Subject Index
A
Absolute misfit indices, 72
ADF estimator, see Asymptotic distribution-free estimator
Ad hoc indices of fit, 69
AIC, see Akaike's Information Criterion
Akaike's Information Criterion (AIC), 69
Alternative models (AM) scenario, 8
AM scenario, see Alternative models scenario
Analysis of covariance structures (factorial equivalence of measuring instrument), 193–226
configural model, 195, 196, 206
error message, 211
expected parameter change statistic, 201
establishing baseline models (elementary and secondary teachers), 198–206
establishing baseline models (general notion), 197–198
testing invariance (configuration model), 206–208
model-specific commands, 211
modification index, 198
Mplus input file specification and output file results, 208–225
testing invariance (measurement model), 212–221
testing invariance (structural model), 221–222
multigroup invariance item reliability and, 195
popular strategy, 198
testing of (across independent samples), 197
testing of (general notion), 194–197
noninvariant unstandardized estimates, 222, 223
partial measurement invariance, 198
scaling correction factor, 217
Asymptotic distribution-free (ADF) estimator, 100, 131
B
Bayes Information Criterion (BIC), 69
Between-cluster effects, 348
Between-level variability, see Within- and between-level variability
BIC, see Bayes Information Criterion
C
categorical variables, 129–133
general analytic strategies, 131–133
underlying assumptions, 131
Causal structure, equivalence of (full structural equation model), 259–282
cross-validation in structural equation modeling, 259–261
Heywood cases, 265
hypothesized model, 262
latent variable covariance matrix, 265
Mplus input file specification and output file results, 264–281
establishing baseline model for calibration group, 264–272
testing multigroup invariance, 273–281
Teacher Stress Scale, 265
testing invariance across calibration and validation samples, 261–262
Causal structure, validity of (full structural equation model), 147–189
asymptotic distribution-free estimator, 131
chi-square-difference values, 168
endogenous variables, 156
estimation difficulties, 131
exogenous variables, 156
expected parameter change statistic, 163, 177
confirmatory factor analyses, 150–152
formulation of indicator variables, 149–150
Mplus input file specification and output file results, 153–167
model misspecification, 163–167
model parameter estimation, 162–163
selected Mplus output (model 6), 177–179
PSI matrix, 159
robust maximum likelihood estimator, 157
warning message, 173
weighted least squares estimator, 131
CFA, see Confirmatory factor analysis
CFI, see Comparative Fit Index
Change, testing of over time (latent growth curve model), 313–344
hypothesized covariate model (age and surgery as predictors of change), 338–343
hypothesized dual-domain LGC model, 316- 321
modeling interindividual differences in change, 320–321
modeling intraindividual change, 317- 320
individual growth parameters, 320
latent growth curve models, 314
measuring change in individual growth over time (general notion), 316
Mplus input file specification and output file results, 321–338
nested model, 343
psychological morbidity, 344
quadratic factor, 317
structural model, 320
unequal twins, 314
Chi-square test
configural model, 279
correlated traits/correlated methods, 292
covariance model, 341
equality constraints model, 281
final model, 336
MLV model, 363
parameter estimation and, 157
residual covariance and, 113
test of model fit, 66, 103, 182
two-factor model, 92
Clustered data, 345
Communalities, 82
Comparative Fit Index (CFI), 69, 152, 354
Configural model, 195, 196, 206
Confirmatory factor analysis (CFA), 5, 22
Congeneric indicators, 31, 126, 140, 289
Construct validity, evidence of (multitrait-multimethod model), 285–311
congeneric indicators, 289
correlated uniquenesses approach to MTMM analyses, 303–310
input file, 305
general CFA approach to MTMM analyses, 286–287
independent measures of different traits, 300
discriminant validity, 300–301
Mplus input file specification and output file results, 288–295
freely correlated traits/uncorrelated methods, 295
no traits/correlated methods, 293
perfectly correlated traits/freely correlated methods, 294
nested model, 300
