Skip to main content
SpringerLink
Log in

An Introduction to (Generalized (Non)Linear Mixed Models

  • Chapter
Explanatory Item Response Models

Part of the book series: Statistics for Social Science and Public Policy ((SSBS))

Abstract

In applied sciences, one is often confronted with the collection of correlated data or otherwise hierarchical data. This generic term embraces a multitude of data structures, such as multivariate observations, clustered data, repeated measurements (called ‘repeated observations’ in this volume), longitudinal data, and spatially correlated data. In particular, studies are often designed to investigate changes in a specific parameter which is measured repeatedly over time in the participating persons. This is in contrast to cross-sectional studies where the response of interest is measured only once for each individual. Longitudinal studies are conceived for the investigation of such changes, together with the evolution of relevant covariates.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Aerts, M., Geys, H., Molenberghs, G., & Ryan, L.M. (2002). Topics in Modeling of Clustered Binary Data. London: Chapman & Hall.

    Book  Google Scholar 

  • Afifi, A., & Elashoff, R. (1966). Missing observations in multivariate statistics I: Review of the literature. Journal of the American Statistical Association, 61, 595–604.

    MathSciNet  Google Scholar 

  • Agresti, A. (1990) Categorical Data Analysis. New York: John Wiley & Sons.

    MATH  Google Scholar 

  • Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, 716–723.

    Article  MathSciNet  MATH  Google Scholar 

  • Altham, P.M.E. (1978). Two generalizations of the binomial distribution. Applied Statistics, 27, 162–167.

    Article  MathSciNet  MATH  Google Scholar 

  • Bahadur, R.R. (1961). A representation of the joint distribution of responses of P dichotomous items. In H. Solomon (Ed.), Studies in Item Analysis and Prediction (pp. 158–168). Stanford, CA: Stanford University Press.

    Google Scholar 

  • Besag, J., Green, P.J., Higdon, D., & Mengersen, K. (1995). Bayesian computation and stochastic systems. Statistical Science, 10, 3–66.

    Article  MathSciNet  MATH  Google Scholar 

  • Böhning, D. (1999). Computer-Assisted Analysis of Mixtures and Applications: Meta-analysis, Disease Mapping and Others. London: Chapman & Hall.

    Google Scholar 

  • Breslow, N.E., & Clayton, D.G. (1993). Approximate inference in generalized linear mixed models. Journal of the American Statistical Association, 88, 9–25.

    MATH  Google Scholar 

  • Breslow, N.E., & Day, N.E. (1987). Statistical Methods in Cancer Research, Volume II. Oxford: Oxford University Press.

    Google Scholar 

  • Conaway, M. (1989). Analysis of repeated categorical measurements with conditional likelihood methods. Journal of the American Statistical Association, 84, 53–62.

    Article  MathSciNet  Google Scholar 

  • Cox, D.R. (1972). The analysis of multivariate binary data. Applied Statistics, 21, 113–120.

    Article  Google Scholar 

  • Cressie, N.A.C. (1991). Statistics for Spatial Data. New York: Wiley.

    MATH  Google Scholar 

  • Dale, J.R. (1986). Global cross-ratio models for bivariate, discrete, ordered responses. Biometrics, 42, 909–917.

    Article  Google Scholar 

  • De Backer, M., De Vroey, C., Lesaffre, E., Scheys, I., & De Keyser, P. (1998). Twelve weeks of continuous oral therapy for toenail onychomycosis caused by dermatophytes: A double-blind comparative trial of terbina-fine 250 mg/day versus itraconazole 200 mg/day. Journal of the American Academy of Dermatology, 38, S57–63.

    Article  Google Scholar 

  • Declerck, L., Aerts, M., & Molenberghs, G. (1998). Behaviour of the likelihood-ratio test statistic under a Bahadur model for exchangeable binary data. Journal of Statistical Computations and Simulations, 61, 15–38.

    Article  MathSciNet  MATH  Google Scholar 

  • Dempster, A.P., Laird, N.M., & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.

    MathSciNet  MATH  Google Scholar 

  • Diggle, P.J. (1983). Statistical Analysis of Spatial Point Patterns. London: Academic Press.

    MATH  Google Scholar 

  • Diggle, P.J. (1990). Time Series: A Bio statistical Introduction. Oxford: Oxford University Press.

    Google Scholar 

  • Diggle, P.J., Heagerty, P., Liang, K-Y., & Zeger, S.L. (2002). Analysis of Longitudinal Data. New York: Oxford University Press.

    Google Scholar 

  • Diggle, P.J., & Kenward, M.G. (1994). Informative drop-out in longitudinal data analysis (with discussion). Applied Statistics, 43, 49–93.

    Article  MATH  Google Scholar 

  • Efron, B. (1986). Double exponential families and their use in generalized linear regression. Journal of the American Statistical Association, 81, 709–721.

