statistika univerzitetni podiplomski Študijski program...

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Statistika Univerzitetni Podiplomski Študijski Program

Univerza v Ljubljani

Multivariatna Analiza Strukturni Modelni-Structural Equation Models (SEM)

Anuška Ferligoj

Vesna Omladič

Germà Coenders

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Objectives

To introduce models that relate variables measured with error. To introduce Structural Equation Models with latent variables (SEM). To learn all stages of fitting these models. To become familiar with the LISREL program. To enable participants to critically read articles in which these models are applied.

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Index 1. History and objectives of SEM. 2. Example. 3. Intuitive explanation of the basics of SEM.

• Interdependence analysis. Path analysis. The regression model from a different perspective.

• Degrees of freedom, residuals and goodness of fit.

• Measurement errors in regression models.

• Confirmatory factor analysis model, reliability, validity.

• Modelling stages.

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4. Theoretical and statistical grounds.

• Specification.

• Identification.

• Estimation. 5. Goodness of fit assessment and model modification. 6. LISREL and PRELIS programs. 7. Results and interpretation.

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8. Example 2: measurement quality evaluation with the multitrait-multimethod model. Extension: ordinal variables. 9. Example 3: Parents’ and children’s values. Extensions:

• Mean structures.

• Missing data.

• Equality constraints.

• Multiple-group models.

• Factor invariance.

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Manuals Batista-Foguet, J.M. & Coenders, G. (1998). Introducción a los modelos estructurales.

Utilización del análisis factorial confirmatorio para la depuración de un cuestionario. In: Renom J. (Ed.), Tratamiento informatizado de datos, (pp. 229-286). Barcelona: Masson.

Batista-Foguet, J.M. & Coenders, G. (2000). Modelos de ecuaciones estructurales. Madrid: La Muralla.

Bollen, K. A. (1989). Structural equations with latent variables. New York: John Wiley & Sons.

Bollen, K.A y Long, J.S. (1993). Testing structural equation models. Thousand Oaks: Sage. Jöreskog, K. G. & Sörbom, D. (1988). PRELIS, a program for multivariate data screening

and data summarization. A preprocessor for LISREL. Chicago: Scientific Software International.

Jöreskog, K. G. & Sörbom, D. (1993a). New features in LISREL8. Chicago: Scientific Software International.

Jöreskog, K. G. & Sörbom, D. (1993b). New features in PRELIS2. Chicago: Scientific Software International.

Jöreskog, K. G. & Sörbom, D. (1993c). Structural equation modeling with the SIMPLIS command language. Chicago: Scientific Software International.

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Jöreskog, K. G., Sörbom, D. du Toit, S. & du Toit, M. (1999). LISREL8: New statistical features. Chicago: Scientific Software International.

Kelloway, E.K. (1998). Using LISREL for structural equation modelling. A Researcher’s Guide. London: Sage.

Raykov, T. & Marcoulides, G.A. (2000). A first course in structural equation modeling. Mahwah: Lawrence Erlbaum.

Saris, W.E. & Stronkhorst, L.H. (1984). Causal modelling in nonexperimental research. Amsterdam: Sociometric Research Foundation.

Schumacker, R. E. & Lomax, R. G. (1996). A beginner’s guide to structural equation modeling.. Mahwah: Lawrence Erlbaum.

du Toit, S., du Toit (2001). Interactive LISREL. User’s Guide. Chicago: Scientific Software International.

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Articles Batista-Foguet, J. M., Coenders, G. & Artés-Ferragud, M. (2001). Using Structural

Equation Models to Evaluate the Magnitude of Measurement Error in Blood Pressure. Statistics in Medicine, 20, 2351-2368.

Batista-Foguet, J. M., Coenders, G. & Sureda, J. (1996). Satisfaction in Catalonia, Spain. In: Saris, W. E., Veenhoven, R., Scherpenzeel, A. C. & Bunting, B. (Eds.), A comparative study of satisfaction with life in Europe, (pp. 155-174). Budapest: Eötvös University Press.

Batista-Foguet, J.M., & Cuadras, C. (1983). Análisis de la causalidad y planteamiento LISREL a partir de los modelos de medida. Qüestiió, 7, 359-383.

