Gaussian graphical models (GGM, aka partial correlation networks) have become increasingly popular in psychological science. The underlying rationale is that a group of variables, say, items from a psychometric scale, are thought to measure a dynamic system that mutually influence and interact with one another. This is distinct from customary approaches, such as structural equation models, where observed items are indicators that are thought to share a common cause represented by an unobserved or latent variable. This dissertation includes four chapters that approach (partial correlation) network psychometrics from a frequentist perspective. The first demonstrates that the advantages of regularization have been (substantially) overstated, and further that good old fashioned, non-regularized, regression works just fine for estimating psychological networks. The second builds upon this general notion and demonstrates that, in many cases, null hypothesis significance routinely outperforms "state-of-art" methodology. The first two chapters focused on detected edges, when another supposed advantage of regularization is that it reduces overfitting, especially compared to non-regularized methods. Accordingly, in chapter 3, I set out to demonstrate that, yet again, this is not really the case in situations common to the network literature (i.e., when there are far more observations $n$ than nodes $p$). That is, although The final chapter then employs frequentist reasoning to shed light upon the topic of replicability in networks.

Measurement invariance (MI) is an important assumption in testing group-mean differences. MI describes the conditions where the measurement model of a latent construct is equivalent across different groups. Despite the importance of establishing full MI, it is unrealistic to fulfill all levels of MI (i.e., pattern, weak, strong, strict) in the real world. Alternatively, researchers may choose to fit partially invariant models. Yet, this popular approach requires the exact identification of non-invariance and leaves the question open about which model fitting strategies should be selected. Thus, the current dissertation seeks to answer two related questions via simulation studies: (1) How to use sparse Bayesian estimation methods (i.e., the Horseshoe prior) for detecting non-invariant items; (2) Can we improve the predictive performance of partially invariant models using the horseshoe (HS) prior? The first study aims to demonstrate how to use the HS prior for detecting non-invariant items. I discuss how to approach the identification of non-invariant items as a variable selection problem and I describe how to identify non-invariant items within a Bayesian framework relying on the HS prior used in Bayesian model selection. A simulation study is introduced to investigate the performance of the HS priors in identifying non-invariant items under various conditions. The simulation conditions include sample size, parameter difference, scale length, and item reliability. The results showed that the HS prior approach almost always accurately identified non-invariant items. A large sample size and a high item-reliability can facilitate the identification of non-invariant items even when the amount of non-invariance was small. For identifying invariant items, the HS prior approach exhibited an almost perfect performance. The second study seeks to improve the predictive performance of partially invariant models. Common model solutions for partially invariant models usually focus on explaining the underlying mechanism of psychological phenomena. Current approaches may not generalize well to new and unseen data since MI is only considering the characteristics of the current sample at hand. Recently, the field has seen an increase in the interest in evaluating psychological assessments based on out-of-sample performance. Here, I evaluate an out-of-sample prediction-focused strategy based on partially invariant items. I employed a HS prior model to mimic the idea of Bayesian-Model-Averaging for improving the predictive performance of partially invariant models. The results of a simulation study indicated that the HS prior model outperformed the other commonly used model fitting strategies (i.e., fully constrained model, partially constrained model, freely estimated model) under most conditions. Sample size, parameter differences and item-reliability showed differential impacts on models’ predictive performance. The third study illustrated with an example how to use the HS prior approach for empirical analyses. First, the DERS-9 scale was assessed for item-level MI between genders and between two measurement occasions with a sample of 300 for each group. Next, a partially invariant SEM model was fitted with a sample of 728 college students, where the partially invariant status of the self-esteem scale between genders was confirmed, and then the HS prior model and the freely estimated model were fitted for comparison where peer victimization was regressed on self-esteem between genders. The results indicated that the HS prior performed well in terms of predictions with empirical data. In conclusion, this dissertation discussed and explored potential solutions of two major issues with measurement invariance: non-invariant item detections, and the predictive performance of partially invariant models. The results of two simulation studies indicated that the HS prior approach is a viable alternative to traditional methods for identifying non-invariant items and fitting partially invariant models. Finally, the implications and limitations of this set of studies, along with recommendations for future studies were discussed.

This dissertation is a collection of three papers I wrote during my time in graduate school. Each proposes a novel way in which a Bayesian statistical technique may be applied or conceptualized for the purpose of better aligning statistical hypotheses and research aims, or improving upon the status quo with respect to the application of statistical methods in psychological science. On a personal note, these articles represent endeavors that pushed my intellectual limits and challenged my grit and mettle. The articles are presented as chapters, and in the chronological order in which they were written. I hope this reflects my thought process and growth throughout my time in graduate school. The first chapter presents a framework for integrating exploratory and confirmatory analyses in psychological network research. It is argued that while network analysis has been proposed as a tool for hypothesis generation, there is untapped potential for confirmatory hypothesis testing. We suggest using Bayesian Gaussian graphical models to first generate and then test ordered hypotheses based on the conditional (in)dependence structure of psychological networks.

The second chapter proposes the use of the Bayesian bootstrap method to estimate various correlation coefficients commonly used in the social-behavioral sciences. We demonstrate how the Bayesian bootstrap can be used to estimate Pearson's, Spearman's, Gaussian rank, Kendall's $\tau$, and polychoric correlations. We also describe a method for comparing correlations and evaluating null associations among the estimated correlations.

Finally, in an effort to provide a more nuanced understanding of individual differences than standard approaches, the third chapter explores the spike-and-slab prior distribution for random effect selection in mixed-effects models. Simulation studies were conducted to evaluate the spike-and-slab prior in accurately distinguishing ``average'' and ``non-average'' individuals. The results highlight the spike-and-slab prior's ability to identify individual differences, even in situations with low between-person variance. This dissertation concludes by offering some discussion on why Bayesian analyses are more flexible than standard approaches, and how this flexibility can lead to higher-quality inferences in psychological science.