This dissertation focuses on issues related to fitting an optimal variance-covariance structure in multilevel linear modeling framework with two Monte Carlo simulation studies. In the first study, the author evaluated the performance of common fit statistics such as Likelihood Ratio Test (LRT), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC) and a new proposed method, standardized root mean square residual (SRMR), for selecting the correct within-subject covariance structure. Results from the simulated data suggested SRMR had the best performance in selecting the optimal covariance structure. A pharmaceutical example was also used to evaluate the performance of these fit statistics empirically. The LRT failed to decide which is a better model because LRT can only be used for nested models. SRMR, on the other hand, had congruent result as AIC and BIC and chose ARMA(1,1) as the optimal variance-covariance structure. In the second study, the author adopted a first-order autoregressive structure as the true within-subject V-C structure with variability in the intercept and slope (estimating [tau]00 and [tau]11 only) and investigated the consequence of misspecifying different levels/types of the V-C matrices simultaneously on the estimation and test of significance for the growth/fixed-effect and random-effect parameters, considering the size of the autoregressive parameter, magnitude of the fixed effect parameters, number of cases, and number of waves. The result of the simulation study showed that the commonly-used identity within-subject structure with unstructured between-subject matrix performed equally well as the true model in the evaluation of the criterion variables. On the other hand, other misspecified conditions, such as Under G & Over R conditions and Generally misspecified G & R conditions had biased standard error estimates for the fixed effect and lead to inflated Type I error rate or lowered statistical power. The two studies bridged the gap between the theory and practical application in the current literature. More research can be done to test the effectiveness of proposed SRMR in searching for the optimal V-C structure under different conditions and evaluate the impact of different types/levels of misspecification with various specifications of the within- and between- level V-C structures simultaneously.

Magister Artium (Psychology) - MA(Psych) / Globally the Subjective Well-Being (SWB) of children is recognized as having a significant effect on the child's psychological and social functioning. Furthermore, not only does children's SWB have effects on childhood well-being research, it has also increased the knowledge of how children view their life that has been determined through the measurement of specific domains that relates to children's lives. The overall aim of this study was to ascertain the SWB of children across three age groups in the Western Cape. Within this process, the study further aimed to fit the structural model depicting the nature of the relationship between global, domain specific and overall life satisfaction across three age groups. The Theory of Model Fit: Goodness of Fit and Fit Indexes was used as the theoretical position conceptualising the study. The sample included 3236 children aged 8, 10, and 12 years selected using stratified random sampling from 29 schools in the Western Cape. The study used Structural Equation Modelling and Multi-group Confirmatory Factor Analysis to address the stated aims and objectives. Ethics principles of informed consent, anonymity, the right to withdraw and privacy were adhered to within the study. Findings of this study indicate that the descriptive statistics depicted high levels of SWB for both measures with mean composite scores ranging between 81.20 to 86.15 for the SLSS; and 83.29 to 84.07 for the PWI-SC. Confirmatory factor analysis showed excellent fit for both the SLSS and the PWI-SC across age groups (multi-group model). The application of Multi-group Confirmatory Factor Analysis in the current study found the measures to be comparable across the three age groups (8, 10 & 12) for the SLSS and two age groups for the PWI-SC (10 & 12). A combined model with two latent constructs, representing different levels of abstraction was also tested. An excellent fit was obtained for this combined model. Appropriate fit statistics was obtained for the overall pooled sample. The standardised regression weights of 0.57 for the PWI-SC and 0.47 for the SLSS point to adequate loadings of the latent constructs onto the OLS. Markedly, it was found that a significant overall mean difference was found between the 10 and 12-year olds and not between the 8 and 10-year olds; while for the domain-specific PWI-SC a similar tendency was noted across the 10 and 12-year olds participants (8 year old group was not applicable in this analysis). / National Research Foundation (NRF)

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.