Batista-Foguet, J.M. & Saris, W. E. (1992). A new measurement procedure for attitudinal research. Analysis of its psychometric and informational properties. Quality & Quantity, 26, 127-146.

Batista-Foguet, J.M. & Saris, W. E. (1997). Tests of stability in attitude research. Quality & Quantity, 31, 269-285.

Casas, F., Coenders, G. & Pascual, S. (2001). Subjective Well-Being and Socially Risky Behaviours of Youth. In Casas, F. & Saurina, C. (Eds.) Proceedings of the Third Conference of the International Society for Quality of Life Studies (pp. 367-384) Girona: Universitat de Girona.

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Coenders, G., Batista-Foguet, J. M. & Satorra, A. (1995). Scale dependence of the true score MTMM model. In: Saris, W. E. & Münnich, Á. (Eds.), The multitrait-multimethod approach to evaluate measurement instruments, (pp. 71-87). Budapest: Eötvös University Press.

Coenders, G. & Saris, W. E. (1995). Categorization and Measurement Quality. The Choice Between Pearson and Polychoric Correlations. In Saris, W. E. & Münnich, Á. (Eds.) The Multitrait-Multimethod Approach to Evaluate Measurement Instruments, (pp. 125-144). Budapest: Eötvös University Press.

Coenders, G. & Saris, W. E. (1998). Relationship between a restricted correlated uniqueness model and a direct product model for multitrait-multimethod data. Metodološki Zvezki, 14, 151-172

Coenders, G. & Saris, W. E. (2000). Testing nested additive, multiplicative and general multitrait-multimethod models. Structural Equation Modeling, 7, 219-250.

Coenders, G. & Saris, W. E. (2000). Systematic and Random Method Effects. Estimating Method Bias and Method Variance. Metodološki Zvezki, 15, 55-74.

Coenders, G., Saris, W. E., Batista-Foguet, J. M. & Andreenkova, A. (1999). Stability of three-wave simplex estimates of reliability. Structural Equation Modeling, 6, 135-157.

Coenders, G., Satorra, A. & Saris, W. E. (1997). Alternative approaches to structural modeling of ordinal data: a Monte Carlo study. Structural Equation Modeling, 4, 261-282.

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Corten, I.W., Saris, W.E., Coenders, G., van der Veld, W., Aalberts, C.E. & Kornelis, C. (2002). Fit of Different Models for Multitrait-Multimethod Experiments. Structural Equation Modeling, 9, 213-232.

Coenders, G., Espinet, J.M. & Saez, M. (in press). Predicting Random Level and Seasonality of Hotel Prices. A Latent Growth Curve Approach. Tourism Analysis.

Kogovšek, T., Ferligoj, A. Coenders, G. & Saris, W. E. (2002). Estimating Reliability and Validity of Personal Support Measures: Full Information ML Estimation with Planned Incomplete Data. Social Networks, 24, 1-20.

O’Loughlin, C. & Coenders, G. (2002). Application of the European Customer Satisfaction Index to Postal Services. Structural Equation Models versus Partial Least Squares. Working Papers of the Department of Economics, University of Girona, 4, 1-28.

Saurina, C. & Coenders, G. (2002). Predicting Overall Service Quality. A Structural Equation Modelling Approach. Metodološki Zvezki, 18, 217-238.

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1. Introduction and History SEM make it possible to:

Fit linear relationships among a large number of variables. Possibly more than one is dependent.

Validate a measurement instrument. Quantify measurement error and prevent its biasing effect.

Freely specify, constrain and test each possible relationship using theoretical knowledge.

Falsify causal theories.

In their most recent versions, they enable researchers to:

Fit the same model to several populations with constraints.

Analyze non-normal, ordinal or binary data.

Treat missing values by maximum likelihood.

Treat hierarchical data such as cluster sample data.

Define qualitative latent variables as finite mixtures.

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1.1. Variance, covariance and correlation The dispersion of a numeric variable is measured by its sample variance:

1

)(1

2

−

−=∑

=

N

vvs

N

ijij

jj The square root of variance is the standard deviation. In order to quantify the degree of linear relationship between variables, the sample covariance can be used:

1

))((1

−

−−=∑

=

N

vvvvs

N

ililjij

jl A correlation is a covariance between two standardized variables (with unit variance).

