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- All Subjects: Statistics
- Creators: Levy, Roy
- Creators: Kamarianakis, Ioannis
Description
Dimensionality assessment is an important component of evaluating item response data. Existing approaches to evaluating common assumptions of unidimensionality, such as DIMTEST (Nandakumar & Stout, 1993; Stout, 1987; Stout, Froelich, & Gao, 2001), have been shown to work well under large-scale assessment conditions (e.g., large sample sizes and item pools; see e.g., Froelich & Habing, 2007). It remains to be seen how such procedures perform in the context of small-scale assessments characterized by relatively small sample sizes and/or short tests. The fact that some procedures come with minimum allowable values for characteristics of the data, such as the number of items, may even render them unusable for some small-scale assessments. Other measures designed to assess dimensionality do not come with such limitations and, as such, may perform better under conditions that do not lend themselves to evaluation via statistics that rely on asymptotic theory. The current work aimed to evaluate the performance of one such metric, the standardized generalized dimensionality discrepancy measure (SGDDM; Levy & Svetina, 2011; Levy, Xu, Yel, & Svetina, 2012), under both large- and small-scale testing conditions. A Monte Carlo study was conducted to compare the performance of DIMTEST and the SGDDM statistic in terms of evaluating assumptions of unidimensionality in item response data under a variety of conditions, with an emphasis on the examination of these procedures in small-scale assessments. Similar to previous research, increases in either test length or sample size resulted in increased power. The DIMTEST procedure appeared to be a conservative test of the null hypothesis of unidimensionality. The SGDDM statistic exhibited rejection rates near the nominal rate of .05 under unidimensional conditions, though the reliability of these results may have been less than optimal due to high sampling variability resulting from a relatively limited number of replications. Power values were at or near 1.0 for many of the multidimensional conditions. It was only when the sample size was reduced to N = 100 that the two approaches diverged in performance. Results suggested that both procedures may be appropriate for sample sizes as low as N = 250 and tests as short as J = 12 (SGDDM) or J = 19 (DIMTEST). When used as a diagnostic tool, SGDDM may be appropriate with as few as N = 100 cases combined with J = 12 items. The study was somewhat limited in that it did not include any complex factorial designs, nor were the strength of item discrimination parameters or correlation between factors manipulated. It is recommended that further research be conducted with the inclusion of these factors, as well as an increase in the number of replications when using the SGDDM procedure.
ContributorsReichenberg, Ray E (Author) / Levy, Roy (Thesis advisor) / Thompson, Marilyn S. (Thesis advisor) / Green, Samuel B. (Committee member) / Arizona State University (Publisher)
Created2013
Description
This simulation study compared the utility of various discrepancy measures within a posterior predictive model checking (PPMC) framework for detecting different types of data-model misfit in multidimensional Bayesian network (BN) models. The investigated conditions were motivated by an applied research program utilizing an operational complex performance assessment within a digital-simulation educational context grounded in theories of cognition and learning. BN models were manipulated along two factors: latent variable dependency structure and number of latent classes. Distributions of posterior predicted p-values (PPP-values) served as the primary outcome measure and were summarized in graphical presentations, by median values across replications, and by proportions of replications in which the PPP-values were extreme. An effect size measure for PPMC was introduced as a supplemental numerical summary to the PPP-value. Consistent with previous PPMC research, all investigated fit functions tended to perform conservatively, but Standardized Generalized Dimensionality Discrepancy Measure (SGDDM), Yen's Q3, and Hierarchy Consistency Index (HCI) only mildly so. Adequate power to detect at least some types of misfit was demonstrated by SGDDM, Q3, HCI, Item Consistency Index (ICI), and to a lesser extent Deviance, while proportion correct (PC), a chi-square-type item-fit measure, Ranked Probability Score (RPS), and Good's Logarithmic Scale (GLS) were powerless across all investigated factors. Bivariate SGDDM and Q3 were found to provide powerful and detailed feedback for all investigated types of misfit.
