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Modèle nonlinéaire à processus latent 741022 Biometrics, December 2006has previously been discussed by Molenberghs and Verbeke(2001).5.5 Multivariate Model versus Univariate ModelsFor each test, the univariate model detected an associationbetween apoE genotype and cognition with a larger p-value(p-value from the likelihood ratio test for apoE × t 2 parameter:p = 0.0043 for MMSE, p = 0.018 for IST, p = 0.020for BVRT, p = 0.054 for the DSSTW) than for the multivariatemodel (p = 0.0018). By using a multivariate modelcompared to four univariate models, we had a gain of powerin assessing the association between apoE genotype and cognition.Moreover, note that interpretation of the associationwith the latent process and with each psychometric test isdifferent.The gain in efficiency can also be evaluated by comparingthe AIC from the multivariate model and the AIC computedby pooling the likelihoods from the four univariate modelswith the total number of parameters in these four models.In our case, even if we added in the multivariate model theconstraint that apoE had a common effect on the four tests,the AIC from the multivariate model was markedly better(39503.1 vs. 40203.8 for the four univariate models).6. DiscussionWe proposed a nonlinear model for multivariate longitudinalnon-Gaussian quantitative outcomes when the outcomes areindirect measures of a common underlying continuous-timeprocess. Such data are very frequent in psychometrics, but themethodology has many other potential areas of application,as, for instance, the study of the course of chronic illnessesevaluated by several biological markers.In this work, psychometric tests are analyzed by consideringtheir sum scores as quantitative variables. This is themost frequent way to consider psychometric tests in gerontology:the summary scores are used to evaluate cognitivelevel and risk of dementia (Hall et al., 2001; Sliwinski etal., 2003; Amieva et al., 2005). From a neuropsychologicalperspective, the alternative approach, which consists of analyzingitem responses using SEM or item response model(Skrondal and Rabe-Hesketh, 2004, Chapter 3), could be usefulif the objective was to understand the underlying componentsof the tests. This methodology is interesting when thetests consist of a limited number of items evaluating differentcognitive domains (such as the MMSE) or exhibiting differentlevels of difficulty (such as the BVRT). On the contrary,this methodology would be difficult to apply to the IST score,which is the number of words cited by the subjects (exceptfor considering the four subscores for each semantic category)and the DSSTW score, which is the count of symbols correctlyassigned to a sequence of numbers.Given that the summary scores are quantitative discretevariables, we could either consider them as continuous variablesor as ordinal variables. However, threshold models forordinal data require estimation of one threshold for each possiblevalue of the scores, which would be very challenging formultivariate modeling of scores with so many different values.In our application including four tests, this would haveimplied estimation of more than 150 additional parameters.Thus, we decided to analyze the scores as continuous variablesand to use nonlinear transformations depending on a limitednumber of parameters as link functions between the Gaussianlatent process and the observed outcomes. Various linkfunctions have been considered, but we found that the betaCDF was flexible enough with only two parameters. Withmany fewer parameters than threshold models, the estimatedcurves provide interesting information on the relationship betweenevolution of the latent cognitive level and evolution ofthe observed scores. Moreover, goodness-of-fit analyses showthat the beta transforms of the four test scores fitted wella Gaussian distribution. Nevertheless, if necessary for otherapplications, it would be easy to include a different familyof continuous transformation for each test. This model couldalso be extended to allow a mixture of continuous, binary, andordinal outcomes with few categories as in Dunson (2000),Dunson (2003), or Rabe-Hesketh et al. (2004).Another asset of this model is the way of accounting forcovariate effects. Using fixed contrasts, we were able