Abstract
Receiver operating characteristic (ROC) curves are commonly used to measure the performance of diagnostic tests. The ROC curve can be estimated empirically without assuming the distributions of the underlying diagnostic test data. Comparison of the accuracy of two diagnostic tests using ROC curves from the two tests are often conducted using fixed sample designs. However, to address the ethics and efficiency concerns of clinical trial studies, there is a need to employ a group sequential design (GSD) and periodically monitor and analyze the accruing data. In this chapter, we incorporate group sequential methods into the design of comparative diagnostic study with respect to the ROC curves. First, we study the difference between sequential empirical ROC curves on the process level. Then we derive the asymptotic distribution theory for the difference between sequential empirical ROC curves and derive the asymptotic covariance structure for comparative ROC statistics. Relating the difference between empirical ROC curves to the Kiefer process, we also show these results can be used to conduct a GSD using standard software.
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Appendix
Appendix
Derivation of the elements a ij in Σ:
The last step is derived by applying the results of sequential empirical process, the compact differentiability of the inverse function and delta method in vander Vaart and Wellner (1996).
The last step is again derived by applying the results of sequential empirical process, the compact differentiability of the inverse function and delta method in vander Vaart and Wellner (1996). Similarly, we can get the following elements of the covariance matrix: \(a_{22}= r_D(\mathit{\rm ROC}_2(t)-\mathit{\rm ROC}_2^2(t))+\lambda \frac{r_D^2}{r_{\bar D}} \left( \frac{f_{2,D}(S_{2,\bar D}^{-1}(t))}{f_{2,\bar D}(S_{2,\bar D}^{-1}(t))} \right)^2(t-t^2),\) \(a_{23}=a_{14},\) \(a_{24}= (r_D\wedge r_D^{\prime})(\mathit{\rm ROC}_2(t)-\mathit{\rm ROC}_2^2(t)) + (r_{\bar D}\wedge r_{\bar D}^{\prime})\lambda \frac{r_D}{r_{\bar D}} \frac{r_D^{\prime}}{r_{\bar D}^{\prime}} \left( \frac{f_{2,D}(S_{2,\bar D}^{-1}(t))}{f_{2,\bar D}(S_{2,\bar D}^{-1}(t))} \right)^2 (t-t^2),\) \(a_{33}=r_D^{\prime}(\mathit{\rm ROC}_1(t)-\mathit{\rm ROC}_1^2(t))+\lambda \frac{r_D^{{\prime} 2}}{r_{\bar D}^{\prime}} \left( \frac{f_{1,D}(S_{1,\bar D}^{-1}(t))}{f_{1,\bar D}(S_{1,\bar D}^{-1}(t))} \right)^2(t-t^2),\)
\(a_{44}= r_D^{\prime}(\mathit{\rm ROC}_2(t)-\mathit{\rm ROC}_2^2(t))+\lambda \frac{r_D^{{\prime}2}}{r_{\bar D}^{\prime}} \left( \frac{f_{2,D}(S_{2,\bar D}^{-1}(t))}{f_{2,\bar D}(S_{2,\bar D}^{-1}(t))} \right)^2(t-t^2).\)
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Ye, X., Tang, L. (2015). Group Sequential Methods for Comparing Correlated Receiver Operating Characteristic Curves. In: Chen, Z., Liu, A., Qu, Y., Tang, L., Ting, N., Tsong, Y. (eds) Applied Statistics in Biomedicine and Clinical Trials Design. ICSA Book Series in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-319-12694-4_6
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DOI: https://doi.org/10.1007/978-3-319-12694-4_6
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