ContributorsCrawford, Aaron (Author) / Levy, Roy (Thesis advisor) / Green, Samuel (Committee member) / Thompson, Marilyn (Committee member) / Arizona State University (Publisher)
Created2014
Description
Missing data are common in psychology research and can lead to bias and reduced power if not properly handled. Multiple imputation is a state-of-the-art missing data method recommended by methodologists. Multiple imputation methods can generally be divided into two broad categories: joint model (JM) imputation and fully conditional specification (FCS) imputation. JM draws missing values simultaneously for all incomplete variables using a multivariate distribution (e.g., multivariate normal). FCS, on the other hand, imputes variables one at a time, drawing missing values from a series of univariate distributions. In the single-level context, these two approaches have been shown to be equivalent with multivariate normal data. However, less is known about the similarities and differences of these two approaches with multilevel data, and the methodological literature provides no insight into the situations under which the approaches would produce identical results. This document examined five multilevel multiple imputation approaches (three JM methods and two FCS methods) that have been proposed in the literature. An analytic section shows that only two of the methods (one JM method and one FCS method) used imputation models equivalent to a two-level joint population model that contained random intercepts and different associations across levels. The other three methods employed imputation models that differed from the population model primarily in their ability to preserve distinct level-1 and level-2 covariances. I verified the analytic work with computer simulations, and the simulation results also showed that imputation models that failed to preserve level-specific covariances produced biased estimates. The studies also highlighted conditions that exacerbated the amount of bias produced (e.g., bias was greater for conditions with small cluster sizes). The analytic work and simulations lead to a number of practical recommendations for researchers.
ContributorsMistler, Stephen (Author) / Enders, Craig K. (Thesis advisor) / Aiken, Leona (Committee member) / Levy, Roy (Committee member) / West, Stephen G. (Committee member) / Arizona State University (Publisher)
Created2015
Description
Many methodological approaches have been utilized to predict student retention and persistence over the years, yet few have utilized a Bayesian framework. It is believed this is due in part to the absence of an established process for guiding educational researchers reared in a frequentist perspective into the realms of Bayesian analysis and educational data mining. The current study aimed to address this by providing a model-building process for developing a Bayesian network (BN) that leveraged educational data mining, Bayesian analysis, and traditional iterative model-building techniques in order to predict whether community college students will stop out at the completion of each of their first six terms. The study utilized exploratory and confirmatory techniques to reduce an initial pool of more than 50 potential predictor variables to a parsimonious final BN with only four predictor variables. The average in-sample classification accuracy rate for the model was 80% (Cohen's κ = 53%). The model was shown to be generalizable across samples with an average out-of-sample classification accuracy rate of 78% (Cohen's κ = 49%). The classification rates for the BN were also found to be superior to the classification rates produced by an analog frequentist discrete-time survival analysis model.
ContributorsArcuria, Philip (Author) / Levy, Roy (Thesis advisor) / Green, Samuel B (Committee member) / Thompson, Marilyn S (Committee member) / Arizona State University (Publisher)
Created2015
Description
Though the likelihood is a useful tool for obtaining estimates of regression parameters, it is not readily available in the fit of hierarchical binary data models. The correlated observations negate the opportunity to have a joint likelihood when fitting hierarchical logistic regression models. Through conditional likelihood, inferences for the regression and covariance parameters as well as the intraclass correlation coefficients are usually obtained. In those cases, I have resorted to use of Laplace approximation and large sample theory approach for point and interval estimates such as Wald-type confidence intervals and profile likelihood confidence intervals. These methods rely on distributional assumptions and large sample theory. However, when dealing with small hierarchical datasets they often result in severe bias or non-convergence. I present a generalized quasi-likelihood approach and a generalized method of moments approach; both do not rely on any distributional assumptions but only moments of response. As an alternative to the typical large sample theory approach, I present bootstrapping hierarchical logistic regression models which provides more accurate interval estimates for small binary hierarchical data. These models substitute computations as an alternative to the traditional Wald-type and profile likelihood confidence intervals. I use a latent variable approach with a new split bootstrap method for estimating intraclass correlation coefficients when analyzing binary data obtained from a three-level hierarchical structure. It is especially useful with small sample size and easily expanded to multilevel. Comparisons are made to existing approaches through both theoretical justification and simulation studies. Further, I demonstrate my findings through an analysis of three numerical examples, one based on cancer in remission data, one related to the China’s antibiotic abuse study, and a third related to teacher effectiveness in schools from a state of southwest US.