    Article  MathSciNet  MATH  Google Scholar 

  • Fahrmeir, L., & Tutz, G. (2001). Multivariate Statistical Modeling Based on Generalized Linear Models. Heidelberg: Springer.

    Book  Google Scholar 

  • Gamerman, D. (1997). Efficient sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7, 57–68.

    Article  Google Scholar 

  • Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (2004). Bayesian Data Analysis. London: Chapman & Hall.

    MATH  Google Scholar 

  • Gilmour, A.R., Anderson, R.D., & Rae, A.L. (1985). The analysis of binomial data by a generalized linear mixed model. Biometrika, 72, 593–599.

    Article  MathSciNet  Google Scholar 

  • Gilula, Z., & Haberman, S. (1994). Conditional log-linear models for analyzing categorical panel data. Journal of the American Statistical Association, 89, 645–656.

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein, H. (1986). Multilevel mixed linear model analysis using iterative generalized least squares. Biometrika, 73, 43–56.

    Article  MathSciNet  MATH  Google Scholar 

  • Goldstein, H. (1991). Nonlinear multilevel models, with an application to discrete response data. Biometrika, 78, 45–51.

    Article  MathSciNet  Google Scholar 

  • Goldstein, H. (1995). Multilevel Statistical Models (2nd ed.). London: Arnold.

    Google Scholar 

  • Goldstein, H. (2003). Multilevel Statistical Models (3rd ed.). London: Arnold.

    MATH  Google Scholar 

  • Goldstein, H., & Rasbash, J. (1996). Improved approximations for multilevel models with binary responses. Journal of the Royal Statistical Society A, 159, 505–513.

    Article  MathSciNet  MATH  Google Scholar 

  • Hartley, H.O., & Hocking, R. (1971). The analysis of incomplete data. Biometrics, 27, 7783–808.

    Article  Google Scholar 

  • Harville, D.A. (1974). Bayesian inference for variance components using only error contrasts. Biometrika, 61, 383–385.

    Article  MathSciNet  MATH  Google Scholar 

  • Hobert, J.P., & Casella, G. (1996). The effect of improper priors on Gibbs sampling in hierarchical linear mixed models. Journal of the American Statistical Association, 91, 1461–1473.

    Article  MathSciNet  MATH  Google Scholar 

  • Kenward, M.G., & Roger, J.H. (1997). Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 53, 983–997.

    Article  MATH  Google Scholar 

  • Kleinman, J. (1973). Proportions with extraneous variance: single and independent samples. Journal of the American Statistical Association, 68, 46–54.

    Google Scholar 

  • Knorr-Held, L. (1997). Hierarchical Modeling of Discrete Longitudinal Data; Applications of Markov Chain Monte Carlo. München: Utz.

    Google Scholar 

  • Kuk, A.Y.C. (1995). Asymptotically unbiased estimation in generalised linear models with random effects. Journal of the Royal Statistical Society B,57, 395–407.

    Google Scholar 

  • Kupper, L.L., & Haseman, J.K. (1978). The use of a correlated binomial model for the analysis of certain toxicology experiments. Biometrics, 34, 69–76.

    Article  MATH  Google Scholar 

  • Lang, J.B., & Agresti, A. (1994). Simultaneously modeling joint and marginal distributions of multivariate categorical responses. Journal of the American Statistical Association, 89, 625–632.

    Article  MATH  Google Scholar 

  • Lavergne, C., & Trottier, C. (2000). Sur l’estimation dans les modèles linéaires généralisés à effets aléatoires. Revue de Statistique Appliquée, 48, 49–67.

    Google Scholar 

  • Lee, Y., & Neider, J.A. (1996). Hierarchical generalized linear models (with discussion). Journal of the Royal Statistical Society, Series B, 58, 619–678.

    MATH  Google Scholar 

  • Liang, K.-Y., & Zeger, S.L. (1986). Longitudinal data analysis using generalized linear models. Biometrika, 73, 13–22.

    Article  MathSciNet  MATH  Google Scholar 

  • Liang, K.Y., Zeger, S.L., & Qaqish, B. (1992). Multivariate regression analyses for categorical data. Journal of the Royal Statistical Society, Series B, 54, 3–40.

    MathSciNet  MATH  Google Scholar 

  • Lin, X., & Breslow, N.E. (1996). Bias correction in generalized linear mixed models with multiple components of dispersion. Journal of the American Statistical Association, 91, 1007–1016.

    Article  MathSciNet  MATH  Google Scholar 

  • Lipsitz, S.R., Laird, N.M., & Harrington, D.P. (1991). Generalized estimating equations for correlated binary data: using the odds ratio as a measure of association. Biometrika, 78, 153–160.

    Article  MathSciNet  Google Scholar 

  • Little, R.J.A., & Rubin, D.B. (1987). Statistical Analysis with Missing Data. New York: Wiley.