Lies between –1 y 1.

Is computed as the covariance divided by the product of standard deviations.

Covariances and correlations are not appropriate when nonlinearities or outliers are present.

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In the population, variances and covariances are:

σσσσjj=E[(Vj-E(Vj))2]

σσσσjl=E[(Vj-E(Vj)) (Vl-E(Vl))]

Some useful properties: cov(v1,v2)=cov(v2,v1)

cov(v1,v1)=var(v1)

var(av1+bv2)=a2var(v1)+b2var(v2)+2abcov(v1,v2)

cov(av1+bv2, cv3+dv4)=accov(v1,v3)+adcov(v1,v4)+bccov(v2,v3)+bdcov(v2,v4)

where a, b, c and d are constants and v1, v2, v3 and v4 variables. These properties hold both in the sample and the population.

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1.2. Correlation and causality

The falsification principle (Popper, 1969) corresponds to what logic calls “modus tollens”.

A hypothesis is rejected if its consequences are not observed in reality. Thus, causal theories can be rejected (falsified) if they are contradicted by data, that is, by covariances and correlations.

On the contrary, theories cannot be statistically confirmed. A correlation can be due to a causal relation or to many other sources.

When studying the relationship between two variables, non-experimental research cannot control (keep fixed) other sources of variation. For this reason, all relevant variables must be in the model.

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1.3. History of models for the study of causality

Analysis of variance (1920-1930): decomposition of the variance of a dependent variable in order to identify the part contributed by an explanatory variable (dependence analysis). Control of third variables.

Macroeconometric models (1940-50): dependence analysis of non-experimental data. All variables must be included in the model.

Path analysis (1920-70): analysis of correlations (interdependence). Otherwise similar to econometric models.

Factor analysis (1900-1970): analysis of correlations among multiple indicators of the same variable. Measurement quality evaluation.

SEM (1970): Goldberger organizes a multidisciplinary conference where econometric models, path analysis and factor analysis are joined together. Relationships among variables measured with error, on non-experimental data from an interdependence analysis perspective. SEM are well suited for microeconometrics (individual data). Aggregated data have smaller measurement errors but other types of problems (autocorrelation) solved by dynamic macroeconometrics (1970-90).

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From 1973, Jöreskog, Bentler, Muthén and then many others developed the statistical theory underlying SEM, optimal estimation methods, robust testing procedures and goodness of fit indices, modelling strategies, and accessible software (LISREL, EQS, MX, AMOS, M-PLUS...). SEM are nowadays very popular (in some journals around half of all published articles use them) because they make it possible to (5 Cs, see Batista & Coenders 2000):

1) Work with Constructs measured through indicators, and evaluate measurement quality.

2) Consider the true Complexity of phenomena, thus abandoning uni and bivariate statistics.

3) Conjointly consider measurement and prediction, factor and path analysis, and thus obtain estimates of relationships among variables that are free of measurement error bias.

4) Introduce a Confirmatory perspective in statistical modelling. Prior to estimation, the researcher must specify a model according to theory.

5) Decompose observed Covariances, and not only variances, from an interdependence analysis perspective.

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2. Example: measurement of quality in a service industry Services have immaterial components, which make it necessary to take the customer’s view into account in order to evaluate quality (Saurina, 1997).

Parasuraman et al. (1985, 1988, 1991) define quality as the gap between consumers’ expectations prior to the service delivery and consumer perceptions during the service delivery.

Parasuraman et al. defined 5 aspects of any service, which can cause a discrepancy between expectations and perceptions and they elaborated the SERVQUAL questionnaire.

Other authors show that relevant aspects can differ from service to service (e.g. Saurina & Coenders, 1999).

It has been suggested that perceptions already imply a comparison with some sort of ideal (Saurina, 1997).

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Saurina & Coenders (1999) studied the banking industry in Girona and concluded that the relevant dimensions were:

• Competence (professionality, fulfilment of agreements and deadlines).

• Information (clear and trustworthy advertising, personal counselling).

