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- All Subjects: Statistics
- Creators: Zheng, Yi
- Creators: Enders, Craig K.
- Creators: Wilson, Jeffrey
- Creators: Zhou, Shuang
- Member of: ASU Electronic Theses and Dissertations
Description
Many longitudinal studies, especially in clinical trials, suffer from missing data issues. Most estimation procedures assume that the missing values are ignorable or missing at random (MAR). However, this assumption leads to unrealistic simplification and is implausible for many cases. For example, an investigator is examining the effect of treatment on depression. Subjects are scheduled with doctors on a regular basis and asked questions about recent emotional situations. Patients who are experiencing severe depression are more likely to miss an appointment and leave the data missing for that particular visit. Data that are not missing at random may produce bias in results if the missing mechanism is not taken into account. In other words, the missing mechanism is related to the unobserved responses. Data are said to be non-ignorable missing if the probabilities of missingness depend on quantities that might not be included in the model. Classical pattern-mixture models for non-ignorable missing values are widely used for longitudinal data analysis because they do not require explicit specification of the missing mechanism, with the data stratified according to a variety of missing patterns and a model specified for each stratum. However, this usually results in under-identifiability, because of the need to estimate many stratum-specific parameters even though the eventual interest is usually on the marginal parameters. Pattern mixture models have the drawback that a large sample is usually required. In this thesis, two studies are presented. The first study is motivated by an open problem from pattern mixture models. Simulation studies from this part show that information in the missing data indicators can be well summarized by a simple continuous latent structure, indicating that a large number of missing data patterns may be accounted by a simple latent factor. Simulation findings that are obtained in the first study lead to a novel model, a continuous latent factor model (CLFM). The second study develops CLFM which is utilized for modeling the joint distribution of missing values and longitudinal outcomes. The proposed CLFM model is feasible even for small sample size applications. The detailed estimation theory, including estimating techniques from both frequentist and Bayesian perspectives is presented. Model performance and evaluation are studied through designed simulations and three applications. Simulation and application settings change from correctly-specified missing data mechanism to mis-specified mechanism and include different sample sizes from longitudinal studies. Among three applications, an AIDS study includes non-ignorable missing values; the Peabody Picture Vocabulary Test data have no indication on missing data mechanism and it will be applied to a sensitivity analysis; the Growth of Language and Early Literacy Skills in Preschoolers with Developmental Speech and Language Impairment study, however, has full complete data and will be used to conduct a robust analysis. The CLFM model is shown to provide more precise estimators, specifically on intercept and slope related parameters, compared with Roy's latent class model and the classic linear mixed model. This advantage will be more obvious when a small sample size is the case, where Roy's model experiences challenges on estimation convergence. The proposed CLFM model is also robust when missing data are ignorable as demonstrated through a study on Growth of Language and Early Literacy Skills in Preschoolers.
ContributorsZhang, Jun (Author) / Reiser, Mark R. (Thesis advisor) / Barber, Jarrett (Thesis advisor) / Kao, Ming-Hung (Committee member) / Wilson, Jeffrey (Committee member) / St Louis, Robert D. (Committee member) / Arizona State University (Publisher)
Created2013
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
It is common in the analysis of data to provide a goodness-of-fit test to assess the performance of a model. In the analysis of contingency tables, goodness-of-fit statistics are frequently employed when modeling social science, educational or psychological data where the interest is often directed at investigating the association among multi-categorical variables. Pearson's chi-squared statistic is well-known in goodness-of-fit testing, but it is sometimes considered to produce an omnibus test as it gives little guidance to the source of poor fit once the null hypothesis is rejected. However, its components can provide powerful directional tests. In this dissertation, orthogonal components are used to develop goodness-of-fit tests for models fit to the counts obtained from the cross-classification of multi-category dependent variables. Ordinal categories are assumed. Orthogonal components defined on marginals are obtained when analyzing multi-dimensional contingency tables through the use of the QR decomposition. A subset of these orthogonal components can be used to construct limited-information tests that allow one to identify the source of lack-of-fit and provide an increase in power compared to Pearson's test. These tests can address the adverse effects presented when data are sparse. The tests rely on the set of first- and second-order marginals jointly, the set of second-order marginals only, and the random forest method, a popular algorithm for modeling large complex data sets. The performance of these tests is compared to the likelihood ratio test as well as to tests based on orthogonal polynomial components. The derived goodness-of-fit tests are evaluated with studies for detecting two- and three-way associations that are not accounted for by a categorical variable factor model with a single latent variable. In addition the tests are used to investigate the case when the model misspecification involves parameter constraints for large and sparse contingency tables. The methodology proposed here is applied to data from the 38th round of the State Survey conducted by the Institute for Public Policy and Michigan State University Social Research (2005) . The results illustrate the use of the proposed techniques in the context of a sparse data set.