    MATH  Google Scholar 

  • Longford, N.T. (1993). Random Coefficient Models. London: Oxford University Press.

    MATH  Google Scholar 

  • McCullagh, P., & Neider, J.A. (1989). Generalized Linear Models. London: Chapman & Hall.

    MATH  Google Scholar 

  • Molenberghs, G., & Lesaffre, E. (1994). Marginal modeling of correlated ordinal data using a multivariate Plackett distribution. Journal of the American Statistical Association, 89, 633–644.

    Article  MATH  Google Scholar 

  • Molenberghs, G., & Lesaffre, E. (1999). Marginal modeling of multivariate categorical data. Statistics in Medicine, 18, 2237–2255.

    Article  Google Scholar 

  • Murray, G.D., & Findlay, J.G. (1988). Correcting for the bias caused by drop-outs in hypertension trials. Statististics in Medicine, 7, 941–946.

    Article  Google Scholar 

  • Neider, J.A., & Wedderburn, R.W.M. (1972). Generalized linear models. Journal of the Royal Statistical Society, Series B, 135, 370–384.

    Article  Google Scholar 

  • Neuhaus, J.M. (1992). Statistical methods for longitudinal and clustered designs with binary responses. Statistical Methods in Medical Research, 1, 249–273.

    Article  Google Scholar 

  • Neuhaus, J.M., Kalbfleisch, J.D., & Hauck, W.W. (1991). A comparison of cluster-specific and population-averaged approaches for analyzing correlated binary data. International Statistical Review, 59, 25–35.

    Article  Google Scholar 

  • Pinheiro, J.C., & Bates, D.M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics, 4, 12–35.

    Google Scholar 

  • Pinheiro, J.C., & Bates D.M. (2000). Mixed Effects Models in S and S-PLUS. New-York: Springer.

    Book  MATH  Google Scholar 

  • Plackett, R.L. (1965). A class of bivariate distributions. Journal of the American Statistical Association, 60, 516–522.

    Article  MathSciNet  Google Scholar 

  • Prentice, R.L. (1988). Correlated binary regression with covariates specific to each binary observation. Biometrics, 44, 1033–1048.

    Article  MathSciNet  MATH  Google Scholar 

  • Press, W.H., Teukolsky, S.A., Vetterling, W.T., & Flannery, B.P. (1992). Numerical recipes in FORTRAN. (2nd ed.). Cambridge, MA: Cambridge University Press.

    MATH  Google Scholar 

  • Raftery, A.E. (1995). Bayesian model selection in social research. Sociological Methodology, 25, 111–163.

    Article  Google Scholar 

  • Raubertas, R.F., Lee, C.I.C., & Nordheim, E.V. (1986). Hypothesis tests for normal means constrained by linear inequalities. Communications in Statistics-Theory and Methods 15, 2809–2833.

    Article  MathSciNet  MATH  Google Scholar 

  • Renard, D., Molenberghs, G., & Geys, H. (2002). A paijwise likelihood approach to estimation in multilevel probit models. Computational Statistics and Data Analysis. Manuscript accepted for publication.

    Google Scholar 

  • Ripley, B.D. (1981). Spatial Statistics. New York: Wiley.

    Book  MATH  Google Scholar 

  • Ripley, B.D. (1987). Stochastic Simulation. New York: Wiley.

    Book  MATH  Google Scholar 

  • Robins, J.M., Rotnitzky, A., & Zhao, L.P. (1995). Analysis of semiparamet-ric regression models for repeated outcomes in the presence of missing data. Journal of the American Statistical Association, 90, 106–121.

    Article  MathSciNet  MATH  Google Scholar 

  • Rodriguez, G., & Goldman, N. (1995). An assessment of estimation procedures for multilevel models with binary responses. Journal of the Royal Statistical Society A, 158, 73–89.

    Article  Google Scholar 

  • Rosner, B. (1984). Multivariate methods in opthalmology with applications to other paired-data. Biometrics, 40, 1025–1035.

    Article  Google Scholar 

  • Rubin, D.B. (1976). Inference and missing data. Biometrika, 63, 581–592.

    Article  MathSciNet  MATH  Google Scholar 

  • Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys. New York: Wiley.

    Book  Google Scholar 

  • Rubin, D.B., Stern, H.S., & Vehovar V. (1995). Handling “don’t know” survey responses: the case of the Slovenian plebiscite. Journal of the American Statistical Association, 90, 822–828.

    Google Scholar 

  • Satterthwaite, F.E. (1941). Synthesis of variance. Psychometrika, 6, 309–316.

    Article  MathSciNet  MATH  Google Scholar 

  • Schall, R. (1991). Estimation in generalised linear models with random effects. Biometrika, 78, 719–727.

    Article  MATH  Google Scholar 

  • Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics, 6, 461–464.