As item response theory (IRT) has developed and is widely applied, investigating the fit of a parametric model becomes an important part of the measurement process when implementing IRT. The usefulness and successes of IRT applications rely heavily on the extent to which the model reflects the data, so it is necessary to evaluate model-data fit by gathering sufficient evidence before any model application. There is a lack of promising solutions on the detection of model misfit in IRT. In addition, commonly used fit statistics are not satisfactory in that they often do not possess desirable statistical properties and lack a means of examining the magnitude of misfit (e.g., via graphical inspections). In this dissertation, a newly-proposed nonparametric approach, RISE was thoroughly and comprehensively studied. Specifically, the purposes of this study are to (a) examine the promising fit procedure, RISE, (b) compare the statistical properties of RISE with that of the commonly used goodness-of-fit procedures, and (c) investigate how RISE may be used to examine the consequences of model misfit. To reach the above-mentioned goals, both a simulation study and empirical study were conducted. In the simulation study, four factors including ability distribution, sample size, test length and model were varied as the factors which may influence the performance of a fit statistic. The results demonstrated that RISE outperformed G2 and S-X2 in that it controlled Type I error rates and provided adequate power under all conditions. In the empirical study, the three fit statistics were applied to one empirical data and the misfitting items were flagged. RISE and S-X2 detected reasonable numbers of misfitting items while G2 detected almost all items when sample size is large. To further demonstrate an advantage of RISE, the residual plot on each misfitting item was shown. Compared to G2 and S-X2, RISE gave a much clearer picture of the location and magnitude of misfit for each misfitting item. Other than statistical properties and graphical displays, the score distribution and test characteristic curve (TCC) were investigated as model misfit consequence. The results indicated that for the given data, there was no practical consequence on classification before and after replacement of misfitting items detected by three fit statistics.

Under item response theory, three types of limited information goodness-of-fit test statistics – M2, Mord, and C2 – have been proposed to assess model-data fit when data are sparse. However, the evaluation of the performance of these GOF statistics under multidimensional item response theory (MIRT) models with polytomous data is limited. The current study showed that M2 and C2 were well-calibrated under true model conditions and were powerful under misspecified model conditions. Mord were not well-calibrated when the number of response categories was more than three. RMSEA2 and RMSEAC2 are good tools to evaluate approximate fit. The second study aimed to evaluate the psychometric properties of the Religious Commitment Inventory-10 (RCI-10; Worthington et al., 2003) within the IRT framework and estimate C2 and its RMSEA to assess global model-fit. Results showed that the RCI-10 was best represented by a bifactor model. The scores from the RCI-10 could be scored as unidimensional notwithstanding the presence of multidimensionality. Two-factor correlational solution should not be used. Study two also showed that religious commitment is a risk factor of intimate partner violence, whereas spirituality was a protecting factor from the violence. More alcohol was related with more abusive behaviors. Implications of the two studies were discussed.

The global loss of biodiversity calls for robust large-scale diversity assessment. Biological diversity is a multi-faceted concept; defined as the “variety of life”, answering questions such as “How much is there?” or more precisely “Have we succeeded in reducing the rate of its decline?” is not straightforward. While various aspects of biodiversity give rise to numerous ways of quantification, we focus on temporal (and spatial) trends and their changes in species diversity. Traditional diversity indices summarise information contained in the species abundance distribution, i.e. each species' proportional contribution to total abundance. Estimated from data, these indices can be biased if variation in detection probability is ignored. We discuss differences between diversity indices and demonstrate possible adjustments for detectability. Additionally, most indices focus on the most abundant species in ecological communities. We introduce a new set of diversity measures, based on a family of goodness-of-fit statistics. A function of a free parameter, this family allows us to vary the sensitivity of these measures to dominance and rarity of species. Their performance is studied by assessing temporal trends in diversity for five communities of British breeding birds based on 14 years of survey data, where they are applied alongside the current headline index, a geometric mean of relative abundances. Revealing the contributions of both rare and common species to biodiversity trends, these "goodness-of-fit" measures provide novel insights into how ecological communities change over time. Biodiversity is not only subject to temporal changes, but it also varies across space. We take first steps towards estimating spatial diversity trends. Finally, processes maintaining biodiversity act locally, at specific spatial scales. Contrary to abundance-based summary statistics, spatial characteristics of ecological communities may distinguish these processes. We suggest a generalisation to a spatial summary, the cross-pair overlap distribution, to render it more flexible to spatial scale.