to distinguishbetween the association with the latent process and thedifferential association with the various psychometric tests.Being able to compare covariate effects over the tests is alsoan advantage of multivariate modeling. With a large numberof tests it may also be possible and advantageous to have theeffects of the covariates on the tests be random rather thanfixed.We assumed the data to be missing at random and thusignorable using a maximum likelihood approach. Even if wehave shown that missing data were associated with randomeffects, this does not preclude missing data from being ignorable.Indeed, if missing data at the last three tests depend onthe observed MMSE score or on the observed evolution of theMMSE, it induces a dependency on the random effects, butthe missing data are ignorable because missing values maybe predicted using observed data. This is an advantage ofmultivariate modeling, in that by using more observed informationit is more robust to missing data. Nevertheless, as itis not excluded that missing data are informative, it could beuseful to jointly model time to dropout using a shared randomeffect model as in Roy and Lin (2002). However, thiswould increase complexity of the estimation process and estimateswould depend on uncheckable parametric assumptions.Another useful extension would be to jointly model dementia,defining dementia diagnosis as the time at which the latentprocess first reaches an estimated threshold (Hashemi,Jacqmin-Gadda, and Commenges, 2003).AcknowledgementsThe authors thank Jean-François Dartigues, Luc Letenneur,and the PAQUID research team for providing the data.ReferencesAmieva, H., Jacqmin-Gadda, H., Orgogozo, J., LeCarret,N., Helmer, C., Letenneur, L., Barberger-Gateau, P.,Fabrigoule, C., and Dartigues, J. (2005). The 9 year cognitivedecline before dementia of the Alzheimer type: Aprospective population-based study. Brain 128, 1093–1101.Arminger, G. and Muthén, B. (1998). A Bayesian approach tononlinear latent variable models using the Gibbs sampler
Modèle nonlinéaire à processus latent 75Nonlinear Model with Latent Process for Cognitive Evolution 1023and the Metropolis-Hastings algorithm. Psychometrika63, 271–300.Dunson, D. (2000). Bayesian latent variable models for clusteredmixed outcomes. Journal of the Royal StatisticalSociety, Series B 62, 355–366.Dunson, D. (2003). Dynamic latent trait models for multidimensionallongitudinal data. Journal of the AmericanStatistical Association 98, 555–563.Fabrigoule, C., Rouch, I., Taberly, A., Letenneur, L.,Commenges, D., Mazaux, J.-M., Orgogozo, J.-M., andDartigues, J.-F. (1998). Cognitive process in preclinicalphase of dementia. Brain 121, 135–141.Farrer, L., Cupples, L., Haines, J., Hyman, B., Kukull, W.,Mayeux, R., Myers, R., Pericak-Vance, M., Risch, N.,and Van Duijn, C. (1997). Effects of age, sex and ethnicityon the association between apolipoprotein E genotypeand Alzheimer disease. A meta-analysis. APOE andAlzheimer Disease Meta Analysis Consortium. Journal ofthe American Medical Association 278, 1349–1356.Gray, S. M. and Brookmeyer, R. (1998). Estimating a treatmenteffect from multidimensional longitudinal data.Biometrics 54, 976–988.Hall, C., Ying, J., Kuo, L., Sliwinski, M., Buschke, H., Katz,M., and Lipton, R. (2001). Estimation of bivariate measurementshaving different change points, with applicationto cognitive ageing. Statistics in Medicine 20, 3695–3714.Harvey, D., Beckett, L., and Mungas, D. (2003). Multivariatemodeling of two associated cognitive outcomes in a longitudinalstudy. Journal of Alzheimer’s Disease 5, 357–365.Hashemi, R., Jacqmin-Gadda, H., and Commenges, D. (2003).A latent process model for joint modeling of events andmarker. Lifetime Data Analysis 9, 331–343.Jacqmin-Gadda, H., Fabrigoule, C., Commenges, D., andDartigues, J.-F. (1997). A 5-year longitudinal study ofthe Mini Mental State Examination in normal aging.American Journal of Epidemiology 145, 498–506.Jöreskog, K. and Yang, F. (1996). Nonlinear structural equationmodels: The Kenny-Judd model with interaction effects.In Advanced Structural Equation Modeling: Issuesand Techniques, G. A. Marcoulides and R. E. Schumacker(eds), 57–88. Mahwah, New Jersey: Lawrence ErlbaumAssociates.Kendall, M. and Stuart, A. (1977). The Advanced Theory ofStatistics, Volume 1. New York: MacMillan Publishing.Lee, S. and Song, X. (2004). Bayesian model comparison ofnonlinear structural equation models with missing continuousand ordinal categorical data. British Journal ofMathematical and Statistical Psychology 57, 131–150.Letenneur, L., Commenges, D., Dartigues, J.