A general approach for conducting power analysis in two-and three-level hierarchical linear models (HLMs) is described. The method can be used to perform power analysis to detect fixed effects at any level of a HLM with dichotomous or continuous covariates. It can easily be extended to perform power analysis for functions of parameters. Important steps in the derivation of this approach are illustrated and numerical examples are provided. Sample code implementing this approach is provided using the free program R.

No study has focussed on the longitudinal effect of APOE-ε4 genotype on the logical memory delayed recall total (LDELTOTAL) score in late-onset Alzheimer’s disease (AD). The LDELTOTAL scores were collected at baseline, 12, 24, 36 and 48 months from 382 participants with AD, 503 with cognitive normal (CN), 1293 with mild cognitive impairment (MCI) in the Alzheimer's Disease Neuroimaging Initiative (ADNI). A linear mixed model (LMM) was used to investigate the effect of APOE-ε4 on the longitudinal changes in the LDELTOTAL scores adjusted for age, gender, education and baseline Mini Mental State Examination score. There were significant differences in LDELTOTAL scores among AD, CN, and MCI (P < 0.0001) and among APOE-ε4 alleles at baseline (P < 0.0001). In the multivariable LMM, elders with 75+ years (P = 0.0051), females (P < 0.0001), lower education (P < 0.0001), AD and MCI (both P values < 0.0001) were associated with decreased LDELTOTAL values, while the individuals with 1 or 2 APOE-ε4 allele revealed significantly lower LDELTOTAL scores (both P values <0.0001) compared with individuals without APOE-ε4 allele. Further, APOE-ε4 alleles had significant interactions with four follow-up visits, where all follow-up visits showed significantly higher LDELTOTAL score. In addition, gender showed interaction with age, education and APOE-ε4 with follow-up visits. Our findings provide the first evidence of the effect of APOE-ε4 genotype on the logical memory declines related to AD. Further, APOE-ε4 alleles showed interactions with gender and follow-up visits.

Objective: Screening for colorectal cancer (CRC) can reduce disease incidence, morbidity, and mortality. However, few studies have investigated the urban-rural differences in social and behavioral factors influencing CRC screening. The objective of the study was to investigate the potential factors across urban-rural groups on the usage of CRC screening. Methods: A total of 38,505 adults (aged ≥40 years) were selected from the 2009 California Health Interview Survey (CHIS) data - the latest CHIS data on CRC screening. The weighted generalized linear mixed-model (WGLIMM) was used to deal with this hierarchical structure data. Weighted simple and multiple mixed logistic regression analyses in SAS ver. 9.4 were used to obtain the odds ratios (ORs) and their 95% confidence intervals (CIs). Results: The overall prevalence of CRC screening was 48.1% while the prevalence in four residence groups - urban, second city, suburban, and town/rural, were 45.8%, 46.9%, 53.7% and 50.1%, respectively. The results of WGLIMM analysis showed that there was residence effect (p < 0.0001) and residence groups had significant interactions with gender, age group, education level, and employment status (p < 0.05). Multiple logistic regression analysis revealed that age, race, marital status, education level, employment stats, binge drinking, and smoking status were associated with CRC screening (p < 0.05). Stratified by residence regions, age and poverty level showed associations with CRC screening in all four residence groups. Education level was positively associated with CRC screening in second city and suburban. Infrequent binge drinking was associated with CRC screening in urban and suburban; while current smoking was a protective factor in urban and town/rural groups. Conclusions: Mixed models are useful to deal with the clustered survey data. Social factors and behavioral factors (binge drinking and smoking) were associated with CRC screening and the associations were affected by living areas such as urban and rural regions.

Linear mixed model (LMM) has the advantage of modeling the corelated data. Alzheimer’s disease (AD) is a chronic neurogenerative disease that affects the brain of the subject. No study was found to study the longitudinal effect of apolipoprotein E epsilon 4 (APOE4) genotype on the logical memory test total score in AD. A longitudinal data of 844 with AD, 2167 with cognitive normal (CN), and 4472 with mild cognitive impairment (MCI) participants who underwent logical memory examination test in the Alzheimer's Disease Neuroimaging Initiative (ADNI) were investigated. Episodic memory of the study participants was monitored based on a short story told to the participants and then participants asked to recall what was told. The multivariate LMM was used to determine the longitudinal changes in the logical memory test total score adjusting for age and sex. The Akaike information criterion (AIC) statistic and the Bayesian information criterion (BIC) statistic were used to select the best covariance structure. The repeated measures longitudinal analysis was performed using PROC MIXED in SAS 9.4. Both AIC and BIC statistics favor the unstructured correlated structure (UN). Using a UN model in the LMM, the APOE gene was is significantly associated with logical memory test total score (pUN covariance structure is the best. This study provided the first evidence of the effect of APOE4 genotype on the logical memory related to AD.

