Use of Area Under the Curve (AUC) from Propensity Model to Estimate Accuracy of the Estimated Effect of Exposure

Objective: To investigate the relationship between the area under the Receiver Operating Characteristic curve (AUC) of the propensity model for exposure and the accuracy of the estimated effect of the exposure on the outcome of interest.Methods: A Monte Carlo simulation study was performed where multiple realizations of three binary variables: outcome, exposure of interest and a covariate were repeatedly generated from the distribution determined by the parameters of the "propensity" and "main" models and the prevalence of the exposure. "Propensity" model was a logistic regression with the exposure of interest as a dependent variable and a single covariate as an "independent" variable. "Main" model was a logistic regression with outcome as a dependent variable, exposure of interest and covariate as "independent" variables. A total of 500 simulations were performed for each considered combination of the model parameters and the prevalence of the exposure. AUC was estimated from the probabilities predicted by the propensity score model. The accuracy of the estimated effect of exposure was primarily assessed with the square root of Mean Square Error (RMSE); the fifth and ninety-fifth percentile of the empirical distribution of the estimator were used to illustrate a range of not unlikely deviations from the true value.Results: The square root of Mean Square Error of the estimated effect of exposure increases as AUC increases from 0.6 to 0.9. Varying values for parameters of the propensity score model or the main effect model does not change the direction of this trend. As the proportion of exposed subjects changes away from 0.5 the RMSE increases, but the effect of AUC on RMSE remains approximately the same. Similarly, as sample size changes from 50 to 100 or 200, the RMSE of effect estimate decreases on average, but the effect of AUC on RMSE remains approximately the same. Also, the rate of change in RMSE increases with increasing AUC; the rate is the lowest when AUC changes from 0.6 to 0.7 and is highest when AUC changes from 0.8 to 0.9.Conclusions: The AUC of the propensity score model for exposure provides a single, relatively easy to compute, and suitable for various kind of data statistic, which can be used as an important indicator of the accuracy of the estimated effect of exposure on the outcome of interest. The public health importance is that it can be considered as an alternative to the previously suggested (Rubin, 2001) simultaneous consideration of the conditions of closeness of means and variances of the propensity scores in the different exposure groups. Our simulations indicate that the estimated effect of exposure is highly unreliable if AUC of the propensity model is larger than 0.8; at the same time AUCs of less than 0.7 are not associated with any substantial increase of inaccuracy of the estimated effect of exposure

Wavelet Galerkin method for fractional elliptic differential equations

Under the guidance of the general theory developed for classical partial differential equations (PDEs), we investigate the Riesz bases of wavelets in the spaces where fractional PDEs usually work, and their applications in numerically solving fractional elliptic differential equations (FEDEs). The technique issues are solved and the detailed algorithm descriptions are provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin method we propose for FEDEs has the striking benefit of efficiency, since the condition numbers of the corresponding stiffness matrixes are small and uniformly bounded; and the Toeplitz structure of the matrix still can be used to reduce cost. Numerical results and comparison with the ordinary Galerkin methods are presented to demonstrate the advantages of the wavelet Galerkin method we provide.Comment: 20 pages, 0 figure

Multi-Soliton solutions to a model equation for shallow water waves

In Soliton theory, Hirota direct method is most efficient tool for seeking one soliton solutions or multi-soliton solutions of integrable nonlinear partial differential equations. The key step of the Hirota direct method is to transform the given equation into its Hirota bilinear form. Once the bilinear form of the given equation is found, we can construct the soliton and multi-soliton solutions of that model. Many interesting characteristics of Pfaffians were discovered through studies of soliton equations. In this thesis, a shallow water wave model and its bilinear equation are investigated. Using Hirota direct method, we obtain the multi-soliton solutions and Pfaffian solutions for a shallow water wave model

Characterization Techniques and Electrolyte Separator Performance Investigation for All Vanadium Redox Flow Battery

The all-vanadium redox flow battery (VRFB) is an excellent prospect for large scale energy storage in an electricity grid level application. High battery performance has lately been achieved by using a novel cell configuration with advanced materials. However, more work is still required to better understand the reaction kinetics and transport behaviors in the battery to guide battery system optimization and new battery material development. The first part of my work is the characterization of the battery systems with flow-through or flow-by cell configurations. The configuration difference between two cell structures exhibit significantly different polarization behavior. The battery output can be increased by higher electrolyte feed rate, but electrolyte utilization was decreased correspondingly. The battery performance can be largely enhanced by non-wetproofed electrode material. The battery cell with higher vanadium crossover has lower energy efficiency and faster capacity decay in cycling test. Secondly, the state of charge (SOC) monitoring is of great importance for battery management. A SOC monitoring method is developed using UV-Vis spectrometric measurements on VRFB electrolyte solutions. The spectrum of the negative electrolyte is linearly dependent on its SOC. In the positive electrolyte, the nonlinear intensity dependence on SOC appears to be caused by formation of complex vanadium-oxygen ion. The characteristic molar UV-Vis spectrum of the complex vanadium-oxygen ion was separated from that of the pure positive vanadium electrolyte components. The SOC of the positive electrolyte can be then calculated from its UV-Vis spectrum by considering the complex vanadium ion equilibrium. Moreover, the understanding of ionic transport mechanism in the electrolyte separator is critical to reduce internal resistance and vanadium crossover in the battery. The properties of Nafion and sulfonated Alder Diels poly(phelynene) (SDAPP) were investigated after equilibration with different electrolyte compositions. Both sulfuric acid and vanadium ion in the membrane can cause membrane conductivity loss. Vanadium-oxygen ion in membrane can slow down proton mobility via an unknown mechanism. Transmission electron microscope imaging showed that SDAPP is a more homogeneous ion exchange polymer with less phase separation than Nafion. The SDAPP membranes have better ion conducting properties than Nafion because of their higher ionic selectivity