ContributorsWang, Bei (Author) / Wilson, Jeffrey R (Thesis advisor) / Kamarianakis, Ioannis (Committee member) / Reiser, Mark R. (Committee member) / St Louis, Robert (Committee member) / Zheng, Yi (Committee member) / Arizona State University (Publisher)
Created2017
Description
Correlation is common in many types of data, including those collected through longitudinal studies or in a hierarchical structure. In the case of clustering, or repeated measurements, there is inherent correlation between observations within the same group, or between observations obtained on the same subject. Longitudinal studies also introduce association between the covariates and the outcomes across time. When multiple outcomes are of interest, association may exist between the various models. These correlations can lead to issues in model fitting and inference if not properly accounted for. This dissertation presents three papers discussing appropriate methods to properly consider different types of association. The first paper introduces an ANOVA based measure of intraclass correlation for three level hierarchical data with binary outcomes, and corresponding properties. This measure is useful for evaluating when the correlation due to clustering warrants a more complex model. This measure is used to investigate AIDS knowledge in a clustered study conducted in Bangladesh. The second paper develops the Partitioned generalized method of moments (Partitioned GMM) model for longitudinal studies. This model utilizes valid moment conditions to separately estimate the varying effects of each time-dependent covariate on the outcome over time using multiple coefficients. The model is fit to data from the National Longitudinal Study of Adolescent to Adult Health (Add Health) to investigate risk factors of childhood obesity. In the third paper, the Partitioned GMM model is extended to jointly estimate regression models for multiple outcomes of interest. Thus, this approach takes into account both the correlation between the multivariate outcomes, as well as the correlation due to time-dependency in longitudinal studies. The model utilizes an expanded weight matrix and objective function composed of valid moment conditions to simultaneously estimate optimal regression coefficients. This approach is applied to Add Health data to simultaneously study drivers of outcomes including smoking, social alcohol usage, and obesity in children.
ContributorsIrimata, Kyle (Author) / Wilson, Jeffrey R (Thesis advisor) / Broatch, Jennifer (Committee member) / Kamarianakis, Ioannis (Committee member) / Kao, Ming-Hung (Committee member) / Reiser, Mark R. (Committee member) / Arizona State University (Publisher)
Created2018
Description
Generalized Linear Models (GLMs) are widely used for modeling responses with non-normal error distributions. When the values of the covariates in such models are controllable, finding an optimal (or at least efficient) design could greatly facilitate the work of collecting and analyzing data. In fact, many theoretical results are obtained on a case-by-case basis, while in other situations, researchers also rely heavily on computational tools for design selection.
Three topics are investigated in this dissertation with each one focusing on one type of GLMs. Topic I considers GLMs with factorial effects and one continuous covariate. Factors can have interactions among each other and there is no restriction on the possible values of the continuous covariate. The locally D-optimal design structures for such models are identified and results for obtaining smaller optimal designs using orthogonal arrays (OAs) are presented. Topic II considers GLMs with multiple covariates under the assumptions that all but one covariate are bounded within specified intervals and interaction effects among those bounded covariates may also exist. An explicit formula for D-optimal designs is derived and OA-based smaller D-optimal designs for models with one or two two-factor interactions are also constructed. Topic III considers multiple-covariate logistic models. All covariates are nonnegative and there is no interaction among them. Two types of D-optimal design structures are identified and their global D-optimality is proved using the celebrated equivalence theorem.
Three topics are investigated in this dissertation with each one focusing on one type of GLMs. Topic I considers GLMs with factorial effects and one continuous covariate. Factors can have interactions among each other and there is no restriction on the possible values of the continuous covariate. The locally D-optimal design structures for such models are identified and results for obtaining smaller optimal designs using orthogonal arrays (OAs) are presented. Topic II considers GLMs with multiple covariates under the assumptions that all but one covariate are bounded within specified intervals and interaction effects among those bounded covariates may also exist. An explicit formula for D-optimal designs is derived and OA-based smaller D-optimal designs for models with one or two two-factor interactions are also constructed. Topic III considers multiple-covariate logistic models. All covariates are nonnegative and there is no interaction among them. Two types of D-optimal design structures are identified and their global D-optimality is proved using the celebrated equivalence theorem.
