This dissertation compares the parameter estimates obtained from two item response theory (IRT) models: the 1-PL IRT model and the MC1-PL IRT model. Several scenarios were explored in which both unidimensional and multidimensional item-level and personal-level data were used to generate the item responses. The Monte Carlo simulations mirrored the real-life application of the two correlated dimensions of Necessary Operations and Calculations in the basic mathematics domain. In all scenarios, the MC1-PL IRT model showed greater precision in the recovery of the true underlying item difficulty values and person theta values along each primary dimension as well as along a second general order factor. The fit statistics that are generally applied to the 1-PL IRT model were not sensitive to the multidimensional item-level structure, reinforcing the requisite assumption of unidimensionality when applying the 1-PL IRT model.

Fit indices and fit measures commonly used to determine the accuracy and desirability of structural equation models are expected to be insensitive to nonlinearity in the data. This includes measures as ubiquitous as the CFI, TLI, RMSEA, SRMR, AIC, and BIC. Despite this, some software will report these measures when certain models are used. Consequently, some researchers may be led to use these fit measures without realizing the impropriety of the act. Alternative fit measures have been proposed, but these measures require further testing. As part of this thesis, a large simulation study was carried out to investigate alternative fit measures and to confirm whether the traditional measures are practically blind to nonlinearity in the data. The results of the simulation provide conclusive evidence that fit statistics and fit indices based on the chi-square distribution or the residual covariance matrix are entirely insensitive to nonlinearity. The posterior predictive p-value was also insensitive to nonlinearity. Only fit measures based on the structural residuals (i.e., HFI and R-squared) showed any sensitivity to nonlinearity. Of these, the R-squared was the only reliable measure of nonlinear model misspecification. This thesis shows that an effective strategy for determining whether a nonlinear model is preferable to a linear one involves using the R-squared to compare models that have been fit to the same data. An R-squared that is much larger for the nonlinear model than the linear model suggests that the linear model may be less desirable than the nonlinear model. The proposed method is intended to be supplementary to substantive theory. It is argued that any dependence on fit indices or fit statistics that places these measures on a higher pedestal than substantive theory will invariably lead to blindness on the part of the researcher. In other words, unwavering adherence to goodness-of-fit measures limits the researchers vision to what the measures themselves can detect.

<p> The Kolmogorov-Smirnov tests of homogeneity or goodness-of-fit and the binomial group tests for eliminating defectives are of different nature. But they are both popular in applications. The former are widely used in nonparametric comparison, while the later are usually adopted in the group screening problems. When the experimenter has k populations, k-sample statistics should be considered for the testing of homogeneity or goodness-of-fit. On the other hand, when there are k experimenters available for performing group testing on a given population, a k-sample group testing procedure should be used.</p> <p> In this thesis, the distribution functions of k-sample analogues of the Kolmogorov-Smirnov statistics have been found under certain conditions and a k-sample group testing procedure has been defined. This procedure has also been shown to be optimal in the sense that it requires a minimum expected number of k-sample group tests for finding a single defective from a binomial population.</p> <p> Our methods are mainly combinatorial: matrix enumeration, tree counting and construction algorithms are explored as a foundation of our study.</p> / Thesis / Doctor of Philosophy (PhD)

In this thesis, we suggest a goodness-ofit test for semi-parametric copula models. We extended the pseudo in-and-out-sample (PIOS) test proposed in [17], which is based on the PIOS test in [28]. The PIOS test is constructed by comparing the pseudo "in-sample" likelihood and pseudo "out-of-sample" likelihood. Our contribution is twoold. First, we use the approximate test statistics instead of the exact test statistics to alleviate the computational burden of calculating the test statistics. Secondly, we propose a parametric bootstrap procedure to approximate the distribution of the test statistic. Unlike the nonparametric bootstrap which resamples from the original data, the parametric procedure resamples the data from the copula model under the null hypothesis. We conduct simulation studies to investigate the performance of the approximate test statistic and parametric bootstrap. The results show that the parametric bootstrap presents higher test power with a well-controlled type I error compared to the nonparametric bootstrap.

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