• Employees (courtesy, confidence, familiarity).

• Design (offices).

Questionnaire items:

Overall quality:

per_qua : “the global assessment of the quality of your bank is ...” in a “very bad” to “very good” scale.

glob_sat: “with respect to the service provided by your bank you are...” in a “very dissatisfied” to “very satisfied” scale.

Behavioural intention:

recomm: “would you recommend your bank to your friends and family?” in a “not at all” to “enthusiastically” scale.

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Employees (in a “totally disagree” to “totally agree” scale):

e_confi: The behaviour of employees instills confidence in customers.

e_neat: Employees appear neat.

e_cour: Employees are consistently courteous with you.

e_knowl: Employees have the knowledge to answer your questions.

e_recogn: Employees recognize you and call you by your name.

Information (in a totally disagree to totally agree scale):

pam_clea: Pamphlets and statements are clear and well explained.

info_ad: Provides appropriate financial and tax information.

adv_real: Advertising of financial products and services reflects reality.

off_conv: Offers you the product that best suits you.

The questionnaire was administered to a systematic stratified random sample of people living in the Girona area (N=310).

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Table 2.1. Covariance matrix

per_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recogn pam_clea info_ad adv_real off_convper_qua 1.423 glob_sat 1.225 1.706 recomm 1.041 1.207 3.058 e_confi 0.777 0.885 0.750 1.593 e_neat 0.478 0.522 0.371 0.785 1.102 e_cour 0.641 0.847 0.577 1.332 0.776 1.743 e_knowl 0.599 0.669 0.559 0.989 0.725 0.956 1.465 e_recogn 0.933 1.147 0.851 1.372 0.947 1.587 0.949 3.540 pam_clea 0.698 0.758 0.573 1.157 0.618 1.047 0.932 1.129 1.961 info_ad 0.653 0.676 0.560 0.906 0.548 0.863 0.981 1.170 1.050 1.634 adv_real 0.834 0.867 0.690 1.041 0.641 0.971 0.854 1.581 1.290 1.174 1.831 off_conv 0.589 0.671 0.639 0.947 0.637 0.882 0.983 1.012 1.007 1.027 0.996 1.570 Data collection was supported by the 1997 Isidre Bonshoms grant, offered by the Girona Savings Bank.

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3 Intuitive explanation of the basics of SEM 3.1 Visual representation of causal theories. The path diagram

e_confi

e_neat

e_cour

e_knowl

e_recogn

pam_clea

info_ad

adv_real

off_conv

emplo quality

informa recom

per_qua

glob_sat

recomm

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3.2. Link between causal relations and covariances. Path analysis 3.2.1. Types of relationships among variables

Path analysis decomposes covariances in order to seek information about underlying causal relationships.

With this goal in mind, one must start in the opposite direction: deriving covariances from the parameters of the causal process.

Drawing a “path diagram” is the first stage in path analysis.

Types of relationship that can make v1 and v2 covary:

v1 causes v2, as implied by a regression model of v2 on v1 represented in the path diagram in Figure 3.1a. They can also covary if v2 causes v1 (Figure 3.1b). In both cases we have direct relationships that can also be reciprocal (Figure 3.1c).

Both have a common cause v3 (spurious relationship, Figure 3.1d).

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Both are related by an intervening variable v3 (indirect relationship, Figure 3.1e).

Joint effect (Figure 3.1f). The difference between Figures 3.1d and 3.1e and Figure 3.1f is that in the latter v1 and v3 are both exogenous so that it is not clear if v3 contributes to the covariance between v1 and v2 through an indirect or spurious mechanism.

v1 v2

Figure 3.1a

v1 v2

Figure 3.1b

v1 v2

Figure 3.1c

v1 v2

Figure 3.1d

v3

v1 v2

Figure 3.1e

v3

v1 v2

Figure 3.1f

v3

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3.2.2. Path analysis decomposition rules The decomposition rules establish the relationship between causal parameters and covariances in an intuitive way:

Variances and covariances of exogenous variables constitute model parameters by themselves.

The covariance between two variables is the sum of all direct, indirect, spurious and joint effects.

