0.1) 0.733 -0.02(0.206) 0.51(0.204) 0.11(0.203)
R(0.15) 0.768 -0.13(0.204) 0.08(0.202) 0.61(0.201)

Table 3: Inference of the ROC curve by the semiparametric least squares based method (OLS), the ROC-GLM method (GLM) and the placement value method (PV). Cases and controls are drawn from normal distributions. (α01) = (1.2, 0.45), ( nD ,Equation) = (100, 100). Relative bias and sampling standard error from 500 simulations are shown. Relative bias = bias/true value × 100%.

In Table 4, we demonstrate performance of the OLS estimator relative to the nonparametric ROC estimator and the parametric ROC estimator assuming normal case and control distributions after Box-Cox transformation. We generate data as Equationand Equationso the resulting ROC curve is the same as above. We further modify the data as following: (i) inflating all data points by two fold (scaled log normal distribution); (ii) increasing all data points by 2 units (shifted log normal distribution); (iii) exp(Y1/3) (non Box-Cox). For the parametric method, Box-Cox transformation is applied to Equationand YD separately before the fitting of a normal distribution. For log normal data or scaled log normal data, the OLS estimator and the parametric estimator combined with Box-Cox transformation have similar performances. For shifted log normal and non Box-Cox data, however, the parametric estimator performs poorly with a large bias even after a Box-Cox transformation. Both the OLS estimator and the nonparametric estimator are unbiased in these scenarios with the former substantially more efficient (Table 4).

Parameter  True value Distribution OLS EMP RES
R(0.2) 0.794 lognormal 0.796(0.039) 0.795(0.044) 0.795(0.036)
R(0.4) 0.861   0.863(0.031) 0.864(0.036) 0.862(0.030)
R(0.7) 0.924   0.926(0.023) 0.926(0.027) 0.924(0.023)
AUC 0.863   0.864(0.027) 0.864(0.027) 0.863(0.026)
 R(0.2) 0.794 lognormal-scaled 0.795(0.039) 0.796(0.043) 0.794(0.036)
R(0.4) 0.861   0.862(0.032) 0.863(0.037) 0.861(0.030)
R(0.7) 0.924   0.925(0.024) 0.925(0.028) 0.923(0.023)
AUC 0.863   0.863(0.028) 0.863(0.028) 0.862(0.026)
 R(0.2) 0.794 lognormal-shift 0.794(0.039) 0.795(0.043) 0.818(0.043)
R(0.4) 0.861   0.861(0.031) 0.861(0.036) 0.900(0.031)
R(0.7) 0.924   0.923(0.023) 0.923(0.027) 0.961(0.017)
AUC 0.863   0.862(0.027) 0.862(0.027) 0.888(0.027)
 R(0.2) 0.794 non Box-Cox 0.796(0.038) 0.796(0.044) 0.764(0.051)
R(0.4) 0.861   0.863(0.031) 0.864(0.036) 0.887(0.037)
R(0.7) 0.924   0.926(0.023) 0.926(0.027) 0.968(0.017)
AUC 0.863   0.864(0.027) 0.864(0.027) 0.862(0.029)

Table 4: Inference of the ROC curve by the semiparametric least squares based method (OLS), the nonparametric method (EMP) and the parametric method (RES). Cases and controls are drawn from log-normal distributions and modified as noted. (α0, α1) = (1.2, 0.45). ( nD ,Equation) = (100, 100). Sample means and sampling standard errors from 1000 simulations are shown.

Lastly, we perform simulation studies to investigate the use of large sample inference for ROC curve and AUC based on the OLS estimator. Table 5 shows the mean estimated asymptotic error and the coverage of corresponding 95% Wald confidence intervals. The variance estimate from the asymptotic theory reflects the actual sampling variance and the coverage of 95% confidence intervals is excellent (Table 5).

