Latent variable models and factor analysis : a unified approach / / David Bartholomew, Martin Knott, Irini Moustaki

• Record Nr.: UNINA9910208838003321
• Autore: Bartholomew David J
• Pubbl/distr/stampa: Hoboken, N.J., : Wiley, 2011
• ISBN: 9786613177698; 1-283-17769-2; 1-119-97059-8; 1-119-97058-X; 1-119-97370-8
• Edizione: [3rd ed.]
• Descrizione fisica: xiii, 277 p. : ill
• Collana: Wiley series in probability and statistics
• Altri autori (Persone): KnottM (Martin); MoustakiIrini
• Disciplina: 519.5/35
• Soggetti: Latent variables; Latent structure analysis; Factor analysis
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di bibliografia: Includes bibliographical references and indexes.
• Nota di contenuto: Intro -- Latent Variable Models and Factor Analysis -- Contents -- Preface -- Acknowledgements -- 1 Basic ideas and examples -- 1.1 The statistical problem -- 1.2 The basic idea -- 1.3 Two examples -- 1.3.1 Binary manifest variables and a single binary latent variable -- 1.3.2 A model based on normal distributions -- 1.4 A broader theoretical view -- 1.5 Illustration of an alternative approach -- 1.6 An overview of special cases -- 1.7 Principal components -- 1.8 The historical context -- 1.9 Closely related fields in statistics -- 2 The general linear latent variable model -- 2.1 Introduction -- 2.2 The model -- 2.3 Some properties of the model -- 2.4 A special case -- 2.5 The sufficiency principle -- 2.6 Principal special cases -- 2.7 Latent variable models with non-linear terms -- 2.8 Fitting the models -- 2.9 Fitting by maximum likelihood -- 2.10 Fitting by Bayesian methods -- 2.11 Rotation -- 2.12 Interpretation -- 2.13 Sampling error of parameter estimates -- 2.14 The prior distribution -- 2.15 Posterior analysis -- 2.16 A further note on the prior -- 2.17 Psychometric ; inference -- 3 The normal linear factor model -- 3.1 The model -- 3.2 Some distributional properties -- 3.3 Constraints on the model -- 3.4 Maximum likelihood estimation -- 3.5 Maximum likelihood estimation by the E-M algorithm -- 3.6 Sampling variation of estimators -- 3.7 Goodness of fit and choice of q -- 3.7.1 Model selection criteria -- 3.8 Fitting without normality assumptions: least squares methods -- 3.9 Other methods of fitting -- 3.10 Approximate methods for estimating -- 3.11 Goodness of fit and choice of q for least squares methods -- 3.12 Further estimation issues -- 3.12.1 Consistency -- 3.12.2 Scale-invariant estimation -- 3.12.3 Heywood cases -- 3.13 Rotation and related matters -- 3.13.1 Orthogonal rotation -- 3.13.2 Oblique rotation -- 3.13.3 Related matters.; 3.14 Posterior analysis: the normal case -- 3.15 Posterior analysis: least squares -- 3.16 Posterior analysis: a reliability approach -- 3.17 Examples -- 4 Binary data: latent trait models -- 4.1 Preliminaries -- 4.2 The logit/normal model -- 4.3 The probit/normal model -- 4.4 The equivalence of the response function and underlying variable approaches -- 4.5 Fitting the logit/normal model: the E-M algorithm -- 4.5.1 Fitting the probit/normal model -- 4.5.2 Other methods for approximating the integral -- 4.6 Sampling properties of the maximum likelihood estimators -- 4.7 Approximate maximum likelihood estimators -- 4.8 Generalised least squares methods -- 4.9 Goodness of fit -- 4.10 Posterior analysis -- 4.11 Fitting the logit/normal and probit/normal models: Markov chain Monte Carlo -- 4.11.1 Gibbs sampling -- 4.11.2 Metropolis-Hastings -- 4.11.3 Choosing prior distributions -- 4.11.4 Convergence diagnostics in MCMC -- 4.12 Divergence of the estimation algorithm -- 4.13 Examples -- 5 Polytomous data: latent trait models -- 5.1 Introduction -- 5.2 A response function model based on the sufficiency principle -- 5.3 Parameter interpretation -- 5.4 Rotation -- 5.5 Maximum likelihood estimation of the polytomous logit model -- 5.6 An approximation to the likelihood -- 5.6.1 One factor -- 5.6.2 More than one factor -- 5.7 Binary data as a special case -- 5.8 Ordering of categories -- 5.8.1 A response function model for ordinal variables -- 5.8.2 Maximum likelihood estimation of the model with ordinal variables -- 5.8.3 The partial credit model -- 5.8.4 An underlying variable model -- 5.9 An alternative underlying variable model -- 5.10 Posterior analysis -- 5.11 Further observations -- 5.12 Examples of the analysis of polytomous data using the logit model -- 6 Latent class models -- 6.1 Introduction.; 6.2 The latent class model with binary manifest variables -- 6.3 The latent class model for binary data as a latent trait model -- 6.4 K latent classes within the GLLVM -- 6.5 Maximum likelihood estimation -- 6.6 Standard errors -- 6.7 Posterior analysis of the latent class model with binary manifest variables -- 6.8 Goodness of fit -- 6.9 Examples for binary data -- 6.10 Latent class models with unordered polytomous manifest variables -- 6.11 Latent class models with ordered polytomous manifest variables -- 6.12 Maximum likelihood estimation -- 6.12.1 Allocation of individuals to latent classes -- 6.13 Examples for unordered polytomous data -- 6.14 Identifiability -- 6.15 Starting values -- 6.16 Latent class models with metrical manifest variables -- 6.16.1 Maximum likelihood estimation -- 6.16.2 Other methods -- 6.16.3 Allocation to categories -- 6.17 Models with ordered latent classes -- 6.18 Hybrid models -- 6.18.1 Hybrid model with binary manifest variables -- 6.18.2 Maximum likelihood estimation -- 7 Models and methods for manifest variables of mixed type -- 7.1 Introduction -- 7.2 Principal results -- 7.3 Other members of the exponential family -- 7.3.1 The binomial distribution -- 7.3.2 The ; Poisson distribution -- 7.3.3 The gamma distribution -- 7.4 Maximum likelihood estimation -- 7.4.1 Bernoulli manifest variables -- 7.4.2 Normal manifest variables -- 7.4.3 A general E-M approach to solving the likelihood equations -- 7.4.4 Interpretation of latent variables -- 7.5 Sampling properties and goodness of fit -- 7.6 Mixed latent class models -- 7.7 Posterior analysis -- 7.8 Examples -- 7.9 Ordered categorical variables and other generalisations -- 8 Relationships between latent variables -- 8.1 Scope -- 8.2 Correlated latent variables -- 8.3 Procrustes methods -- 8.4 Sources of prior knowledge -- 8.5 Linear structural relations models.; 8.6 The LISREL model -- 8.6.1 The structural model -- 8.6.2 The measurement model -- 8.6.3 The model as a whole -- 8.7 Adequacy of a structural equation model -- 8.8 Structural relationships in a general setting -- 8.9 Generalisations of the LISREL model -- 8.10 Examples of models which are indistinguishable -- 8.11 Implications for analysis -- 9 Related techniques for investigating dependency -- 9.1 Introduction -- 9.2 Principal components analysis -- 9.2.1 A distributional treatment -- 9.2.2 A sample-based treatment -- 9.2.3 Unordered categorical data -- 9.2.4 Ordered categorical data -- 9.3 An alternative to the normal factor model -- 9.4 Replacing latent variables by linear functions of the manifest variables -- 9.5 Estimation of correlations and regressions between latent variables -- 9.6 Q-Methodology -- 9.7 Concluding reflections of the role of latent variables in statistical modelling -- Software appendix -- References -- Author index -- Subject index.