-F., andBarberger-Gateau, P. (1994). Incidence of dementia andAlzheimer’s disease in the elderly community residents ofsouth-western France. International Journal of Epidemiology23, 577–590.Longford, N. and Muthén, B. (1992). Factor analysis for clusteredobservations. Psychometrika 57, 581–597.Marquardt, D. (1963). An algorithm for least-squares estimationof nonlinear parameters. SIAM Journal on AppliedMathematics 11, 431–441.Molenberghs, G. and Verbeke, G. (2001). A review on linearmixed models for longitudinal data, possibly subject todropout. Statistical Modelling 1, 235–269.Muthén, B. (2002). Beyond SEM: General latent variablemodeling. Behaviormetrika 29, 81–117.Rabe-Hesketh, S., Skrondal, A., and Pickles, A. (2004). Generalizedmultilevel structural equation modelling. Psychometrika69, 167–190.Roy, J. and Lin, X. (2000). Latent variable models for longitudinaldata with multiple continuous outcomes. Biometrics56, 1047–1054.Roy, J. and Lin, X. (2002). Analysis of multivariate longitudinaloutcomes with nonignorable dropouts and missingcovariates: Changes in methadone treatment practices.Journal of the American Statistical Association 97, 40–52.Salthouse, T., Hancock, H., Meinz, E., and Hambrick, D.(1996). Interrelations of age, visual acuity, and cognitivefunctioning. Journal of Gerontology: PsychologicalSciences 51B, 317–330.Sammel, M. and Ryan, L. (1996). Latent variable models withfixed effects. Biometrics 52, 650–663.Sánchez, B. N., Budtz-Jorgensen, E., Ryan, L. M., and Hu,H. (2005). Structural equation models: A review withapplications to environmental epidemiology. Journal ofthe American Statistical Association 100, 1443–1455.Skrondal, A. and Rabe-Hesketh, S. (2004). Generalized LatentVariable Modelling: Multilevel, Longitudinal and StructuralEquation Models. Boca Raton, Florida: Chapman& Hall/CRC.Sliwinski, M., Hofer, S., and Hall, C. (2003). Correlated andcoupled cognitive change in older adults with or withoutpreclinical dementia. Psychology and Aging 18, 672–683.Song, X. and Lee, S. (2004). Bayesian analysis of two-levelnonlinear structural equation models with continuousand polytomous data. British Journal of Mathematicaland Statistical Psychology 57, 29–52.Wall, M. and Amemiya, Y. (2000). Estimation for polynomialstructural equation models. Journal of the American StatisticalAssociation 95, 929–940.Winnock, M., Letenneur, L., Jacqmin-Gadda, H.,Dallongeville, J., Amouyel, P., and Dartigues, J.(2002). Longitudinal analysis of the effect of apolipoproteinE ɛ4 and education on cognitive performancein elderly subjects: The PAQUID study. Journal ofNeurology, Neurosurgery and Psychiatry 72, 794–797.Yalcin, I. and Amemiya, Y. (2001). Nonlinear factor analysisas a statistical method. Statistical Science 16, 275–294.Received January 2005. Revised December 2005.Accepted January 2006.AppendixDetails on the TransformationsFor subject i and test k, we give the expressions of the functionand its Jacobian for the linear transformation, the combinationof a linear and a logit transformation, and the WeibullCDF.The linear transformation is defined for y ∈ R,η 1k ∈ R, andη 2k ∈ R ∗ as
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Université Victor Segalen Bordeaux
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3RemerciementsA Monsieur Jean-Louis
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5Un immense merci à tous ceux que
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TABLE DES MATIÈRES 92.3.2 Estimati
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Introduction 13ans atteintes d’un
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Chapitre 7BibliographieAmieva, H.,
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Bibliographie 1661, S19-25.Brown, E
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Chapitre 8Annexes8.1 Liste des publ
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Annexes 200Abstract :When investiga
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Annexes 224MMSE 0.0037 * 0.0013,0.0
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