  Bias SSE ASE CP
  (nD, nD) = (100, 100)
a0 2.4% 0.163 0.151 94.8%
a1 2.2% 0.085 0.083 95.4%
R(0.2) 0.4% 0.039 0.039 94.6%
R(0.4) 0.3% 0.032 0.031 93.4%
R(0.7) 0.1% 0.025 0.023 90.8%
  (nD, nD) = (100, 50)
a0 0.9% 0.159 0.157 95.0%
a1 2.9% 0.088 0.088 94.6%
R(0.2) -0.3% 0.041 0.042 94.6%
R(0.4) -0.1% 0.033 0.033 94.6%
R(0.7) -0.1% 0.025 0.024 93.8%
  (nD, nD) = (50, 100)
a0 1.1% 0.205 0.208 96.4%
a1 0.9% 0.101 0.113 97.2%
R(0.2) -0.2% 0.053 0.053 93.8%
R(0.4) -0.2% 0.041 0.043 94.4%
R(0.7) -0.3% 0.030 0.033 94.6%

Table 5: Result of 1000 simulations to evaluate the application of inference based on asymptotic theory to finite sample studies. (α0, α1) = (1.2, 0.45). Relative bias = bias / true value x 100%; SSE: sampling standard error; ASE: mean estimated standard error using asymptotic theory; CP: coverage of 95% Wald-confidence intervals using estimated asymptotic error.

Application to DPOAE Data Set

The DPOAE data set was first published by Stover et al. [13]. DPOAE stands for distortion product otoacoustic emission, which is an audiology test used to separate normal-hearing from hearingimpaired ears.

The test is administrated under nine different auditory stimulus conditions with three levels of frequency (1001, 1416 and 2002 Hz) and three levels of intensity (55, 60 and 65 dB SPL). A total of 210 subjects were included in the study. The subjects were considered cases with hearing impairment at a given frequency if their audiometric threshold exceeds 20dB HL measured by a behavior test (gold standard). Each subject was tested in only one ear. Test result is the negative signal-tonoise ratio, -SNR, with higher value being more indicative of hearing impairment. The objective of the analysis is to determine the optimal setting for the clinical use of DPOAE to separate normal from hearingimpaired ears, but bear in mind an ear may be determined to be hearing impaired or normal at different frequencies.

We partition the data into nine subsets, corresponding to the nine test settings. Data is analyzed by the method specified in (8) where both intercept and slope parameters vary across subsets. The set T is chosen by the midpoint method from (0.0001, 0.9999). We choose the reference subset (subset 1) to be the test setting with frequency value of 1001Hz and intensity value of 55 dB SPL. Let (β1, γ1) be the intercept and slope estimates for the ROC curve for setting (1001, 55) and the subsequent βk and γk, with k = 2,…,9, represent the differences in the intercept and slope parameters of the ROC curves between subset k and subset 1. None of the Equationis statistically significantly different from 0 at α = 0.05 α = 0.05 level (data are not shown).

We also develop a χ2 test statistic Equationwhere γ = (γ2,…γ9) and Equation is the estimated covariance matrix of Equation Write the null hypothesis as H0: γ = 0 and compare the above statistic with a χ2 distribution with 8 degrees of freedom gives a P-value of 0.84, suggesting insignificant slope terms consistent with the result from testing the significance of each γk separately.

We re-analyze the data with the slope terms omitted and Table 6 summarizes the results. We can see that among the nine test settings, the setting (1416, 55) (β4) has the largest intercept estimate and the difference from the intercept of the reference setting (1001, 55) is statistically significant (p=0.0041). The p values for other parameters (β2 β3 β5 to β9) are not significant. The estimated ROC curve for setting (1416, 55) is ROC(t) = Φ(2.54 + 0.82Φ−1(t)) with estimated AUC value of 0.975, which is the largest among all settings. The performance of the test declines with increasing intensity at each fixed frequency value. This analysis suggests (1416Hz, 55 dB SPL) is a better test setting than the reference setting. It has also shown that at fixed false positive level of 10%, the expected sensitivities for the nine test settings range from 62.5% to 93.2%, demonstrating the importance of choosing an optimal test setting (Table 6).

