Human microbial research has become increasingly popular in biomedical areas due to the importance of role of human microbiome in human health. One purpose of studying human microbiome is to detect differentially abundant features from a limited group of subjects across biological conditions. Metagenomic analyses of the human microbial communities are extensively used for biomedical applications due to its reliable and evident comparative discoveries across more than one metagenomes when multiple communities are taken into consideration. Next-generation sequencing technology helps to detect taxonomic compositions of specific features/species contained in human microbial communities. Statistical analysis often starts by generating the Operational Taxonomic Units (OTUs) using taxonomic compositions to classify groups of closely associated human microbiomes. Oftentimes, the counts of features are observed as matched count data with excess zeros. Such data lead some differential abundance analysis methods to apply Zero-Inflated Poisson (ZIP) or Zero-Inflated Negative Binomial (ZINB) regression for modeling the microbial abundance. However, over-dispersion as well as within-subject variation and correlation of matched count data render the standard ZIP and ZINB regression inadequate. To account for the inherent within-subject variation and correlation, independent random effect terms are commonly included in the regressions. Therefore, a robust method that accounts the effect of matched samples and correlated random effects while considering over-dispersion and excess zeros of count data is need for statistical analysis. In this paper, a statistical method, the two-part correlated ZINB model with correlated random effects (cZINB), is proposed for testing the matched samples with repeated measurements.

This dissertation investigates the interaction of population invariance, equating assumptions, and equating accuracy with group differences. In addition, matched samples equating methods are considered as a possible way to improve equating accuracy with large group differences. Data from one administration of four mixed-format Advanced Placement (AP) Exams were used to create pseudo old and new forms sharing common items. Population invariance analyses were conducted based on levels of examinee parental education using a single group equating design. Old and new form groups with common item effect sizes (ESs) ranging from 0 to 0.75 were created by sampling examinees based on their level of parental education. Equating was conducted for four common item nonequivalent group design equating methods: frequency estimation, chained equipercentile, IRT true score, and IRT observed score. Additionally, groups with ESs greater than zero were matched using three different matching techniques including exact matching on parental education level and propensity score matching with several other background variables. The accuracy of equating results was evaluated by comparing each equating relationship with an ES greater than zero to the equating relationship where the ES equaled zero. Differences between comparison and criterion equating relationships were quantified using the root expected mean squared difference (REMSD) statistic, classification consistency, and standard errors of equating (SEs). The accuracy of equating results and the adequacy of equating assumptions was compared for unmatched and matched samples. As ES increased, equating results tended to become less accurate and less consistent across equating methods. However, there was relatively little population dependence of equating results, despite large subgroup performance differences. Large differences between criterion and comparison equating relationships appeared to be caused instead by violations of equating assumptions. As group differences increased, the degree to which frequency estimation and chained equipercentile assumptions held decreased. In addition, all four AP Exams showed some evidence of multidimensionality. Because old and new form groups were selected to differ in terms of their respective levels of parental education, the matching methods that included parental education appeared to improve equating accuracy and the degree to which equating assumptions held, at least for very large ESs.