Acknowledgements xv

1 Basic Ideas and Examples 1

1.1 The statistical problem 1

1.2 The basic idea 3

1.3 Two Examples 4

1.4 A broader theoretical view 6

1.5 Illustration of an alternative approach 8

1.6 An overview of special cases 10

1.7 Principal components 11

1.8 The historical context 12

1.9 Closely related fields in Statistics 17

2 The General Linear Latent Variable Model 19

2.1 Introduction 19

2.2 The model 19

2.3 Some properties of the model 20

2.4 A special case 21

2.5 The sufficiency principle 22

2.6 Principal special cases 24

2.7 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 30

2.11 Rotation 33

2.12 Interpretation 35

2.13 Sampling error of parameter estimates 38

2.14 The prior distribution 39

2.15 Posterior analysis 41

2.16 A further note on the prior 43

2.17 Psychometric Inference 44

3 The Normal Linear Factor Model 47

3.1 The model 47

3.2 Some distributional properties 48

3.3 Constraints on the model 50

3.4 Maximum likelihood estimation 50

3.5 Maximum likelihood estimation by the E-M algorithm 53

3.6 Sampling variation of estimators 55

3.7 Goodness of fit and choice of q 58

3.8 Fitting without normality assumptions: Least squares methods 59

3.9 Other methods of fitting 61

3.10 Approximate methods for estimating 62

3.11 Goodness-of-fit and choice of q for least squares methods 63

3.12 Further estimation issues 64

3.13 Rotation and related matters 69

3.14 Posterior analysis: The normal case 67

3.15 Posterior analysis: least squares 72

3.16 Posterior analysis: a reliability approach 74

3.17 Examples 74

4 Binary Data: Latent Trait Models 83

4.1 Preliminaries 83

4.2 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 90

4.6 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 97

4.10 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 109

4.13 Examples 109

5 Polytomous Data: Latent Trait Models 119

5.1 Introduction 119

5.2 A response function model based on the sufficiency principle 120

5.3 Parameter interpretation 124

5.4 Rotation 124

5.5 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 136

5.9 An alternative underlying variable model 144

5.10 Posterior analysis 147

5.11 Further observations 148

5.12 Examples of the analysis of polytomous data using the logit model 149

6 Latent Class Models 157

6.1 Introduction 157

6.2 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 161

6.5 Maximum likelihood estimation 162

6.6 Standard errors 164

6.7 Posterior analysis of the latent class model with binary manifest variables 166

6.8 Goodness of Fit 167

6.9 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 171

6.12 Maximum likelihood estimation 172

6.13 Examples for unordered polytomous data 174

6.14 Identifiability 178

6.15 Starting values 180

6.16 Latent class models with metrical manifest variables 180

6.17 Models with ordered latent classes 181

6.18 Hybrid models 182

7 Models and Methods for Manifest Variables of Mixed Type 191

7.1 Introduction 191

7.2 Principal results 192

7.3 Other members of the exponential family 193

7.4 Maximum likelihood estimation 195

7.5 Sampling properties and Goodness of Fit 201

7.6 Mixed latent class models 202

7.7 Posterior analysis 203

7.8 Examples 204

7.9 Ordered categorical variables and other generalisations 208

8 Relationships Between Latent Variables 213

8.1 Scope 213

8.2 Correlated latent variables 213

8.3 Procrustes methods 215

8.4 Sources of prior knowledge 215

8.5 Linear structural relations models 216

8.6 The LISREL model 218

8.7 Adequacy of a structural equation model 221

8.8 Structural relationships in a general setting 222

8.9 Generalisations of the LISREL model 223

8.10 Examples of models which are indistinguishable 224

8.11 Implications for analysis 227

9 Related Techniques for Investigating Dependency 229

9.1 Introduction 229

9.2 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 240

9.6 Q-Methodology 242

9.7 Concluding reflections of the role of latent variables in statistical modelling 244

References 247

Software appendix 247

References 249

Author Index 265

Subject Index 271