Each effect is a possible way of joining both variables on the path diagram, from an arbitrary origin and following the arrows.

The origin can be one of both variables (direct and indirect effects), a third variable (spurious effects), or a covariance among two exogenous variables (joint effects).

Effects are computed as the product of the origin variance or covariance and all parameters associated to the arrows followed.

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The variance of a dependent variable is the variance of the disturbance plus the explained variance. The explained variance is the sum for all variables with a direct effect on a dependent variable of all products of the direct effect and the covariance with the dependent variable.

These decomposition rules are cumbersome for large models and are not able to deal with reciprocal relationships. The structural equation system expresses each element of the population covariance matrix Σ as a function of model parameters. These model parameters thus impose a structure on Σ. SEM are also called covariance structure models. Σ=Σ(π) where:

Σ: Population covariance matrix (with variances on the main diagonal). π: vector containing all parameters (e.g. effects, disturbance variances, variances and covariances of exogenous variables).

Path analysis is useful for obtaining an insight into a causal process and into the possible effects explaining a covariance. Unfortunately this information is often insufficient. Many models can explain the same set of covariances. The choice among them cannot be statistical but theoretical.

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3.3. Examples and intuitive introduction of basic concepts 3.3.1. Simple linear regression model. Introduction to interdependence analysis The specification of a SEM consists in a set of assumptions regarding the behaviour of the variables involved.

Substantive part: it requires translating verbal theories into equations.

Statistical part: it is needed for the eventual estimation and testing of the model. The assumptions regard the distribution of the variables involved.

v2 v1 d2

β21

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Substantive assumptions:

v2=ββββ21v1+d2 (3.1)

Linearity.

β21 : by how much will the expected value of v2 increase following a unit increase in v1?

Standardized β21 : by how many standard deviations will the expected value of v2 increase following a standard deviation increase in v1?

d2 collects the effect of omitted explanatory variables, measurement error in v2 and the random and unpredictable part of v2.

v1 is assumed to be free of measurement error.

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Statistical assumptions:

→

22

11

2

1

00

,00

ψφ

Ndv

(3.2)

Two additional parameters: the variances of v1 (φφφφ11) and d2 (ψ22).

Bivariate normal distribution of v1 and d2,

Variables are mean-centred.

Uncorrelation of v1 and d2. If this holds, the variance of v2 can be additively decomposed into explained variance and disturbance variance. R2 is the explained percentage.

Identical and independent distribution for all cases. The fact that ψ22 is constant for all cases is called homoskedasticity.

Equations 3.1 and 3.2 exhaustively describe the joint distribution of v1 and v2 as a function of 3 parameters.

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In order to derive the structural equation system Σ=Σ(π) we can apply either the path analysis decomposition rules or the properties of variances and covariances:

=Σ

2221

1211

σσσσ

(3.3)

For a model with k observed variables, the number of distinct elements in Σ is (k+1)k/2.

π=(φφφφ11, ψ22, β21) (3.4)

+=+===

221112221212222

211121

1111

βφψβσψσβφσ

φσ

(3.5)

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It is possible to solve the system as it contains an equal number of equations (distinct elements of Σ) and unknowns (elements of π) ⇒⇒⇒⇒ exactly identified:

−=

=

=

11

221

2222

11

2121

1111

σσσψ

σσβ

σφ

(3.6)

We can estimate Σ from a sample covariance matrix:

=

2221

1211

ssss

S (3.7)

and estimate ( )212211ˆ,ˆ,ˆ βψφ=p : by solving the system Σ(p)=S:

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−=

=

=

11

221

2222

11

2121

1111

ˆ

ˆ

ˆ

sss

ss

s

ψ

βφ

(3.8)

In our example, (v2=recomm) can be explained by overall satisfaction (v1=glob_sat):

recomm=ββββ21glob_sat+d2 (3.9)

=

=

=

058.3207.1207.1706.1

,,_

,__,_

2221

1211

recommrecommrecommsatglob

recommsatglobsatglobsatglob

ssss

ssss

S (3.10)

=−=−=

===

==

204.2706.1207.1058.3ˆ

708.0706.1207.1ˆ

706.1ˆ

2

_,_

2,_

,22

_,_

,_21

_,_11

satglobsatglob

recommsatglobrecommrecomm

satglobsatglob

recommsatglob

satglobsatglob

sss

ss

s

ψ

β

φ

(3.11)

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21β̂ is identical to the ordinary least squares estimation (dependence analysis).