convergent validity, 301
discriminant validity, 302–303
self-perceptions, 310
Correlated traits and correlated methods (CTCM), 288
Correlated traits/uncorrelated methods (CTUM), 295
Correlated uniquenesses (CU) model, 306
Cross-loading, 87, 105, 247, 355
CTCM, see Correlated traits and correlated methods
CTUM, see Correlated traits/uncorrelated methods
CU model, see Correlated uniquenesses model
D
Dependent variable, 16
Diagonally weighted least squares (DWLS) estimator, 132
Disturbance terms, 17
DWLS estimator, see Diagonally weighted least squares estimator
E
Ecological analysis, 371
Ecological fallacy, 348
ECVI, see Expected Cross-Validation Index
EFA, see Exploratory factor analysis
EM algorithms, see Expectation maximization algorithms
Endogenous variables, 156
EPC statistic, see Expected parameter change statistic
Error uniqueness, 11
Estimator(s), 62
asymptotic distribution-free, 131
development of, 128
diagonally weighted least squares, 132
maximum likelihood, 52, 53, 104
MLMV, 121
primary, 132
robust maximum likelihood, 99, 100, 101, 104, 157
statistical, 24
unweighted least squares, 132
weighted least squares, 131
Exogenous variables, 156
Expectation maximization (EM) algorithms, 349
Expected Cross-Validation Index (ECVI), 72
Expected parameter change (EPC) statistic, 86, 87, 163, 177, 201
Exploratory factor analysis (EFA), 5
F
Factor analytic model, 5
Factors, 17
First-order confirmatory factor analysis model, factorial validity of scores from measuring instrument, 95–124
cross-loading, 105
kurtosis values, 99
maximum likelihood estimator, 104
measuring instrument under study, 96
MLMV estimator, 121
Mplus input file specification and output file results, 101–121
comparison of ML and MLM output, 102–106
input file 1, 101
input file 2, 107
model misspecification, 104–106
testing validity of model 2, 107
testing validity of model 3, 110–121
robust maximum likelihood estimator, 99, 100, 101, 104
First-order confirmatory factor analysis model, factorial validity of theoretical construct, 43–93
absolute misfit indices, 72
absolute model fit index, 69
ad hoc indices of fit, 69
Akaike's Information Criterion, 69
Bayes Information Criterion, 69
communalities, 82
Comparative Fit Index, 69
comparative model fit index, 69
confirmatory mode, 88
cross-loading, 87
“error of approximation” index, 74
estimator, 62
expected parameter change statistic, 86, 87
exploratory mode, 88
Heywood cases, 78
incremental model fit index, 69
independence model, 67
lambda matrix, 48
maximum likelihood estimator, 52, 53
model fit indices, 69
Mplus input file specification and output file results, 48–89
appropriateness of standard errors, 78
assessment of individual parameter estimates, 77–82
assessment of model as a whole, 64–77
estimation process, 65
feasibility of parameter estimates, 77–78
goodness-of-fit statistics, 66–77
input file, 89
input file specification, 48–54
issue of statistical significance, 64–65
model-fitting process, 64
statistical significance of parameter estimates, 78–82
summary of model and analysis specifications, 56–64
noncentrality parameter, 74
nontarget loadings, 45
null model, 67
overfitted model, 88
parsimony-corrective indices of fit, 70
practical indices of fit, 69
predictive indices of fit, 70
psi matrix, 48
secondary factor loading, 87
self-concept as one-factor structure, 91–92
self-concept as two-factor structure, 89
specification searches, 88
subjective indices of fit, 69
target loading, 45
theta matrix, 48
Tucker-Lewis Fit Index, 69
z-statistic, 78
Fixed factor method, 34
Full latent variable model, 6–7
Full structural equation model, 6–7, 38–39
Full structural equation model, equivalence of causal structure, 259–282
cross-validation in structural equation modeling, 259–261
Heywood cases, 265
hypothesized model, 262
latent variable covariance matrix, 265
Mplus input file specification and output file results, 264–281
establishing baseline model for calibration group, 264–272
testing multigroup invariance, 273–281
Teacher Stress Scale, 265
testing invariance across calibration and validation samples, 261–262
Full structural equation model, validity of causal structure, 147–189
asymptotic distribution-free estimator, 131
chi-square-difference values, 168
