Lloyd J. Edwards, PhD, Charles R. Katholi, PhD, Byron C. Jaeger, PhD

UAB Department of Biostatistics

Title: Properties of $R^2_\beta$ and Tests of Hypotheses for Fixed Effects in the Linear Mixed Model

Abstract:

Properties of the $R^2$ statistic used in linear regression with least squares estimation are well developed. Additionally, the $R^2$ statistic used in linear regression has a corresponding test of hypoyhesis associated with it. Similar properties are developed for $R^2_\beta$, an $R^2$ statistic for fixed effects in the linear mixed model. The central and non-central Beta distributions are used to approximate the distribution of $R^2_\beta$ under corresponding null and alternative test hypotheses. The asymptotic expectation and variance of $R^2_\beta$ are derived. Simulations are used to demonstrate the performance of the Beta approximation. Test statistics are proposed that are based on estimators of the derived expectation and variance. Simulations are used to compare the test statistics to the overall $F$ test for fixed affects in the linear mixed model. Using simulations, the Type I error rate of the proposed $R^2_\beta$ test statistics is shown to be equivalent to the Type I error rate for the overall $F$ test.

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Title: Bounds for Preperiodic Points for Maps with Good Reduction

Abstract:

Let $K$ be a number field and let $\phi$ in $K(z)$ be a rational function of degree $d\geq 2$. Let $S$ be the set of places of bad reduction for $\phi$ (including the archimedean places). Let $\Per(\phi,K)$, $\PrePer(\phi, K)$, and $\Tail(\phi,K)$ be the set of $K$-rational periodic, preperiodic, and purely preperiodic points of $\phi$, respectively.

In this talk, we will present two main results. Firstly, assuming that $|\Per(\phi,K)| \geq 4$ (resp.\$|\Tail(\phi,K)| \geq 3$), we prove bounds for $|\Tail(\phi,K)|$ (resp.\$|\Per(\phi,K)|$) that depend only on the number of places of bad reduction $|S|$ (and not on the degree $d$). We show that the hypotheses of this result are sharp, giving counterexamples to any possible result of this form when $|\Per(\phi,K)| < 4$ (resp.\$|\Tail(\phi,K)| < 3$).

For the second result, a bound for $|\PrePer(\phi,K)|$ in terms of the number of places of bad reduction $|S|$ and the degree $d$ of the rational function $\phi$ is obtained. This bound significantly improves a previous result.

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Title: A Dynamic Model of Free Fatty Acids, Glucose, and Insulin Metabolism

Abstract:

The role of free fatty acids (FFA) on the progression of type 2 diabetes (T2D) has been widely studied. Prior studies suggest that individuals with shared familiar genetic predispositions to metabolic-related diseases may be vulnerable to dysfunctional regulation of plasma FFA. A vicious cycle may arise when FFA is not regulated properly leading to the development of insulin resistance, a key indicator for T2D as prolonged insulin resistance results in hyperglycemia. We propose a hypothesis-driven model to quantitatively study the role of FFA on the progression of insulin resistance. The nonlinear dynamics among glucose, insulin, and FFA are modeled using delay differential equations and compared to the well-known minimal model consisting of ordinary differential equations. Model validation and parameter estimation utilizing clinical data of patients who underwent bariatric surgery, serve as the quantitative measures used to evaluate the regulation of FFA production by insulin action within a heterogeneous population (nondiabetic to diabetic). Results show that several metabolic factors for insulin, glucose, and FFA regulation improved post-bariatric surgery, and these results were supported with prior clinical findings.

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