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This project was dedicated to the problem of improving a non-invasive blood glucose monitor being developed by the VivaScan Corporation. The company has made some progress in the non-invasive blood glucose device development and approached WPI for a statistical assistance in the improvement of their model in order to predict the glucose level more accurately. The main goal of this project was to improve the ability of the non-invasive blood glucose monitor to predict the glucose values more precisely. The goal was achieved by finding and implementing the best regression model. The methods included ordinary least squared regression, partial least squares regression, robust regression method, weighted least squares regression, local regression, and ridge regression. VivaScan calibration data for seven patients were analyzed in this project. For each of these patients, the individual regression models were built and compared based on the two factors that evaluate the model prediction ability. It was determined that partial least squares and ridge regressions are two best methods among the others that were considered in this work. Using these two methods gave better glucose prediction. The additional problem of data reduction to minimize the data collection time was also considered in this work.

Doctor of Philosophy / Department of Statistics / Weixing Song / Kernel based non-parametric regression is a popular statistical tool to identify the relationship between response and predictor variables when standard parametric regression models are not appropriate. The efficacy of kernel based methods depend both on the kernel choice and the smoothing parameter. With insufficient smoothing, the resulting regression estimate is too rough and with excessive smoothing, important features of the underlying relationship is lost. While the choice of the kernel has been shown to have less of an effect on the quality of regression estimate, it is important to choose kernels to best match the support set of the underlying predictor variables. In the past few decades, there have been multiple efforts to quantify the properties of asymmetric kernel density and regression estimators. Unlike classic symmetric kernel based estimators, asymmetric kernels do not suffer from boundary problems. For example, Beta kernel estimates are especially suitable for investigating the distribution structure of predictor variables with compact support. In this dissertation, two types of Beta kernel based non parametric regression estimators are proposed and analyzed. First, a Nadaraya-Watson type Beta kernel estimator is introduced within the regression setup followed by a local linear regression estimator based on Beta kernels. For both these regression estimators, a comprehensive analysis of its large sample properties is presented. Specifically, for the first time, the asymptotic normality and the uniform almost sure convergence results for the new estimators are established. Additionally, general guidelines for bandwidth selection is provided. The finite sample performance of the proposed estimator is evaluated via both a simulation study and a real data application. The results presented and validated in this dissertation help advance the understanding and use of Beta kernel based methods in other non-parametric regression applications.

Cancer is a highly prevalent disease worldwide, claiming hundreds of lives each year. In the field of cancer research, it is customary to conduct surveys in which patients are asked to self-report and assess their symptoms and overall health. In such research, it is essential for patients to respond promptly to questionnaires to avoid recall bias and for a representative patient sample to respond to avoid biased sampling. This report aims to investigate the factors that impact response rate and response time using logistic regression and Poisson regression. The study focuses on a dataset of patients with colorectal cancer, with the response rate of patients with pancreatic cancer serving as a reference. By analyzing variables such as gender, age, place of residence, and the method of survey notification, the conclusion is that patients over the age of 80 who received their survey login codes on paper are the least responsive and underrepresented subgroup of the sample. In the analysis of the response time using Poisson regression, the conclusion is that the notification channel has the most significant impact on response rate. / Cancer är en mycket utbredd sjukdom världen över och kräver hundratals liv varje år. Inom cancerforskningen är det vanligt att genomföra undersökningar där patienter ombeds att självrapportera och bedöma sina symtom och övergripande hälsa. I sådana undersökningar är det avgörande att patienterna svarar snabbt på enkäter för att undvika minnesbias och för att få fram en representativ patientgrupp och undvika snedvriden urvalsprocess. Syftet med denna rapport är att undersöka faktorer som påverkar svarsfrekvensen och svarstiden genom att använda logistisk regression och Poisson-regression. Studien fokuserar på en dataset av patienter med tjocktarmscancer, där svarsfrekvensen hos patienter med bukspottkörtelcancer används som referens. Genom att analysera variabler som kön, ålder, bostadsort och metod för undersökningsmeddelande dras slutsatsen att patienter över 80 år som fick sina inloggningskoder på papper är den minst responsiva och mest underrepresenterade undergruppen av urvalet. I analysen av svarstiden med hjälp av Poisson-regression dras slutsatsen att undersökningskanalen har den största påverkan på svarsfrekvensen.

Differential item functioning (DIF) detection rates were examined for the logistic regression and analysis of variance (ANOVA) DIF detection methods. The methods were applied to simulated data sets of varying test length (20, 40, and 60 items) and sample size (200, 400, and 600 examinees) for both equal and unequal underlying ability between groups as well as for both fixed and varying item discrimination parameters. Each test contained 5% uniform DIF items, 5% non-uniform DIF items, and 5% combination DIF (simultaneous uniform and non-uniform DIF) items. The factors were completely crossed, and each experiment was replicated 100 times. For both methods and all DIF types, a test length of 20 was sufficient for satisfactory DIF detection. The detection rate increased significantly with sample size for each method. With the ANOVA DIF method and uniform DIF, there was a difference in detection rates between discrimination parameter types, which favored varying discrimination and decreased with increased sample size. The detection rate of non-uniform DIF using the ANOVA DIF method was higher with fixed discrimination parameters than with varying discrimination parameters when relative underlying ability was unequal. In the combination DIF case, there was a three-way interaction among the experimental factors discrimination type, relative ability, and sample size for both detection methods. The error rate for the ANOVA DIF detection method decreased as test length increased and increased as sample size increased. For both methods, the error rate was slightly higher with varying discrimination parameters than with fixed. For logistic regression, the error rate increased with sample size when relative underlying ability was unequal between groups. The logistic regression method detected uniform and non-uniform DIF at a higher rate than the ANOVA DIF method. Because the type of DIF present in real data is rarely known, the logistic regression method is recommended for most cases.

In microarray analysis, people are interested in those features that have different characters in diseased samples compared to normal samples. The usual p-value method of selecting significant genes either gives too many false positives or cannot detect all the significant features. The False Discovery Rate (FDR) method controls false positives and at the same time selects significant features. We introduced Benjamini's method and Storey's method to control FDR, applied the two methods to human Meningioma data. We found that Benjamini's method is more conservative and that, after the number of the tests exceeds a threshold, increase in number of tests will lead to decrease in number of significant genes. In the second chapter, we investigate ways to search interesting gene expressions that cannot be detected by linear models as t-test or ANOVA. We propose a novel approach to use quadratic logistic regression to detect genes in Meningioma data that have non-linear relationship within phenotypes. By using quadratic logistic regression, we can find genes whose expression correlates to their phenotypes both linearly and quadratically. Whether these genes have clinical significant is a very interesting question, since these genes most likely be neglected by traditional linear approach.