endogenous variables, 156
estimation difficulties, 131
exogenous variables, 156
expected parameter change statistic, 163, 177
confirmatory factor analyses, 150–152
formulation of indicator variables, 149–150
Mplus input file specification and output file results, 153–167
model misspecification, 163–167
model parameter estimation, 162–163
selected Mplus output (model 6), 177–179
PSI matrix, 159
robust maximum likelihood estimator, 157
warning message, 173
weighted least squares estimator, 131
G
Goodness-of-fit
adequacy of sample data, 6
alternative indices, 69
baseline model, 233
causal structure, 269
CFA structure, 355
configural model, 279
correlated uniquenesses model, 304, 306
covariance model, 341
equality constraints model, 281
exclusive reliance on indices, 77
inadequate, 64
incremental fit indices and, 73
invariance testing and, 300
model cross-validation, 262
Mplus output information, 25
MTMM models, 299
NTCM model, 293
postulated relations among variables, 3
residual and, 7
revised baseline model, 272
sample data and, 76
Satorra-Bentler-scaled χ2
statistic and, 132
statistics, 137
category characteristics, 70
first-order confirmatory factor analysis model, 66–77
InvModel.2, 214
InvModel.4, 218
InvModel.6, 222
link between the χ2 statistic and, 68
MBI three-factor structure, 198
ML estimation, 103
two-factor model, 92
H
Hypothesized model
full structural equation model
equivalence of causal structure, 262
validity of causal structure, 147–152
latent growth curve model, 338–343
means and covariance structures, 231
measuring instrument, factorial equivalence of (analysis of covariance structures), 197–208
measuring instrument, factorial validity of scores
first-order confirmatory factor analysis model), 96–101
second-order confirmatory factor analysis model), 126
multitrait-multimethod model, 287
theoretical construct, factorial validity of (first-order confirmatory factor analysis model), 43–48
I
ICCs, see Intraclass correlation coefficients
Independence model, 67
Indicator variables, formulation of, 149–150
Intraclass correlation coefficients (ICCs), 354
Lambda matrix, 48
Latent factor means, equivalence of (means and covariance structures), 227–257
cross-loading, 247
hypothesized model, 231
modification indices, 233
moment matrix, 228
Mplus input file specification and output file results (testing invariance), 241–254
common residual covariance, 244–246
multigroup equivalence, original technique, 256
testing latent mean structures (basic notion), 227–231
factor identification, 230
model parameterization, 229–230
testing multigroup invariance, 231–240
establishing baseline models, 233–240
testing multigroup invariance (other considerations), 254–256
partial measurement invariance, 254–255
statistical versus practical evaluative criteria, 255–256
Latent growth curve (LGC) model, testing change over time, 313–344
hypothesized covariate model (age and surgery as predictors of change), 338–343
hypothesized dual-domain LGC model, 316–321
interindividual differences in change, 320–321
intraindividual change, 317–320
individual growth parameters, 320
latent growth curve models, 314
measuring change in individual growth over time (general notion), 316
Mplus input file specification and output file results, 321–338
nested model, 343
psychological morbidity, 344
quadratic factor, 317
structural model, 320
unequal twins, 314
Latent variables
exogenous versus, 5
LGC model, see Latent growth curve model, testing change over time
Longitudinal analyses, 345
M
MACS, see Means and covariance structures
Manifest variables, 4
Maslach Burnout Inventory (MBI), 152, 194
Maximum likelihood (ML) estimation, 102, 126
MBI, see Maslach Burnout Inventory Means and covariance structures (MACS), 193, 227–257
cross-loading, 247
hypothesized model, 231
modification indices, 233
moment matrix, 228
Mplus input file specification and output file results (testing invariance), 241–254
common residual covariance, 244–246
multigroup equivalence, original technique, 256
testing latent mean structures (basic notion), 227–231
factor identification, 230
model parameterization, 229–230
testing multigroup invariance, 231–240
establishing baseline models, 233–240