ContributorsMilovanovic, Jelena (Author) / Young, Dennis (Thesis advisor) / Reiser, Mark R. (Thesis advisor) / Wilson, Jeffrey (Committee member) / Eubank, Randall (Committee member) / Yang, Yan (Committee member) / Arizona State University (Publisher)
Created2011
Description
Although the issue of factorial invariance has received increasing attention in the literature, the focus is typically on differences in factor structure across groups that are directly observed, such as those denoted by sex or ethnicity. While establishing factorial invariance across observed groups is a requisite step in making meaningful cross-group comparisons, failure to attend to possible sources of latent class heterogeneity in the form of class-based differences in factor structure has the potential to compromise conclusions with respect to observed groups and may result in misguided attempts at instrument development and theory refinement. The present studies examined the sensitivity of two widely used confirmatory factor analytic model fit indices, the chi-square test of model fit and RMSEA, to latent class differences in factor structure. Two primary questions were addressed. The first of these concerned the impact of latent class differences in factor loadings with respect to model fit in a single sample reflecting a mixture of classes. The second question concerned the impact of latent class differences in configural structure on tests of factorial invariance across observed groups. The results suggest that both indices are highly insensitive to class-based differences in factor loadings. Across sample size conditions, models with medium (0.2) sized loading differences were rejected by the chi-square test of model fit at rates just slightly higher than the nominal .05 rate of rejection that would be expected under a true null hypothesis. While rates of rejection increased somewhat when the magnitude of loading difference increased, even the largest sample size with equal class representation and the most extreme violations of loading invariance only had rejection rates of approximately 60%. RMSEA was also insensitive to class-based differences in factor loadings, with mean values across conditions suggesting a degree of fit that would generally be regarded as exceptionally good in practice. In contrast, both indices were sensitive to class-based differences in configural structure in the context of a multiple group analysis in which each observed group was a mixture of classes. However, preliminary evidence suggests that this sensitivity may contingent on the form of the cross-group model misspecification.
ContributorsBlackwell, Kimberly Carol (Author) / Millsap, Roger E (Thesis advisor) / Aiken, Leona S. (Committee member) / Enders, Craig K. (Committee member) / Mackinnon, David P (Committee member) / Arizona State University (Publisher)
Created2011
Description
Designing studies that use latent growth modeling to investigate change over time calls for optimal approaches for conducting power analysis for a priori determination of required sample size. This investigation (1) studied the impacts of variations in specified parameters, design features, and model misspecification in simulation-based power analyses and (2) compared power estimates across three common power analysis techniques: the Monte Carlo method; the Satorra-Saris method; and the method developed by MacCallum, Browne, and Cai (MBC). Choice of sample size, effect size, and slope variance parameters markedly influenced power estimates; however, level-1 error variance and number of repeated measures (3 vs. 6) when study length was held constant had little impact on resulting power. Under some conditions, having a moderate versus small effect size or using a sample size of 800 versus 200 increased power by approximately .40, and a slope variance of 10 versus 20 increased power by up to .24. Decreasing error variance from 100 to 50, however, increased power by no more than .09 and increasing measurement occasions from 3 to 6 increased power by no more than .04. Misspecification in level-1 error structure had little influence on power, whereas misspecifying the form of the growth model as linear rather than quadratic dramatically reduced power for detecting differences in slopes. Additionally, power estimates based on the Monte Carlo and Satorra-Saris techniques never differed by more than .03, even with small sample sizes, whereas power estimates for the MBC technique appeared quite discrepant from the other two techniques. Results suggest the choice between using the Satorra-Saris or Monte Carlo technique in a priori power analyses for slope differences in latent growth models is a matter of preference, although features such as missing data can only be considered within the Monte Carlo approach. Further, researchers conducting power analyses for slope differences in latent growth models should pay greatest attention to estimating slope difference, slope variance, and sample size. Arguments are also made for examining model-implied covariance matrices based on estimated parameters and graphic depictions of slope variance to help ensure parameter estimates are reasonable in a priori power analysis.
