The advantages of Multilevel Item Response Theory (MLIRT) model have been studied by several researchers, and even the impact of ignoring a higher level of data structure in multilevel analysis has been studied and discussed. However, due to the technical complexity of modeling and the shortage in function of dealing with multilevel data in traditional IRT packages (e.g., BILOG and PARSCALE), researchers may not be able to analyze the multilevel IRT data accurately. The impact of this type of misspecification, especially for MLIRT models, has not yet been thoughtfully examined. This dissertation consists of two studies: one is a Monte Carlo study that investigates the impact of this type of misspecification and the other one is a study with real-world data to validate the results obtaining from the simulation study. In Study One (the simulation study), we investigate the potential impact of several factors, including: intra-class correlation (ICC), sample size, cluster size and test length, on the parameter estimates and corresponding test of significance under two situations: when the higher level nesting structure is appropriately modeled (i.e., true model condition) versus inappropriately modeled (i.e., misspecified model condition). Three-level straightly hierarchical data (i.e., items are nested within students who are further nested within schools) were generated. Two person-related and school-related covariates were added at the second level (i.e., person-level) and the third level (i.e., school-level), respectively. The results of simulation studies showed that both parameter estimates and their corresponding standard errors would be biased if the higher level nesting structure was ignored. In Study Two, a real data from the Programme for International Student Assessment with purely hierarchical structure were analyzed by comparing parameter estimates when inappropriate versus appropriate IRT models are specified. The findings mirrored the results obtained from the first study. The implication of this dissertation to researchers is that it is important to model the multilevel data structure even in item response theory models. Researchers should interpret their results in caution when ignoring a higher level nesting structure in MLIRT models. What's more, the findings may help researchers determine when MLIRT should be used to get an unbiased result. Limitations concerning about some of the constraints of the simulation study could be relaxed. For instance, although this study used only dichotomous items, the MLIRT could also be used with polytomous items. The test length could be longer and more variability could be introduced into the item parameters’ values.

This thesis shows highly encouraging results as the gain of accuracy reached 18.4% when the pairwise comparisons method was used instead of the direct method for comparing random shapes. The thesis describes a heuristic for generating random but nice shapes, called placated shapes. Random, but visually nice shapes, are often needed for cognitive experiments and processes. These shapes are produced by applying the Gaussian blur to randomly generated polygons. Afterwards, the threshold is set to transform pixels to black and white from di erent shades of gray. This transformation produces placated shapes for easier estimation of areas. Randomly generated placated shapes are used to perform the Monte Carlo method to test the accuracy of cognitive processes by using pairwise comparisons. An on-line questionnaire has been implemented and participants were asked to estimate the areas of ve shapes using a provided unit of measure. They were also asked to compare the shapes in pairs. Such Monte Carlo experiment has never been conducted for 2D case. The received results are of considerable importance.

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A generalized model which describes a family of antiferromagnetic Heisenberg magnets on a three-dimensional stacked triangular lattice is introduced. The model contains a constraint parameter which changes the details of the interactions but not the symmetry of the model. We investigate the question of whether a first or second order phase transition occurs in these systems using a short time dynamics method. This method does not suffer from the problem of critical slowing down which occurs in the usual equilibrium Monte Carlo simulations. The effective critical exponents are determined as a function of the constraint parameter. Our results provide strong evidence that the phase transition is first order. In addition, for a particular value of the constraint parameter, the model corresponds to an antiferromagnet on a stacked Kagome lattice. In this case, our results are not inconsistent with the existence of a finite temperature first order phase transition.

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