    Article  MathSciNet  MATH  Google Scholar 

  • Shapiro, A. (1988). Towards a unified theory of inequality constrained testing in multivariate analysis. International Statistical Review 56, 49–62.

    Article  MathSciNet  MATH  Google Scholar 

  • Silvapulle, M.J., & Silvapulle, P. (1995). A score test against one-sided alternatives. Journal of the American Statistical Association 90, 342–349.

    Article  MathSciNet  MATH  Google Scholar 

  • Skellam, J.G. (1948). A probability distribution derived from the binomial distribution by regarding the probability of success as variable between the sets of trials. Journal of the Royal Statistical Society, Series B, 10, 257–261.

    MathSciNet  MATH  Google Scholar 

  • Spiessens, B., Lesaffre, E., Verbeke, G., & Kim, K. (2002). Group sequential methods for an ordinal logistic random-effects model under misspecifica-tion. Biometrics, 58, 569–575.

    Article  MathSciNet  MATH  Google Scholar 

  • Spiessens, B., Lesaffre, E., Verbeke, G., & Kim, K. (2003). The use of mixed models for longitudinal count data when the random-effects distribution is misspecified. Manuscript submitted for publication.

    Google Scholar 

  • Stiratelli, R., Laird, N., & Ware, J.H. (1984). Random-effects model for serial observations with binary response. Biometrics, 40, 961–971.

    Article  Google Scholar 

  • Stram, D.O., & Lee, J.W. (1994). Variance components testing in the longitudinal mixed-effects model. Biometrics, 50, 1171–1177.

    Article  MATH  Google Scholar 

  • Stram, D.A., & Lee, J.W. (1995). Correction to: Variance components testing in the longitudinal mixed-effects model. Biometrics, 51, 1196.

    Google Scholar 

  • Ten Have, T.R., Landis, R., & Weaver, S.L. (1995). Association models for periodontal disease progression: A comparison of methods for clustered binary data. Statistics in Medicine, 14, 413–429.

    Article  Google Scholar 

  • Thélot, C. (1985). Lois logistiques à deux dimensions. Annales de l’Insée, 58, 123–149.

    Google Scholar 

  • Verbeke, G., & Lesaffre, E. (1996). A linear mixed-effects model with heterogeneity in the random-effects population. Journal of the American Statistical Association, 91, 217–221.

    Article  MATH  Google Scholar 

  • Verbeke, G., & Molenberghs, G. (1997). Linear Mixed Models in Practice: A SAS-Oriented Approach. Lecture Notes in Statistics 126. New York: Springer.

    Book  MATH  Google Scholar 

  • Verbeke, G., & Molenberghs, G. (2000). Linear Mixed Models for Longitudinal Data. New York: Springer.

    MATH  Google Scholar 

  • Verbeke, G., & Molenberghs, G. (2003). The use of score tests for inference on variance components. Biometrics, 59, 254–262.

    Article  MathSciNet  MATH  Google Scholar 

  • Wedderburn, R.W.M. (1974). Quasi-likelihood functions, generalized linear models, and the Gauss-Newton method. Biometrika, 61, 439–447.

    MathSciNet  MATH  Google Scholar 

  • Wolfinger, R. (1998). Towards practical application of generalized linear mixed models. In B. Marx and H. Friedl (Eds), Proceedings of the 13th International Workshop on Statistical Modeling (pp. 388–395). New Orleans, Louisiana.

    Google Scholar 

  • Wolfinger, R., & O’Connell, M. (1993). Generalized linear mixed models: A pseudo-likelihood approach. Journal of Statistical Computing and Simulation, 48, 233–243.

    Article  MATH  Google Scholar 

  • Zeger, S.L., & Karim, M.R. (1991). Generalised linear models with random effects: a Gibbs sampling approach. Journal of the American Statistical Association, 86, 79–102.

    Article  MathSciNet  Google Scholar 

  • Zhao, L.P., & Prentice, R.L. (1990). Correlated binary regression using a quadratic exponential model. Biometrika, 77, 642–648.

    Article  MathSciNet  Google Scholar 

Download references

Authors
  1. Geert Molenberghs
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Geert Verbeke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Psychology, K.U. Leuven, Tiensestraat 102, B-3000, Leuven, Belgium

    Paul De Boeck

  2. Graduate School of Education, University of California, Berkeley, 94720-1670, Berkeley, CA, USA

    Mark Wilson

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Science+Business Media New York

About this chapter

Cite this chapter

Molenberghs, G., Verbeke, G. (2004). An Introduction to (Generalized (Non)Linear Mixed Models. In: De Boeck, P., Wilson, M. (eds) Explanatory Item Response Models. Statistics for Social Science and Public Policy. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3990-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-1-4757-3990-9_4

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4419-2323-3

  • Online ISBN: 978-1-4757-3990-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation