Latent Variable Models and Factor Analysis

David Bartholomew, Martin Knott and Irini Moustaki, Department of Statistics, The London School of Economics and Political Science, London, UK

• Preface xi  Acknowledgements xv 1 Basic Ideas and Examples 1 1.1 The statistical problem 1 1.2 The basic idea 31.3 Two Examples 41.4 A broader theoretical view 6 1.5 Illustration of an alternative approach 81.6 An overview of special cases 101.7 Principal components 111.8 The historical context 121.9 Closely related fields in Statistics 17 2 The General Linear Latent Variable Model 19 2.1 Introduction 192.2 The model 19 2.3 Some properties of the model 20 2.4 A special case 212.5 The sufficiency principle 22 2.6 Principal special cases 242.7 Latent variable models with non-linear terms 25 2.8 Fitting the models 27 2.9 Fitting by maximum likelihood 29 2.10 Fitting by Bayesian methods 302.11 Rotation 332.12 Interpretation 35 2.13 Sampling error of parameter estimates 38 2.14 The prior distribution 39 2.15 Posterior analysis 412.16 A further note on the prior 43 2.17 Psychometric Inference 443 The Normal Linear Factor Model 47 3.1 The model 473.2 Some distributional properties 48 3.3 Constraints on the model 503.4 Maximum likelihood estimation 50 3.5 Maximum likelihood estimation by the E-M algorithm 53 3.6 Sampling variation of estimators 553.7 Goodness of fit and choice of q 58 3.8 Fitting without normality assumptions: Least squares methods 59 3.9 Other methods of fitting 613.10 Approximate methods for estimating 62 3.11 Goodness-of-fit and choice of q for least squares methods 633.12 Further estimation issues 643.13 Rotation and related matters 69 3.14 Posterior analysis: The normal case 67 3.15 Posterior analysis: least squares 723.16 Posterior analysis: a reliability approach 74 3.17 Examples 744 Binary Data: Latent Trait Models 83 4.1 Preliminaries 834.2 The logit/normal model 84 4.3 The probit/normal model 86 4.4 The equivalence of the response function and underlying variable approaches 88 4.5 Fitting the logit/normal model: the E-M algorithm 904.6 Sampling properties of the maximum likelihood estimators 94 4.7 Approximate maximum likelihood estimators 95 4.8 Generalised least squares methods 96 4.9 Goodness of fit 974.10 Posterior analysis 100 4.11 Fitting the logit/normal and probit/normal models: Markov Chain Monte Carlo 102 4.12 Divergence of the estimation algorithm 1094.13 Examples 1095 Polytomous Data: Latent Trait Models 119  5.1 Introduction 1195.2 A response function model based on the sufficiency principle 120 5.3 Parameter interpretation 1245.4 Rotation 1245.5 Maximum likelihood estimation of the polytomous logit model 125 5.6 An approximation to the likelihood 126 5.7 Binary data as a special case 134 5.8 Ordering of categories 1365.9 An alternative underlying variable model 144 5.10 Posterior analysis 1475.11 Further observations 1485.12 Examples of the analysis of polytomous data using the logit model 1496 Latent Class Models 1576.1 Introduction 1576.2 The latent class model with binary manifest variables 158 6.3 The latent class model for binary data as a latent trait model 159 6.4 Latent Classes within the GLLVM 1616.5 Maximum likelihood estimation 1626.6 Standard errors 1646.7 Posterior analysis of the latent class model with binary manifest variables 166 6.8 Goodness of Fit 1676.9 Examples for binary Data 167 6.10 Latent class models with unordered polytomous manifest variables 170 6.11 Latent class models with ordered polytomous manifest variables 1716.12 Maximum likelihood estimation 1726.13 Examples for unordered polytomous data 174 6.14 Identifiability 1786.15 Starting values 1806.16 Latent class models with metrical manifest variables 1806.17 Models with ordered latent classes 1816.18 Hybrid models 1827 Models and Methods for Manifest Variables of Mixed Type 191 7.1 Introduction 1917.2 Principal results 192 7.3 Other members of the exponential family 193 7.4 Maximum likelihood estimation 1957.5 Sampling properties and Goodness of Fit 201 7.6 Mixed latent class models 2027.7 Posterior analysis 2037.8 Examples 2047.9 Ordered categorical variables and other generalisations 208 8 Relationships Between Latent Variables 2138.1 Scope 2138.2 Correlated latent variables 2138.3 Procrustes methods 2158.4 Sources of prior knowledge 215 8.5 Linear structural relations models 216 8.6 The LISREL model 2188.7 Adequacy of a structural equation model 221 8.8 Structural relationships in a general setting 2228.9 Generalisations of the LISREL model 2238.10 Examples of models which are indistinguishable 224 8.11 Implications for analysis 2279 Related Techniques for Investigating Dependency 229 9.1 Introduction 2299.2 Principal Components Analysis, (PCA) 229 9.3 An alternative to the normal factor model 236 9.4 Replacing latent variables by linear functions of the manifest variables 238 9.5 Estimation of correlations and regressions between latent variables 2409.6 Q-Methodology 2429.7 Concluding reflections of the role of latent variables in statistical modelling 244 References 247Software appendix 247 References 249Author Index 265Subject Index 271

“Latent Variable Models and Factor Analysis provides a comprehensive and unified approach to factor analysis and latent variable modeling from a statistical perspective.”  (Mathematical Reviews, 2012)"Statistical techniques to study the nature and interpretation of a latent variable should be highly useful for researchers and practitioners across several fields. The third edition of this book is comprehensive and provides a solid foundation for understanding these techniques, and is strongly recommended." (Book Pleasures, 2012)