Compare diagnostic tests using transformation-invariant smoothed ROC curves().

Receiver operating characteristic (ROC) curve, plotting true positive rates against false positive rates as threshold varies, is an important tool for evaluating biomarkers in diagnostic medicine studies. By definition, ROC curve is monotone increasing from 0 to 1 and is invariant to any monotone transformation of test results. And it is often a curve with certain level of smoothness when test results from the diseased and non-diseased subjects follow continuous distributions. Most existing ROC curve estimation methods do not guarantee all of these properties. One of the exceptions is Du and Tang (2009) which applies certain monotone spline regression procedure to empirical ROC estimates. However, their method does not consider the inherent correlations between empirical ROC estimates. This makes the derivation of the asymptotic properties very difficult. In this paper we propose a penalized weighted least square estimation method, which incorporates the covariance between empirical ROC estimates as a weight matrix. The resulting estimator satisfies all the aforementioned properties, and we show that it is also consistent. Then a resampling approach is used to extend our method for comparisons of two or more diagnostic tests. Our simulations show a significantly improved performance over the existing method, especially for steep ROC curves. We then apply the proposed method to a cancer diagnostic study that compares several newly developed diagnostic biomarkers to a traditional one.

[1]  Rob J Hyndman,et al.  Improved methods for bandwidth selection when estimating ROC curves , 2003 .

[2]  Xiao-Hua Zhou,et al.  Semiparametric Inferential Procedures for Comparing Multivariate ROC Curves with Interaction Terms , 2008 .

[3]  D. Dorfman,et al.  Maximum-likelihood estimation of parameters of signal-detection theory and determination of confidence intervals—Rating-method data , 1969 .

[4]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[5]  Xiao-Hua Zhou,et al.  A Flexible Method for Estimating the ROC Curve , 2004 .

[6]  M. Wells,et al.  Quantile Comparison Functions in Two-Sample Problems, with Application to Comparisons of Diagnostic Markers , 1996 .

[7]  Rob J Hyndman,et al.  Nonparametric confidence intervals for receiver operating characteristic curves , 2004 .

[8]  Chong Gu,et al.  Smoothing spline Gaussian regression: more scalable computation via efficient approximation , 2004 .

[9]  Gang Li,et al.  A Unified Approach to Nonparametric Comparison of Receiver Operating Characteristic Curves for Longitudinal and Clustered Data , 2008, Journal of the American Statistical Association.

[10]  Mitchell H. Gail,et al.  A family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data , 1989 .

[11]  D. McClish,et al.  Comparing the Areas under More Than Two Independent ROC Curves , 1987, Medical decision making : an international journal of the Society for Medical Decision Making.

[12]  Pang Du,et al.  Transformation‐invariant and nonparametric monotone smooth estimation of ROC curves , 2009, Statistics in medicine.

[13]  S. Geer.  Empirical Processes in M-Estimation , 2000 .

[14]  C A Roe,et al.  Statistical Comparison of Two ROC-curve Estimates Obtained from Partially-paired Datasets , 1998, Medical decision making : an international journal of the Society for Medical Decision Making.

[15]  K. Zou,et al.  Smooth non-parametric receiver operating characteristic (ROC) curves for continuous diagnostic tests. , 1997, Statistics in medicine.

[16]  Xiao-Hua Zhou,et al.  Statistical Methods in Diagnostic Medicine , 2002 .

[17]  Liang Peng,et al.  Local linear smoothing of receiver operating characteristic (ROC) curves , 2004 .

[18]  Chris Lloyd,et al.  Using Smoothed Receiver Operating Characteristic Curves to Summarize and Compare Diagnostic Systems , 1998 .

[19]  Debashis Kushary,et al.  Bootstrap Methods and Their Application , 2000, Technometrics.

[20]  I Hinberg,et al.  Receiver operator characteristic (ROC) curves and non-normal data: an empirical study. , 1990, Statistics in medicine.

[21]  Chong Gu.  Smoothing Spline Anova Models , 2002 .

[22]  Chris Lloyd,et al.  Kernel estimators of the ROC curve are better than empirical , 1999 .

[23]  G. Wahba.  Smoothing noisy data with spline functions , 1975 .

[24]  B. Turnbull,et al.  NONPARAMETRIC AND SEMIPARAMETRIC ESTIMATION OF THE RECEIVER OPERATING CHARACTERISTIC CURVE , 1996 .

[25]  Y. Benjamini,et al.  Controlling the false discovery rate: a practical and powerful approach to multiple testing , 1995 .