testing multigroup invariance (other considerations), 254–256
partial measurement invariance, 254–255
statistical versus practical evaluative criteria in determining evidence of invariance, 255–256
Measurement error, 11
Measuring instrument, factorial equivalence of (analysis of covariance structures), 193–226
configural model, 195, 196, 206
error message, 211
expected parameter change statistic, 201
establishing baseline models (elementary and secondary teachers), 198–206
establishing baseline models (general notion), 197–198
testing invariance (configuration model), 206–208
model-specific commands, 211
modification index, 198
Mplus input file specification and output file results, 208–225
testing invariance (measurement model), 212–221
testing invariance (structural model), 221–222
multigroup invariance
item reliability and, 195
popular strategy, 198
testing of (across independent samples), 197
testing of (general notion), 194–197
noninvariant unstandardized estimates, 222, 223
partial measurement invariance, 198
scaling correction factor, 217
Measuring instrument, factorial validity of scores from (first-order confirmatory factor analysis model), 95–124
cross-loading, 105
kurtosis values, 99
maximum likelihood estimator, 104
measuring instrument under study, 96
MLMV estimator, 121
Mplus input file specification and output file results, 101–121
comparison of ML and MLM output, 102–106
input file 1, 101
input file 2, 107
model misspecification, 104–106
testing validity of model 2, 107
testing validity of model 3, 110–121
robust maximum likelihood estimator, 99, 100, 101, 104
Measuring instrument, factorial validity of scores from (second-order confirmatory factor analysis model), 125–146
analysis of categorical data, 126–133
categorical variables, 129–133
general analytic strategies, 131–133
underlying assumptions, 131
asymptotic distribution-free estimator, 131
congeneric indicators, 126, 140
development of estimators, 128
diagonally weighted least squares estimator, 132
hypothesized model, 126 Mplus input file specification and output file results, 133–146
primary estimators, 132
unweighted least squares estimator, 132
weighted least squares estimator, 131
MG scenario, see Model generating scenario
ML estimation, see Maximum likelihood estimation
MLM estimation, see Robust maximum likelihood estimation
MLV model, see Multilevel model
Model generating (MG) scenario, 8
Model identification, 30
degrees of freedom, 136
just-identification, 32, 93, 136
overidentification, 32, 33, 229, 360
underidentification, 32, 33, 212, 252, 277
Model-specific commands, 211
Modification Indices (MIs), 152, 198, 233
Moment matrix, 228
Mplus notation and input file components and structure, 19–26
confirmatory factor analysis model, 22
congeneric indicators, 31
dependent variable, 38
fixed factor method, 34
latent variable scaling, 30
model identification, 30, 31–39
full structural equation model, 38–39
latent variable scaling, 33–34
model specification from two perspectives, 27–31
Mplus language generator, 26
Mplus notation and input file components and structure, 19–26
ANALYSIS command, 24
DATA command, 21
DEFINE command, 24
MONTECARLO command, 26
SAVEDATA command, 25
TITLE command, 21
VARIABLE command, 22
observed variables, variance-covariance matrix, 32
overview of remaining chapters, 39
reference variable method, 33
residuals, 28
statistical estimators, 24
statistical identification, 30
MTMM model, see Multitrait-multimethod model, evidence of construct validity
Multigroup invariance
item reliability and, 195
original technique, 256
popular strategy, 198
across independent samples, 197
establishing baseline models, 233–240
Multilevel (MLV) model, 345–371
between-cluster effects, 348
clustered data, 345
confirmatory factor analysis model, 355
cross-loading, 355
ecological analysis, 371
ecological fallacy, 348
error message text, 360
expectation maximization algorithms, 349
exploratory factor analysis, 351
current approach to analyses of hypothesized MLV model, 353–354
estimation, 353
item scaling, 353
sample size, 353
intraclass correlation coefficients, 354
latent growth curve modeling, 345
longitudinal analyses, 345
modification indices, 355
Mplus input file specification and output file results, 354–370