ContributorsVan Vleet, Bethany Lucía (Author) / Thompson, Marilyn S. (Thesis advisor) / Green, Samuel B. (Committee member) / Enders, Craig K. (Committee member) / Arizona State University (Publisher)
Created2011
Description
Researchers are often interested in estimating interactions in multilevel models, but many researchers assume that the same procedures and interpretations for interactions in single-level models apply to multilevel models. However, estimating interactions in multilevel models is much more complex than in single-level models. Because uncentered (RAS) or grand mean centered (CGM) level-1 predictors in two-level models contain two sources of variability (i.e., within-cluster variability and between-cluster variability), interactions involving RAS or CGM level-1 predictors also contain more than one source of variability. In this Master’s thesis, I use simulations to demonstrate that ignoring the four sources of variability in a total level-1 interaction effect can lead to erroneous conclusions. I explain how to parse a total level-1 interaction effect into four specific interaction effects, derive equivalencies between CGM and centering within context (CWC) for this model, and describe how the interpretations of the fixed effects change under CGM and CWC. Finally, I provide an empirical example using diary data collected from working adults with chronic pain.
ContributorsMazza, Gina L (Author) / Enders, Craig K. (Thesis advisor) / Aiken, Leona S. (Thesis advisor) / West, Stephen G. (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
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
Quadratic growth curves of 2nd degree polynomial are widely used in longitudinal studies. For a 2nd degree polynomial, the vertex represents the location of the curve in the XY plane. For a quadratic growth curve, we propose an approximate confidence region as well as the confidence interval for x and y-coordinates of the vertex using two methods, the gradient method and the delta method. Under some models, an indirect test on the location of the curve can be based on the intercept and slope parameters, but in other models, a direct test on the vertex is required. We present a quadratic-form statistic for a test of the null hypothesis that there is no shift in the location of the vertex in a linear mixed model. The statistic has an asymptotic chi-squared distribution. For 2nd degree polynomials of two independent samples, we present an approximate confidence region for the difference of vertices of two quadratic growth curves using the modified gradient method and delta method. Another chi-square test statistic is derived for a direct test on the vertex and is compared to an F test statistic for the indirect test. Power functions are derived for both the indirect F test and the direct chi-square test. We calculate the theoretical power and present a simulation study to investigate the power of the tests. We also present a simulation study to assess the influence of sample size, measurement occasions and nature of the random effects. The test statistics will be applied to the Tell Efficacy longitudinal study, in which sound identification scores and language protocol scores for children are modeled as quadratic growth curves for two independent groups, TELL and control curriculum. The interpretation of shift in the location of the vertices is also presented.
ContributorsYu, Wanchunzi (Author) / Reiser, Mark R. (Thesis advisor) / Barber, Jarrett (Committee member) / Kao, Ming-Hung (Committee member) / St Louis, Robert D (Committee member) / Wilson, Jeffrey (Committee member) / Arizona State University (Publisher)
Created2015