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Composite Likelihood Estimation for Latent Variable Models with Ordinal and Continuous, or Ranking Variables
Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The estimation of latent variable models with ordinal and continuous, or ranking variables is the research focus of this thesis. The existing estimation methods are discussed and a composite likelihood approach is developed. The main advantages of the new method are its low computational complexity which remains unchanged regardless of the model size, and that it yields an asymptotically unbiased, consistent, and normally distributed estimator.

The thesis consists of four papers. The first one investigates the two main formulations of the unrestricted Thurstonian model for ranking data along with the corresponding identification constraints. It is found that the extra identifications constraints required in one of them lead to unreliable estimates unless the constraints coincide with the true values of the fixed parameters.

In the second paper, a pairwise likelihood (PL) estimation is developed for factor analysis models with ordinal variables. The performance of PL is studied in terms of bias and mean squared error (MSE) and compared with that of the conventional estimation methods via a simulation study and through some real data examples. It is found that the PL estimates and standard errors have very small bias and MSE both decreasing with the sample size, and that the method is competitive to the conventional ones.

The results of the first two papers lead to the next one where PL estimation is adjusted to the unrestricted Thurstonian ranking model. As before, the performance of the proposed approach is studied through a simulation study with respect to relative bias and relative MSE and in comparison with the conventional estimation methods. The conclusions are similar to those of the second paper.

The last paper extends the PL estimation to the whole structural equation modeling framework where data may include both ordinal and continuous variables as well as covariates. The approach is demonstrated through an example run in R software. The code used has been incorporated in the R package lavaan (version 0.5-11).

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2013. , 31 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Social Sciences, ISSN 1652-9030 ; 86
Keyword [en]
latent variable models, factor analysis, structural equation models, Thurstonian model, item response theory, composite likelihood estimation, pairwise likelihood estimation, maximum likelihood, weighted least squares, ordinal variables, ranking variables, lavaan
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
URN: urn:nbn:se:uu:diva-188342ISBN: 978-91-554-8571-9 (print)OAI: oai:DiVA.org:uu-188342DiVA: diva2:577664
Public defence
2013-02-15, Hörsal 2, Ekonomikum, Kyrkogårdsgatan 10, Uppsala, 10:15 (English)
Opponent
Supervisors
Available from: 2013-01-23 Created: 2012-12-15 Last updated: 2013-02-11Bibliographically approved
List of papers
1. On the identification of the unrestricted Thurstonian model for ranking data
Open this publication in new window or tab >>On the identification of the unrestricted Thurstonian model for ranking data
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The identification issues of the unrestricted Thurstonian model for ranking data is the focus of the current paper. Within the Thurstonian framework, each object among those to be ranked is associated with a latent continuous variable, often interpreted as the utility of the object. The unrestricted Thurstonian model, due to the discrete and comparative nature of ranking data, faces more serious identification problems than the indeterminacy of the latent scale origin and unit. Most researchers resort to the study of the unrestricted model referring to the differences of the object utilities but then the inference on object utilities becomes tricky. Maydeu-Olivares and Böckenholt (2005) suggest a strategy to overcome the identification problem of the unrestricted model referring to object utilities but this requires many extra identification constraints, additional to the ones needed for defining the scale origin and unit. In the current paper, we study the general applicability of the suggested identification approach. Our simulation study indicates that the estimates obtained can be seriously biased with relatively large mean squared errors (MSE) when the extra constraints deviate from the true values of the parameters. Besides, the bias and MSE do not seem to decrease with increase in the sample size, and the effect of the constraints is not uniform on all estimated parameters.