The linear mixed model is widely implemented in the analysis of longitudinal data. Inference techniques and information criteria are available and well-studied for goodness-of-fit within the linear mixed model setting. Predictive approaches such as R-squared, PRESS, and CCC are available for the linear mixed model but require more research (Edward, 2005). This project used simulation to investigate the performance of R-squared, PRESS, CCC, Pseudo F-test and information criterion for goodness-of-fit within the linear mixed model framework. Marginal and conditional approaches for these predictive statistics were studied under different variance-covariance structures. For compound symmetry structure, the success rates for all 17 statistics (marginal and conditional R-squared, PRESS, CCC, F test, AIC and BIC) were high. The study suggested using marginal rather than conditional residuals for PRESS, CCC and R-squared. It suggested using REML likelihood function which has the determinant term for AIC and BIC. For CCC, R-squared, and the information criterion, there was no difference for the various parameter number adjustments. For autoregressive order 1 plus random effect, the study suggested using conditional residuals for PRESS, marginal residuals for CCC and R-squared, and using REML function with the determinant term for AIC and BIC. Also there was no difference for the different parameter number adjustments. The F-test performed well for all covariance structures. The study also indicated that characteristics of the data, such as the covariance structure, parameter values, and sample size, can greatly impact performance of various statistics. No one criterion is consistently better than the others in terms of selection performance in the simulation study.

Chemotherapy is a method used to treat cancer but it has a number of side-effects. Research conducted by the Department of Chemical Engineering at BYU involves a new method of administering chemotherapy using ultrasound waves and water-soluble capsules. The goal is to reduce the side-effects by localizing the delivery of the medication. As part of this research, a two-factor experiment was conducted on rats to test if the water-soluble capsules and ultrasound waves by themselves have an effect on tumor growth or patient survival. Our project emphasizes the usage of Bayesian Hierarchical Models and Win-BUGS to jointly model the survival data and the longitudinal data—mass. The results of the joint analysis indicate that the use of ultrasound and water-soluble microcapsules have no negative effect on survival. In fact, there appears to be a positive effect on the survival since the rats in the ultrasound-capsule group had higher survival rates than the rats in other treatment groups. From these results, it does appear that the new technology involving ultrasound waves and microcapsules is a promising way to reduce the side-effects of chemotherapy. It is strongly advocated that the formulation of a joint model for any longitudinal and survival data be performed. For future work for the ultrasound-microcapsule data it is recommended that joint modeling of the mass, tumor volume, and survival data be conducted to obtain additional information.

Spline smoothers as implemented in common mixed model software provide a familiar framework for estimating semi-parametric and non-parametric models. Following a review of literature on splines and mixed models, details for implementing mixed model splines are presented. The examples use an experiment in the health sciences to demonstrate how to use mixed models to generate the smoothers. The first example takes a simple one-group case, while the second example fits an expanded model using three groups simultaneously. The second example also demonstrates how to fit confidence bands to the three-group model. The examples use mixed model software as implemented in lme() in R. Following the examples a discussion of the method is presented.

The purpose of the study was to examine the relationship between media as family leisure and family functioning among families with at least one adolescent child. Specifically, this study examined the relationship between family functioning and media use, media connection, and media monitoring over time. Furthermore, because the data were nested in families, and because most family leisure research has been limited to individual-level analyses, this study incorporated mixed modeling into its analysis which accounted for family-level and individual-level variance. The sample consisted of 500 families participating in the Flourishing Families (FFP) Project, a longitudinal study of inner-family life involving families with a child between the ages of 11 and 16. Multiple regression analysis indicated there was a significant negative relationship between media use and family functioning. Mixed model analysis further indicated there was a significant positive relationship between media connection, parental media monitoring, and family functioning, and this relationship was stable over time. These relationships were significant even when accounting for the variance explained by depression, anxiety, conflict, and other demographic variables. Findings support existing media effects and family leisure research. This research, however, goes beyond existing research in its mixed level analysis that accounted for family-level variance and in its analysis of time in the stability of the relationship between media variables and family functioning. Findings further suggest the importance in parental involvement in adolescent media use when explaining variance in family functioning.

