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A Finite Mixture of Nonlinear Random Coefficient Models for Continuous Repeated Measures Data

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Abstract

Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation–maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.

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Notes

  1. The data were obtained from Robert Cudeck who acquired them from Scott Chaiken of the Armstrong Laboratory, Brooks Air Force Base. A more detailed explanation of the origins of the data can be found in Cudeck (1996).

  2. \(\hbox {MD}=\Delta =\sqrt{( {\varvec{\upalpha } }_1 -{\varvec{\upalpha } }_2 {)}'{\varvec{\Phi } }^{-1}({\varvec{\upalpha } }_1 -{\varvec{\upalpha } }_2 )},\) where \({\varvec{\upalpha } }_1 \) and \({\varvec{\upalpha } }_2 \) denote the mean vector of fixed / population effects or class 1 and class 2, respectively, and \({\varvec{\Phi } }\) denotes the variance—covariance matrix of random effects.

  3. Results from a preliminary simulation study in which MD values were much small (i.e., less distributional separation) can be found at http://www.cehd.umn.edu/edpsych/people/Faculty/Kohli.html. These results showed poor convergence to a proper solution and large bias of model parameters—particularly the variance–covariance components of the linear–linear NRMM.

  4. We ran a separate simulation study where we empirically generated the starting values. The results from that study can be found on the following webpage: http://www.cehd.umn.edu/edpsych/people/Faculty/Kohli.html

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Authors and Affiliations

  1. Quantitative Methods in Education Program, Department of Educational Psychology, University of Minnesota, 161 Education Sciences Bldg., 56 East River Road, Minneapolis, MN, 55455 , USA

    Nidhi Kohli

  2. University of Maryland, College Park, MD, USA

    Jeffrey R. Harring

  3. University of Miami, Coral Gables, FL, USA

    Cengiz Zopluoglu

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  1. Nidhi Kohli
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  2. Jeffrey R. Harring
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  3. Cengiz Zopluoglu
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Correspondence to Nidhi Kohli.

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Kohli, N., Harring, J.R. & Zopluoglu, C. A Finite Mixture of Nonlinear Random Coefficient Models for Continuous Repeated Measures Data. Psychometrika 81, 851–880 (2016). https://doi.org/10.1007/s11336-015-9462-0

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