In statistical analysis, a function of residuals (e.g. the sum of squares) is used as:

A criterion function to minimize during estimation.

A goodness of fit measure.

In a dependence analysis, a residual is v2-β21v1 .

In an interdependence analysis residuals are differences between covariances fitted by the model parameters ΣΣΣΣ(p) and sample covariances S.

They are arranged in the S-ΣΣΣΣ(p) residual matrix.

In an exactly identified model they are zero.

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3.3.2. Model with two dependent variables and an indirect effect. Identification, goodness of fit and specification errors

v2=ββββ21v1+d2

v3=ββββ32v2+d3 (3.12)

→

33

22

11

3

2

1

000000

,000

ψψ

φN

ddv

(3.13)

Σ is 3××××3 and contains 4××××3/2=6 non-duplicated elements. ππππ has 5 elements (φφφφ11, ψ22, ψ33, β21, β32). The difference is the number of degrees of freedom (g) of the model.

d3 v1 v2

β21 v3

β32 d2

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Structural equation system:

+===

+===

32323333

322232

32211131

21212222

211121

1111

βσψσβσσ

ββφσβσψσ

βφσφσ

(3.14)

EXERCISE 1: Derive Equation 3.14 using both path analysis and the properties of variances and covariances.

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The existence of degrees of freedom has three interesting consequences:

Degrees of freedom introduce restrictions in the covariance space. Equation 3.14 implies:

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3221322132211131 σ

σσβσββφσ === (3.15)

This derives from many explicit or implicit restrictions of our model.

The existence of degrees of freedom implies higher parsimony.

The existence of degrees of freedom affects estimation. Due to sampling fluctuations, the S matrix will never exactly fulfil restrictions (in our case 22322131 ssss = ).

In general, no p vector of estimates will exactly satisfy ΣΣΣΣ(p)=S. A naive estimation method could be to drop g equations. Different estimates will be obtained depending on which equations are dropped.

Estimation consists in finding a p vector that leads to an S-ΣΣΣΣ(p) matrix with small values. A function of all elements in S-ΣΣΣΣ(p) called fit function is minimized.

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The existence of degrees of freedom makes it possible to test the model fit. A model with g=0 leads to a p vector that always fulfils ΣΣΣΣ(p)=S or S-ΣΣΣΣ(p)=0 and thus perfectly fits any data set.

Since all models are false, it is not possible to obtain a correct one, no matter how elaborate. Complex models are a sign of mediocrity (Box, 1976).

In a correct model with g>0, population covariances corresponding to the surplus equations must also fulfil the restrictions and thus ΣΣΣΣ must equal Σ(π)Σ(π)Σ(π)Σ(π). This equality is also applicable to the sample covariances, albeit only approximately. If S-ΣΣΣΣ(p), contains large values, we can say that some of the restrictions are false.

If assumptions are fulfilled and the model is correct, then a transformation of the minimum value of the fit function follows a khi-squared (or χχχχ2) distribution, which makes it possible to test the model restrictions.

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Example: we try to explain the behavioural intention variable (v3=recomm) from global satisfaction (v2=glob_sat), and the latter from confidence with employees v1=e_confi.

glob_sat=ββββ21e_confi+d2

recomm=ββββ32glob_sat+d3 (3.16)

We can estimate the model by dropping one of the surplus equations corresponding to any of the restricted covariances:

satglobsatglobrecommsatglobsatglobconfieconfierecomm ssss _,_,__,__, = .

