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Essays on Estimation Methods for Factor Models and Structural Equation Models
Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis which consists of four papers is concerned with estimation methods in factor analysis and structural equation models. New estimation methods are proposed and investigated.

In paper I an approximation of the penalized maximum likelihood (ML) is introduced to fit an exploratory factor analysis model. Approximated penalized ML continuously and efficiently shrinks the factor loadings towards zero. It naturally factorizes a covariance matrix or a correlation matrix. It is also applicable to an orthogonal or an oblique structure.

Paper II, a simulation study, investigates the properties of approximated penalized ML with an orthogonal factor model. Different combinations of penalty terms and tuning parameter selection methods are examined. Differences in factorizing a covariance matrix and factorizing a correlation matrix are also explored. It is shown that the approximated penalized ML frequently improves the traditional estimation-rotation procedure.

In Paper III we focus on pseudo ML for multi-group data. Data from different groups are pooled and normal theory is used to fit the model. It is shown that pseudo ML produces consistent estimators of factor loadings and that it is numerically easier than multi-group ML. In addition, normal theory is not applicable to estimate standard errors. A sandwich-type estimator of standard errors is derived.

Paper IV examines properties of the recently proposed polychoric instrumental variable (PIV) estimators for ordinal data through a simulation study. PIV is compared with conventional estimation methods (unweighted least squares and diagonally weighted least squares). PIV produces accurate estimates of factor loadings and factor covariances in the correctly specified confirmatory factor analysis model and accurate estimates of loadings and coefficient matrices in the correctly specified structure equation model. If the model is misspecified, robustness of PIV depends on model complexity, underlying distribution, and instrumental variables.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2015. , 29 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Social Sciences, ISSN 1652-9030 ; 111
Keyword [en]
shrinkage, factor rotation, penalized maximum likelihood, pseudo-maximum likelihood, multi-group analysis, ordinal data, robustness
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
URN: urn:nbn:se:uu:diva-247292ISBN: 978-91-554-9199-4 (print)OAI: oai:DiVA.org:uu-247292DiVA: diva2:795680
Public defence
2015-05-08, Hörsal 2, Ekonomikum, Kyrkogårdsgatan 10, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2015-04-16 Created: 2015-03-17 Last updated: 2015-07-07
List of papers
1. Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis
Open this publication in new window or tab >>Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-247287 (URN)
Available from: 2015-03-17 Created: 2015-03-17 Last updated: 2015-07-07
2. Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis: A Simulation Study of an Orthogonal Case
Open this publication in new window or tab >>Approximated Penalized Maximum Likelihood for Exploratory Factor Analysis: A Simulation Study of an Orthogonal Case
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-247288 (URN)
Available from: 2015-03-17 Created: 2015-03-17 Last updated: 2015-07-07
3. Asymptotic Efficiency of the Pseudo-Maximum Likelihood Estimator in Multi-Group Factor Models with Pooled Data
Open this publication in new window or tab >>Asymptotic Efficiency of the Pseudo-Maximum Likelihood Estimator in Multi-Group Factor Models with Pooled Data
2016 (English)In: British Journal of Mathematical & Statistical Psychology, ISSN 0007-1102, E-ISSN 2044-8317, Vol. 69, no 1, 20-42 p.Article in journal (Refereed) Published
Abstract [en]

A multi-group factor model is suitable for data originating from different strata. However, it often requires a relatively large sample size to avoid numerical issues such as non-convergence and non-positive definite covariance matrices. An alternative is to pool data from different groups in which a single-group factor model is fitted to the pooled data using maximum likelihood. In this paper, properties of pseudo-maximum likelihood (PML) estimators for pooled data are studied. The pooled data are assumed to be normally distributed from a single group. The resulting asymptotic efficiency of the PML estimators of factor loadings is compared with that of the multi-group maximum likelihood estimators. The effect of pooling is investigated through a two-group factor model. The variances of factor loadings for the pooled data are underestimated under the normal theory when error variances in the smaller group are larger. Underestimation is due to dependence between the pooled factors and pooled error terms. Small-sample properties of the PML estimators are also investigated using a Monte Carlo study.

Keyword
sandwich estimator; robustness; normal theory; measurement invariance
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:uu:diva-247289 (URN)10.1111/bmsp.12056 (DOI)000368697000002 ()
Funder
Swedish Research Council, 421-2011-1712
Available from: 2015-03-17 Created: 2015-03-17 Last updated: 2017-12-04Bibliographically approved
4. A Simulation Study of Polychoric Instrumental Variable Estimation in Structural Equation Models
Open this publication in new window or tab >>A Simulation Study of Polychoric Instrumental Variable Estimation in Structural Equation Models
2016 (English)In: Structural Equation Modeling, ISSN 1070-5511, E-ISSN 1532-8007, Vol. 23, no 5, 680-694 p.Article in journal (Refereed) Published
Abstract [en]

Data collected from questionnaires are often in ordinal scale. Unweighted least squares (ULS), diagonally weighted least squares (DWLS) and normal-theory maximum likelihood (ML) are commonly used methods to fit structural equation models. Consistency of these estimators demands no structural misspecification. In this article, we conduct a simulation study to compare the equation-by-equation polychoric instrumental variable (PIV) estimation with ULS, DWLS, and ML. Accuracy of PIV for the correctly specified model and robustness of PIV for misspecified models are investigated through a confirmatory factor analysis (CFA) model and a structural equation model with ordinal indicators. The effects of sample size and nonnormality of the underlying continuous variables are also examined. The simulation results show that PIV produces robust factor loading estimates in the CFA model and in structural equation models. PIV also produces robust path coefficient estimates in the model where valid instruments are used. However, robustness highly depends on the validity of instruments.

Keyword
ordinal data, factor analysis, robustness, model misspecification
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-247291 (URN)10.1080/10705511.2016.1189334 (DOI)000383882200004 ()
Funder
Swedish Research Council, 421-2011-1727
Available from: 2015-03-17 Created: 2015-03-17 Last updated: 2017-12-04Bibliographically approved

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