Due to recent advances in technology that facilitate acquisition of multi-parameter defined phenotypes, new opportunities have arisen for predicting patient outcomes based on individual specific cell subset changes. The data resulting from these trials can be a challenge to analyze, as predictors may be highly correlated with each other or related to outcome within levels of other predictor variables. As a result, applying traditional methods like simple linear models and univariate approaches such as odds ratios may be insufficient. In this dissertation, we describe potential solutions including tree-based methods, ridge regression, mixed modeling, and a new estimator called a mixed ridge estimator with expectation-maximization (EM) algorithm. Data examples are provided. In particular, flow cytometry is a method of measuring a large number of particle counts at once by suspending them in a fluid and shining a beam of light onto the fluid. This is specifically relevant in the context of studying human immunodeficiency virus (HIV), where there exists a great potential to draw from the rich array of data on host cell-mediated response to infection and drug exposures, to inform and discover patient level determinants of disease progression and/or response to anti-retroviral therapy (ART). The data sets collected are often high dimensional with correlated columns, which can be challenging to analyze. We demonstrate the application and comparative interpretations of three tree-based algorithms for the analysis of data arising from flow cytometry in the first chapter of this manuscript. Specifically, we consider the question of what best predicts CD4 T-cell recovery in HIV-1 infected persons starting antiretroviral therapy with CD4 count between 200-350 cell/μl. The tree-based approaches, namely, classification and regression trees (CART), random forests (RF) and logic regression (LR), were designed specifically to uncover complex structure in high dimensional data settings. While contingency table analysis and RFs provide information on the importance of each potential predictor variable, CART and LR offer additional insight into the combinations of variables that together are predictive of the outcome. Specifically, application of tree-based methods to our data suggest that a combination of baseline immune activation states, with emphasis on CD8 T cell activation, may be a better predictor than any single T cell/innate cell subset analyzed. In the following chapter, tree-based methods are compared to each other via a simulation study. Each has its merits in particular circumstances; for example, RF is able to identify the order of importance of predictors regardless of whether there is a tree-like structure. It is able to adjust for correlation among predictors by using a machine learning algorithm, analyzing subsets of predictors and subjects over a number of iterations. CART is useful when variables are predictive of outcome within levels of other variables, and is able to find the most parsimonious model using pruning. LR also identifies structure within the set of predictor variables, and nicely illustrates relationship among variables. However, due to the vast number of combinations of predictor variables that would need to be analyzed in order to find the single best LR tree, an algorithm is used that only searches a subset of potential combinations of predictors. Therefore, results may be different each time the algorithm is used on the same data set. Next we use a regression approach to analyzing data with correlated predictors. Ridge regression is a method of accounting for correlated data by adding a shrinkage component to the estimators for a linear model. We perform a simulation study to compare ridge regression to linear regression over various correlation coefficients and find that ridge regression outperforms linear regression as correlation increases. To account for collinearity among the predictors along with longitudinal data, a new estimator that combines the applicability of ridge regression and mixed models using an EM algorithm is developed and compared to the mixed model. We find from a simulation study comparing our mixed ridge (MR) approach with a traditional mixed model that our new mixed ridge estimator is able to handle collinearity of predictor variables better than the mixed model, while accounting for random within-subject effects that regular ridge regression does not take into account. As correlation among predictors increases, power decreases more quickly for the mixed model than MR. Additionally, type I error rate is not significantly elevated when the MR approach is taken. The MR estimator gives us new insight into flow cytometry data and other data sets with correlated predictor variables that our tree-based methods could not give us. These methods all provide unique insight into our data that more traditional methods of analysis do not offer.

The thesis addresses the mixed model assembly line sequencing and outbound logistics planning problems in the automotive industry at the operational level. Different from the sequential decision-making procedure used in practice, the thesis proposes a scheme that integrates production sequencing and logistics planning. Mixed integer programs are established for the production sequencing, logistics planning, and integrated problems. The integrated model cannot be solved by commercial solvers in a reasonable amount of time. After studying the optimality properties of the product mode, the thesis proposes a modified integrated model. The results of numerical experiments and simulations demonstrate the benefit of the integration by comparing the modified integrated model with two sequential schemes, the Production-First-Scheme and the Logistics-First-Scheme.