CFA of FV scale structure, 354–358
MLV model of FV scale structure, 359–367
potential analytic extensions, 367–370
nested data, 345
overview of multilevel modeling, 346–350
disaggregation approach, 347
MLV model-testing approaches, 349–350
multilevel analyses of hierarchical data, 348–350
single-level analyses of hierarchical data, 347
saturated model, 363
within-cluster effects, 348
Multitrait-multimethod (MTMM) model, evidence of construct validity, 285–311
congeneric indicators, 289
correlated uniquenesses approach to MTMM analyses, 303–310
input file, 305
general CFA approach to MTMM analyses, 286–287
independent measures of different traits, 300
discriminant validity, 300–301
Mplus input file specification and output file results, 288–295
freely correlated traits/uncorrelated methods, 295
no traits/correlated methods, 293
perfectly correlated traits/freely correlated methods, 294
nested model, 300
convergent validity, 301
discriminant validity, 302–303
self-perceptions, 310
N
NCP, see Noncentrality parameter
Nested data, 345
Nested model, 69, 71, 300, 343
CFA MTMM, 300
CFI values, 256
comparison, 343
descriptions, 288
indicators of comparisons, 298
taxonomy of comparisons, 286
Noncentrality parameter (NCP), 74
Nonrecursive model, 7
Nontarget loadings, 45
No traits/correlated methods (NTCM), 293
NTCM, see No traits/correlated methods
Null model, 67
O
Observed variables, 4
variance-covariance matrix, 32
Overfitted model, 88
P
Parsimony-corrective indices of fit, 70
Path diagrams, 9
PCTCM, see Perfectly correlated traits/freely correlated methods
Peer activity, 13
Peer network, 11
Perfectly correlated traits/freely correlated methods (PCTCM), 294
Practical indices of fit, 69
Predictive indices of fit, 70
Psi matrix, 48
Psychological morbidity, 344
Q
Quadratic factor, 317
R
Random measurement error, 11
Recursive model, 7
Reference variable method, 33
Residual error, 11
RMR, see Root Mean Square Residual
RMSEA, see Root Mean Square Error of Approximation
Robust maximum likelihood (MLM) estimation, 102, 126, 152
Root Mean Square Error of Approximation (RMSEA), 72
category, 72
CFA structure, 355
configural model, 279
correlated traits/uncorrelated methods, 295
correlated uniquenesses model, 307
covariate model, 341
discrepancy measured by, 73
equality constraints model, 281
final model, 116, 182, 244, 336
first proposal, 73
inferior goodness-of-fit and, 91
invariance of MBI, 225
InvModel.1, 212
MI results, 168
model complexity and, 74
MTMM models, 299
no traits/correlated methods, 293
parsimonious model, 334
residual covariance and, 113
respecified means model, 253
tests of model fit, 66
two-factor model, 92
Root Mean Square Residual (RMR), 76
S
2, Saturated model, 70, 93, 169, 256, 298, 363
SC, see Self-concept
SC scenario, see Strictly confirmatory scenario
Secondary factor loading, 87
Second-order confirmatory factor analysis model, factorial validity of scores from measuring instrument, 125–146
analysis of categorical data, 126–133
categorical variables, 129–133
general analytic strategies, 131–133
underlying assumptions, 131
asymptotic distribution-free estimator, 131
congeneric indicators, 126, 140
development of estimators, 128
diagonally weighted least squares estimator, 132
hypothesized model, 126
Mplus input file specification and output file results, 133–146
primary estimators, 132
unweighted least squares estimator, 132
weighted least squares estimator, 131
Self-concept (SC), 38
SEM, see Structural equation modeling
Specification searches, 88
SRMR, see Standardized Root Mean Square Residual
Standardized Root Mean Square Residual (SRMR), 72
category, 72
CFA structure, 355
configural model, 279
correlated traits/uncorrelated methods, 295
correlated uniquenesses model, 307
covariate model, 341
equality constraints model, 281
final model, 116, 182, 244, 336
invariance of MBI, 225
InvModel.1, 212
MI results, 168
MLM model, 104
MTMM models, 299
no traits/correlated methods, 293
parsimonious model, 334
residual covariance and, 113
respecified means model, 253
two-factor model, 92
Statistical identification, 30
degrees of freedom, 136
just-identification, 32, 93, 136
overidentification, 32, 33, 229, 360
underidentification, 32, 33, 212, 252, 277