ContributorsWang, Zhongsheng (Author) / Stufken, John (Thesis advisor) / Kamarianakis, Ioannis (Committee member) / Kao, Ming-Hung (Committee member) / Reiser, Mark R. (Committee member) / Zheng, Yi (Committee member) / Arizona State University (Publisher)
Created2018
Description
The Pearson and likelihood ratio statistics are well-known in goodness-of-fit testing and are commonly used for models applied to multinomial count data. When data are from a table formed by the cross-classification of a large number of variables, these goodness-of-fit statistics may have lower power and inaccurate Type I error rate due to sparseness. Pearson's statistic can be decomposed into orthogonal components associated with the marginal distributions of observed variables, and an omnibus fit statistic can be obtained as a sum of these components. When the statistic is a sum of components for lower-order marginals, it has good performance for Type I error rate and statistical power even when applied to a sparse table. In this dissertation, goodness-of-fit statistics using orthogonal components based on second- third- and fourth-order marginals were examined. If lack-of-fit is present in higher-order marginals, then a test that incorporates the higher-order marginals may have a higher power than a test that incorporates only first- and/or second-order marginals. To this end, two new statistics based on the orthogonal components of Pearson's chi-square that incorporate third- and fourth-order marginals were developed, and the Type I error, empirical power, and asymptotic power under different sparseness conditions were investigated. Individual orthogonal components as test statistics to identify lack-of-fit were also studied. The performance of individual orthogonal components to other popular lack-of-fit statistics were also compared. When the number of manifest variables becomes larger than 20, most of the statistics based on marginal distributions have limitations in terms of computer resources and CPU time. Under this problem, when the number manifest variables is larger than or equal to 20, the performance of a bootstrap based method to obtain p-values for Pearson-Fisher statistic, fit to confirmatory dichotomous variable factor analysis model, and the performance of Tollenaar and Mooijaart (2003) statistic were investigated.
ContributorsDassanayake, Mudiyanselage Maduranga Kasun (Author) / Reiser, Mark R. (Thesis advisor) / Kao, Ming-Hung (Committee member) / Wilson, Jeffrey (Committee member) / St. Louis, Robert (Committee member) / Kamarianakis, Ioannis (Committee member) / Arizona State University (Publisher)
Created2018
Description
In the presence of correlation, generalized linear models cannot be employed to obtain regression parameter estimates. To appropriately address the extravariation due to correlation, methods to estimate and model the additional variation are investigated. A general form of the mean-variance relationship is proposed which incorporates the canonical parameter. The two variance parameters are estimated using generalized method of moments, negating the need for a distributional assumption. The mean-variance relation estimates are applied to clustered data and implemented in an adjusted generalized quasi-likelihood approach through an adjustment to the covariance matrix. In the presence of significant correlation in hierarchical structured data, the adjusted generalized quasi-likelihood model shows improved performance for random effect estimates. In addition, submodels to address deviation in skewness and kurtosis are provided to jointly model the mean, variance, skewness, and kurtosis. The additional models identify covariates influencing the third and fourth moments. A cutoff to trim the data is provided which improves parameter estimation and model fit. For each topic, findings are demonstrated through comprehensive simulation studies and numerical examples. Examples evaluated include data on children’s morbidity in the Philippines, adolescent health from the National Longitudinal Study of Adolescent to Adult Health, as well as proteomic assays for breast cancer screening.
ContributorsIrimata, Katherine E (Author) / Wilson, Jeffrey R (Thesis advisor) / Kamarianakis, Ioannis (Committee member) / Kao, Ming-Hung (Committee member) / Reiser, Mark R. (Committee member) / Stufken, John (Committee member) / Arizona State University (Publisher)
Created2018
Description
Investigation of measurement invariance (MI) commonly assumes correct specification of dimensionality across multiple groups. Although research shows that violation of the dimensionality assumption can cause bias in model parameter estimation for single-group analyses, little research on this issue has been conducted for multiple-group analyses. This study explored the effects of mismatch in dimensionality between data and analysis models with multiple-group analyses at the population and sample levels. Datasets were generated using a bifactor model with different factor structures and were analyzed with bifactor and single-factor models to assess misspecification effects on assessments of MI and latent mean differences. As baseline models, the bifactor models fit data well and had minimal bias in latent mean estimation. However, the low convergence rates of fitting bifactor models to data with complex structures and small sample sizes caused concern. On the other hand, effects of fitting the misspecified single-factor models on the assessments of MI and latent means differed by the bifactor structures underlying data. For data following one general factor and one group factor affecting a small set of indicators, the effects of ignoring the group factor in analysis models on the tests of MI and latent mean differences were mild. In contrast, for data following one general factor and several group factors, oversimplifications of analysis models can lead to inaccurate conclusions regarding MI assessment and latent mean estimation.
ContributorsXu, Yuning (Author) / Green, Samuel (Thesis advisor) / Levy, Roy (Committee member) / Thompson, Marilyn (Committee member) / Arizona State University (Publisher)
Created2018