(Frequency,intensity)          
  Parameter Estimate Std.Err P value AUC ROC(0.1)
(1001,55) β1 1.73 0.26 < 0.0001    
(1001,55) γ1 0.82 0.12 < 0.0001 0.910 0.751
(1001,60) β2 -0.20 0.14 0.1455 0.882 0.684
(1001,65) β3 -0.36 0.19 0.0607 0.855 0.625
(1416,55) β4 0.81 0.28 0.0041* 0.975 0.932
(1416,60) β5 0.36 0.29 0.2172 0.947 0.851
(1416,65) β6 0.19 0.28 0.5064 0.931 0.808
(2002,55) β7 0.47 0.29 0.1052 0.956 0.875
(2002,60) β8 0.42 0.28 0.1293 0.952 0.864
(2002,65) β9 0.10 0.27 0.7019 0.921 0.782

Table 6: Covariates effects on the ROC curves estimated by the the OLS method for the DPOAE test.

Concluding Remarks

This manuscript proposes a semiparametric OLS method to estimate and compare performance of diagnostic tests and more generally, to assess potential covariate effects on the test performance. The asymptotic distribution theory for the OLS estimator is developed for the ROC curve estimation, in which the estimators are shown to be consistent and asymptotically normally distributed. For modeling covariate effects, we recommend boostrap resampling for variance estimation.

Our proposed estimator provides useful addition to the field of rank-based semiparametric ROC modeling. Those semiparametric approaches are more robust than parametric approaches, by assuming a functional form on the ROC curve itself but not the test results and thus invariant to monotone transformation of the test results. At the same time, they offer better efficiency compared to nonparametric method. We have done extensive simulations to compare the proposed OLS estimator with two other commonly used semiparametric ROC modeling methods (the ROC-GLM and placement value based method), and found the OLS estimator has comparable performance in general and slightly better performance in some scenarios [12]. The OLS estimator, however is much more intuitive compared to other estimators and very easy to implement using standard linear regression software, which could make it particularly appealing to clinical audience.

In summary, the proposed linear regression framework provides an unified approach for the ROC curve analysis. It can be used to estimate the ROC curve, as well as model covariate effect. The application of ROC curve goes beyond the medical diagnostic field and it can be used for evaluating any discrimination tools. It is, and will continue to be an important and exciting area to engage in research.

Acknowledgements

The authors would like to acknowledge Margaret S. Pepe for her invaluable contribution to the development of this manuscript. The work is partially funded by NIH grant GM54438 and P30CA015704.

References

Appendix

Appendix I: Proof of Equation (4)

Hsieh and Turnbull(1996, Theorem 2.1) showed, as Equation

       Equation                                                 (9)

They also showed (Theorem 2.2) for independent observations, under the above conditions, there exists a probability space on which one can define sequences of two independent versions of Brownian bridges Equation, and the following statement holds:

Equation                                                                                                                             (10)

a.s. uniformly on [a,b], 0 < a < b <1.

The above two theorems stated the strong consistency and strong approximation properties for the ROC curve

Fix t[a,b], by intermediate value theorem,

Equation (11)

where

Equation                                      (12)

Therefore, Equationby (9)

From continuous mapping theorem:

Equation                                         (13)

Notice

Equation            (14)

Then we have

Equation            (15)

Equation            (16)

a.s. uniformly on [a,b], 0 < a < b <1.

Let

Equation                                  (17)

Equation    (18)

Equation (16) implies Equation

Equation (4) resulted from the fact that V(t) is the sum of two independent Brownian Bridges.

Appendix II: Proof of Theorems 1

Theorem 1 is a direct consequence of the equation (4), namely, asymptotic normality of the empirical ROC estimates.

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