Strictly confirmatory (SC) scenario, 8
Structural equation modeling (SEM), 3–17
alternative models scenario, 8
exogenous versus endogenous latent variables, 5
full latent variable model, 6–7
general purpose and process of statistical modeling, 7–8
latent versus observed variables, 4–5
confirmatory factor analysis, 5
dependent variable, 16
disturbance terms, 17
error uniqueness, 11
exploratory factor analysis, 5
factors, 17
general Mplus structural equation model, 15–17
generically labeled model, 17
general structural equation model, 9–15
basic composition, 14
formulation of covariance and mean structures, 14–15
nonvisible components of model, 13
symbol notation, 9
manifest variables, 4
measurement error, 11
model generating scenario, 8
model-testing procedure, primary task, 7
nonexistence of certain parameters, 13
nonrandom measurement error, 11
nonrecursive model, 7
observed variable, direct measurement of, 4
peer activity, 13
peer network, 11
random measurement error, 11
recursive model, 7
residual error, 11
residuals, 11
strictly confirmatory scenario, 8
structural model, 14
Structural model, 14
Subjective indices of fit, 69
T
Target loading, 45
Teacher Stress Scale (TSS), 265
Theoretical construct, factorial validity of (first-order confirmatory factor analysis model), 43–93
absolute misfit indices, 72
absolute model fit index, 69
ad hoc indices of fit, 69
Akaike's Information Criterion, 69
Bayes Information Criterion, 69
communalities, 82
Comparative Fit Index, 69
comparative model fit index, 69
confirmatory mode, 88
cross-loading, 87
“error of approximation” index, 74
estimator, 62
expected parameter change statistic, 86, 87
exploratory mode, 88
Heywood cases, 78
incremental model fit index, 69
independence model, 67
lambda matrix, 48
maximum likelihood estimator, 52, 53
model fit indices, 69
Mplus input file specification and output file results, 48–89
appropriateness of standard errors, 78
assessment of individual parameter estimates, 77–82
assessment of model as a whole, 64–77
estimation process, 65
feasibility of parameter estimates, 77–78
goodness-of-fit statistics, 66–77
input file, 89
input file specification, 48–54
issue of statistical significance, 64–65
model-fitting process, 64
statistical significance of parameter estimates, 78–82
summary of model and analysis specifications, 56–64
noncentrality parameter, 74
nontarget loadings, 45
null model, 67
overfitted model, 88
parsimony-corrective indices of fit, 70
practical indices of fit, 69
predictive indices of fit, 70
psi matrix, 48
secondary factor loading, 87
self-concept as one-factor structure, 91–92
self-concept as two-factor structure, 89
specification searches, 88
subjective indices of fit, 69
target loading, 45
theta matrix, 48
Tucker-Lewis Fit Index, 69
z-statistic, 78
Theta matrix, 48
TLI, see Tucker-Lewis Fit Index
TSS, see Teacher Stress Scale
Tucker-Lewis Fit Index (TLI), 69, 158
U
ULS estimator, see Unweighted least squares estimator
Unweighted least squares (ULS) estimator, 132
V
Variability, see Within- and between-level variability
Variance-covariance matrix, 32
W
Weighted least squares (WLS) estimator, 131, 132
Within- and between-level variability (multilevel model), 345–371
between-cluster effects, 348
clustered data, 345
confirmatory factor analysis model, 355
cross-loading, 355
ecological analysis, 371
ecological fallacy, 348
error message text, 360
expectation maximization algorithms, 349
exploratory factor analysis, 351
current approach to analyses of, 353–354
estimation, 353
item scaling, 353
sample size, 353
intraclass correlation coefficients, 354
latent growth curve modeling, 345
longitudinal analyses, 345
modification indices, 355
Mplus input file specification and output file results, 354–370
CFA of FV scale structure, 354–358
MLV model of FV scale structure, 359–367
potential analytic extensions, 367–370
nested data, 345
overview of multilevel modeling, 346–350
disaggregation approach, 347
MLV model-testing approaches, 349–350
multilevel analyses of hierarchical data, 348–350
single-level analyses of hierarchical data, 347
saturated model, 363
within-cluster effects, 348
Within-cluster effects, 348
WLS estimator, see Weighted least squares estimator
Z
z-statistic, 78