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Generalized linear models with clustered data: fixed and random effects models
Umeå University, Faculty of Social Sciences, Department of Statistics.
Umeå University, Faculty of Social Sciences, Department of Statistics.
2011 (English)In: Computational Statistics & Data Analysis, ISSN 0167-9473, E-ISSN 1872-7352, Vol. 55, no 12, 3123-3134 p.Article in journal (Refereed) Published
Abstract [en]

The statistical analysis of mixed effects models for binary and count data is investigated. In the statistical computing environment R, there are a few packages that estimate models of this kind. The packagelme4 is a de facto standard for mixed effects models. The packageglmmML allows non-normal distributions in the specification of random intercepts. It also allows for the estimation of a fixed effects model, assuming that all cluster intercepts are distinct fixed parameters; moreover, a bootstrapping technique is implemented to replace asymptotic analysis. The random intercepts model is fitted using a maximum likelihood estimator with adaptive Gauss–Hermite and Laplace quadrature approximations of the likelihood function. The fixed effects model is fitted through a profiling approach, which is necessary when the number of clusters is large. In a simulation study, the two approaches are compared. The fixed effects model has severe bias when the mixed effects variance is positive and the number of clusters is large.

Place, publisher, year, edition, pages
2011. Vol. 55, no 12, 3123-3134 p.
Keyword [en]
Bernoulli distribution, Gauss-Hermite quadrature, Laplace approximation, Implicit derivation, Profiling, Poisson distribution
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
URN: urn:nbn:se:umu:diva-40018DOI: 10.1016/j.csda.2011.06.011OAI: oai:DiVA.org:umu-40018DiVA: diva2:397338
Funder
Riksbankens Jubileumsfond, 2005-0488
Available from: 2011-02-14 Created: 2011-02-14 Last updated: 2017-12-11Bibliographically approved
In thesis
1. Generalised linear models with clustered data
Open this publication in new window or tab >>Generalised linear models with clustered data
2010 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In situations where a large data set is partitioned into many relativelysmall clusters, and where the members within a cluster have some common unmeasured characteristics, the number of parameters requiring estimation tends to increase with sample size if a fixed effects model is applied. This fact causes the assumptions underlying asymptotic results to be violated. The first paper in this thesis considers two possible solutions to this problem, a random intercepts model and a fixed effects model, where asymptoticsare replaced by a simple form of bootstrapping. A profiling approach is introduced in the fixed effects case, which makes it computationally efficient even with a huge number of clusters. The grouping effect is mainly seen as a nuisance in this paper.

In the second paper the effect of misspecifying the distribution of the random effects in a generalised linear mixed model for binary data is studied. One problem with mixed effects models is that the distributional assumptions about the random effects are not easily checked from real data. Models with Gaussian, logistic and Cauchy distributional assumptions are used for parameter estimation on data simulated using the same three distributions. The effect of these assumptions on parameter estimation is presented. Two criteria for model selection are investigated, the Akaike information criterion and a criterion based on a chi-square statistic. The estimators for fixed effects parameters are quite robust against misspecification of the random effects distribution, at least with the distributions used in this paper. Even when the true random effects distribution is Cauchy, models assuming a Gaussian or a logistic distribution regularly produce estimates with less bias.

Place, publisher, year, edition, pages
Umeå: Statistiska institutionen, Umeå Universitet, 2010. 21 p.
Series
Statistical studies, ISSN 1100-8989 ; 43
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:umu:diva-39972 (URN)978-91-7459-039-5 (ISBN)
Presentation
2009-09-01, Umeå universitet, Umeå, 10:00 (English)
Opponent
Supervisors
Available from: 2011-02-17 Created: 2011-02-11 Last updated: 2011-02-17Bibliographically approved
2. Generalized linear models with clustered data
Open this publication in new window or tab >>Generalized linear models with clustered data
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In situations where a large data set is partitioned into many relatively small groups, and where the members within a group have some common unmeasured characteristics, the number of parameters requiring estimation tends to increase with sample size if a fixed effects model is applied. This fact causes the assumptions underlying asymptotic results to be violated.

The first paper in this thesis considers two possible solutions to this problem, a random intercepts model and a fixed effects model, where asymptotics are replaced by a simple form of bootstrapping. A profiling approach is introduced in the fixed effects case, which makes it computationally efficient even with a huge number of groups. The grouping effect is mainly seen as a nuisance in this paper.

In the second paper the effect of misspecifying the distribution of the random effects in a generalized linear mixed model for binary data is studied. One problem with mixed effects models is that the distributional assumptions about the random effects are not easily checked from real data. Models with Gaussian, logistic and Cauchy distributional assumptions are used for parameter estimation on data simulated using the same three distributions. The eect of these assumptions on parameter estimation is presented. Two criteria for model selection are investigated, the Akaike information criterion and a criterion based on a X2 statistic. The estimators for fixed effects parameters are quite robust against misspecification of the random effects distribution, at least with the distributions used in this paper. Even when the true random effects distribution is Cauchy, models assuming a Gaussian or a logistic distribution regularly produce estimates with less bias.

In the third paper the results from the first two papers are applied to infant mortality data. We found that there was significant clustering of infant mortality in the Skellefteå region in the years 1831-1890. An "ad hoc" method for comparing the magnitude of unexplained clustering after a model is applied is also presented.

The last paper of this thesis is concerned with the problem of testing for spatial clustering caused by autocorrelation. A test that is robust against heteroscedasticity is proposed. In a simulation study the properties of the proposed statistic, K, are investigated. The power of the test based on K is compared to that of Moran's I in the simulation study. Both tests are then applied to mortality data from Swedish municipalities.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2012. 25 p.
Series
Statistical studies, ISSN 1100-8989 ; 46
National Category
Probability Theory and Statistics
Research subject
Statistics
Identifiers
urn:nbn:se:umu:diva-52902 (URN)978-91-7459-378-5 (ISBN)
Public defence
2012-03-30, Norra Beteendevetarhuset, HS1031, Umeå universitet, Umeå, 13:00 (English)
Opponent
Supervisors
Available from: 2012-03-09 Created: 2012-03-05 Last updated: 2016-03-04Bibliographically approved

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