Estimating the Area under a Receiver Operating Characteristic Curve For Repeated Measures Design

The receiver operating characteristic (ROC) curve is widely used for diagnosing as well as for judging the discrimination ability of different statistical models. Although theories about ROC curves have been established and computation methods and computer software are available for cross-sectional design, limited research for estimating ROC curves and their summary statistics has been done for repeated measure designs, which are useful in many applications, such as biological, medical and health services research. Furthermore, there is no published statistical software available that can generate ROC curves and calculate summary statistics of the area under a ROC curve for data from a repeated measures design. Using generalized linear mixed model (GLMM), we estimate the predicted probabilities of the positivity of a disease or condition, and the estimated probability is then used as a bio-marker for constructing the ROC curve and computing the area under the curve. The area under a ROC curve is calculated using the Wilcoxon non-parametric approach by comparing the predicted probability of all discordant pairs of observations. The ROC curve is constructed by plotting a series of pairs of true positive rate (sensitivity) and false positive rate (1- specificity) calculated from varying cuts of positivity escalated by increments of 0.005 in predicted probability. The computation software is written in SAS/IML/MACRO v8 and can be executed in any computer that has a working SAS v8 system with SAS/IML/MACRO.

Galerkin approximations for the optimal control of nonlinear delay differential equations

Optimal control problems of nonlinear delay differential equations (DDEs) are considered for which we propose a general Galerkin approximation scheme built from Koornwinder polynomials. Error estimates for the resulting Galerkin-Koornwinder approximations to the optimal control and the value function, are derived for a broad class of cost functionals and nonlinear DDEs. The approach is illustrated on a delayed logistic equation set not far away from its Hopf bifurcation point in the parameter space. In this case, we show that low-dimensional controls for a standard quadratic cost functional can be efficiently computed from Galerkin-Koornwinder approximations to reduce at a nearly optimal cost the oscillation amplitude displayed by the DDE's solution. Optimal controls computed from the Pontryagin's maximum principle (PMP) and the Hamilton-Jacobi-Bellman equation (HJB) associated with the corresponding ODE systems, are shown to provide numerical solutions in good agreement. It is finally argued that the value function computed from the corresponding reduced HJB equation provides a good approximation of that obtained from the full HJB equation.Comment: 29 pages. This is a sequel of the arXiv preprint arXiv:1704.0042

Sample Size Calculation and Power Analysis of Time-Averaged Difference

Little research has been done on sample size and power analysis under repeated measures design. With detailed derivation, we have shown sample size calculation and power analysis equations for timeaveraged difference to allow unequal sample sizes between two groups for both continuous and binary measures and explored the relative importance of number of unique subjects and number of repeated measurements within each subject on statistical power through simulation

Estimating the Area under a Receiver Operating Characteristic Curve For Repeated Measures Design

The receiver operating characteristic (ROC) curve is widely used for diagnosing as well as for judging the discrimination ability of different statistical models. Although theories about ROC curves have been established and computation methods and computer software are available for cross-sectional design, limited research for estimating ROC curves and their summary statistics has been done for repeated measure designs, which are useful in many applications, such as biological, medical and health services research. Furthermore, there is no published statistical software available that can generate ROC curves and calculate summary statistics of the area under a ROC curve for data from a repeated measures design. Using generalized linear mixed model (GLMM), we estimate the predicted probabilities of the positivity of a disease or condition, and the estimated probability is then used as a bio-marker for constructing the ROC curve and computing the area under the curve. The area under a ROC curve is calculated using the Wilcoxon non-parametric approach by comparing the predicted probability of all discordant pairs of observations. The ROC curve is constructed by plotting a series of pairs of true positive rate (sensitivity) and false positive rate (1- specificity) calculated from varying cuts of positivity escalated by increments of 0.005 in predicted probability. The computation software is written in SAS/IML/MACRO v8 and can be executed in any computer that has a working SAS v8 system with SAS/IML/MACRO

Galerkin-Koornwinder approximations of delay differential equations for physicists

Formulas for Galerkin-Koornwinder (GK) approximations of delay differential equations are summarized. The functional analysis ingredients (semigroups, operators, etc.) are intentionally omitted to focus instead on the formulas required to perform GK approximations in practice

Optimal Parameterizing Manifolds for Anticipating Tipping Points and Higher-order Critical Transitions

A general, variational approach to derive low-order reduced systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes the more classical notions of invariant or slow manifold when breakdown of ''slaving'' occurs, i.e. when the unresolved variables cannot be expressed as an exact functional of the resolved ones anymore. The OPM provides, within a given class of parameterizations of the unresolved variables, the manifold that averages out optimally these variables as conditioned on the resolved ones. The class of parameterizations retained here is that of continuous deformations of parameterizations rigorously valid near the onset of instability. These deformations are produced through integration of auxiliary backward-forward (BF) systems built from the model's equations and lead to analytic formulas for parameterizations. In this modus operandi, the backward integration time is the key parameter to select per scale/variable to parameterize in order to derive the relevant parameterizations which are doomed to be no longer exact, away from instability onset, due to breakdown of slaving typically encountered e.g. for chaotic regimes. The selection criterion is then made through data-informed minimization of a least-square parameterization defect. It is thus shown, through optimization of the backward integration time per scale/variable to parameterize, that skilled OPM reduced systems can be derived for predicting with accuracy higher-order critical transitions or catastrophic tipping phenomena, while training our parameterization formulas for regimes prior to these transitions take place