Keyword
unrestricted Thurstonian model; identification; ranking data
National Category
Social Sciences
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-187598 (URN)
Available from: 2012-12-09 Created: 2012-12-09 Last updated: 2013-02-11
2. Pairwise likelihood estimation for factor analysis models with ordinal data
Open this publication in new window or tab >>Pairwise likelihood estimation for factor analysis models with ordinal data
2012 (English)In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 56, no 12, 4243-4258 p.Article in journal (Refereed) Published
Abstract [en]

A pairwise maximum likelihood (PML) estimation method is developed for factor analysis models with ordinal data and fitted both in an exploratory and confirmatory set-up. The performance of the method is studied via simulations and comparisons with full information maximum likelihood (FIML) and three-stage limited information estimation methods, namely the robust unweighted least squares (3S-RULS) and robust diagonally weighted least squares (3S-RDWLS). The advantage of PML over FIML is mainly computational. Unlike PML estimation, the computational complexity of FIML estimation increases either with the number of factors or with the number of observed variables depending on the model formulation. Contrary to 3S-RULS and 3S-RDWLS estimation, PML estimates of all model parameters are obtained simultaneously and the PML method does not require the estimation of a weight matrix for the computation of correct standard errors. The simulation study on the performance of PML estimates and estimated asymptotic standard errors investigates the effect of different model and sample sizes. The bias and mean squared error of PML estimates and their standard errors are found to be small in all experimental conditions and decreasing with increasing sample size. Moreover, the PML estimates and their standard errors are found to be very close to those of FIML.

Keyword
Composite maximum likelihood, Factor analysis, Ordinal data, Pairwise likelihood, Three-stage estimation, Item response theory approach
National Category
Social Sciences
Identifiers
urn:nbn:se:uu:diva-181384 (URN)10.1016/j.csda.2012.04.010 (DOI)000307483100034 ()
Available from: 2012-09-28 Created: 2012-09-24 Last updated: 2017-12-07Bibliographically approved
3. Composite likelihood estimation for Thurstonian models with ranking data
Open this publication in new window or tab >>Composite likelihood estimation for Thurstonian models with ranking data
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A composite likelihood estimation based on trinary rankings (TCL) is developed for Thurstonian models with ranking data. The merits of the proposed method are that: it gives an asymptotically unbiased, consistent, and normally distributed estimator; it is of low computationally complexity regardless of the model size; it estimates all model parameters in a single step; and it does not require the estimation of a weight matrix to compute correct standard errors. Via a simulation study, the performance of the proposed method is evaluated in terms of relative bias and relative mean squared error (MSE) and in comparison with the performance of the three-stage robust diagonally weighted least squares (3S-RDWLS) and three-stage robust unweighted least squares (3S-RULS), both as applied within structural equation models (SEM) with ordinal variables. The simulation results indicate that TCL yields estimates and standard errors with very small relative bias and small MSE even for small sample sizes and large models. Both relative bias and MSE are decreasing with increases in the sample size. Moreover, TCL performs similarly to 3S-RULS and 3S-RDWLS.

Keyword
unrestricted Thurstonian model, ranking data, composite maximum likelihood, estimation, trinary rankings
National Category
Social Sciences
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-187599 (URN)
Available from: 2012-12-15 Created: 2012-12-09 Last updated: 2013-02-11
4. Pairwise likelihood estimation for structural equation modeling with ordinal and continuous variables
Open this publication in new window or tab >>Pairwise likelihood estimation for structural equation modeling with ordinal and continuous variables
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A pairwise likelihood (PL) estimation method is developed for structural equation models (SEM) with continuous and ordinal observed variables where covariates may also be included. The proposed methodology starts off with the development of PL for estimating the mean vector and the covariance matrix of a variable vector consisting of continuous and ordinal variables. Later on a parametric structure is given to the mean vector and the covariance matrix according to a general SEM. The suggested method is demonstrated using an example with empirical data. An R code has been written which has been implemented in the R package lavaan (version 0.5-11). Maximum likelihood estimation, as implemented in Mplus (version 5.21) and in LISREL (version 9.10), is not feasible for our example due to its high computational complexity.

Keyword
structural equation modeling, pairwise likelihood estimation, continuous and ordinal variables, lavaan
National Category
Social Sciences
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-187600 (URN)
Available from: 2012-12-15 Created: 2012-12-09 Last updated: 2013-02-11

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