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If for instance we drop srecomm,e_confi:

=

=

=

058.3207.1750.0207.1706.1885.0750.0885.0593.1

,,_,_

,__,__,_

,__,__,_

332313

232221

131211

recommrecommrecommsatglobrecommconfie

recommsatglobsatglobsatglobsatglobconfie

recommconfiesatglobconfieconfieconfie

sssssssss

sssssssss

S

=×−=−=

===

=×−=−=

===

==

204.2708.0207.1058.3ˆˆ

708.0706.1207.1ˆ

214.1556.0885.0706.1ˆˆ

556.0593.1885.0ˆ

593.1ˆ

32,_,33

_,_

,_32

21_,__,_22

_,_

_,_21

_,_11

βψ

β

βψ

β

φ

recommsatglobrecommrecomm

satglobsatglob

recommsatglob

satglobconfiesatglobsatglob

confieconfie

satglobconfie

confieconfie

sss

sss

ss

s

(3.17)

All residuals equal zero except that corresponding to srecomm,e_confi :

123.0708.0556.0593.1750.0ˆˆˆ750.0 322111 =××−=− ββφ

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If we drop sglob_sat, recomm estimates are different:

=×−=−=

===

=×−=−=

===

==

834.1847.0706.1058.3ˆˆ

847.0885.0750.0ˆ

214.1556.0885.0706.1ˆˆ

556.0593.1885.0ˆ

593.1ˆ

2232_,_,33

_,_

_,32

21_,__,_22

_,_

_,_21

_,_11

βψ

β

βψ

β

φ

satglobsatglobrecommrecomm

satglobconfie

confierecomm

satglobconfiesatglobsatglob

confieconfie

satglobconfie

confieconfie

sss

sss

ss

s

(3.18)

EXERCISE 2: Repeat estimation when dropping se_confi,glob_sat.

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Errors such as the omission of important explanatory variables or the inclusion of wrong restrictions are known as specification errors.

There are far more incorrect models than correct models. Specification errors are frequent.

In general, a specification error can bias any parameter estimate.

If the model in Equations 3.12 and 3.13 is incorrect because v3 receives a direct effect from v1:

v2=ββββ21v1+d2

v3=ββββ31v1+ββββ32v2+d3 (3.19)

and we apply path analysis, then we observe that the new parameter affects σ31 y σ33:

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++==

+=+=

==

313132323333

322232

311132211131

21212222

211121

1111

βσβσψσβσσ

βφββφσβσψσ

βφσφσ

(3.20)

If we fit the model in Equations 3.12 and 3.13 to the covariances in Equation 3.20, we find σ31 and σ33 values that are biased by the absent β31 parameter but they will be fitted only by the present parameters.

42

Attempts must be made to detect specification errors by all means, both statistical and theoretical:

Specification errors are undetectable in any model with g=0.

It can happen that many models with different interpretations have a similarly good fit, even an exactly equal fit (equivalent models).

Let us assume that the model in Equations 3.12 and 3.13 is correct, and thus population covariances fulfil Equation 3.15. The following model has a completely different causal interpretation:

v1=ββββ12v2+d1

v2=ββββ23v3+d2 (3.21)

→

33

22

11

3

2

1

000000

,000

φψ

ψN

vdd

(3.22)

EXERCISE 3: Derive the Σ=Σ(π)Σ=Σ(π)Σ=Σ(π)Σ=Σ(π) system for this model and show that it also leads to Equation 3.15 and is thus equivalent to the previous model.

43

If we estimate a general model that includes Equations 3.12 y 3.21 as particular cases:

v1=ββββ12v2+d1 v2=ββββ21v1+ββββ23v3+d2 v3=ββββ32v2+d3 (3.23)

→

33

22

11

3

2

1

000000

,000

ψψ

ψN

ddd

(3.24)

then the parameter vector includes 7 elements π=( ψ 11, ψ22, ψ33, β12, β21, β23, β32) versus 6 equations: infinite number of solutions (underidentified model).

44

3.3.3 Simple regression model with errors in the explanatory variable. Introduction to models with measurement error The observed explanatory variable (v1) is measured with error (e1). The unobservable error-free value f1 is called factor or latent variable. f2 is observed because e2 is for the moment assumed to be zero.

Two equation types:

1) Relating factors to one another:

f2=β21f1+d2 (3.25)

2) Relating factors to observed variables or indicators:

v1=f1+e1 v2=f2 (3.26)

β21

f1 f2

d2

v2 1

v1 e1 1

45

Assumptions:

• Measurement errors are uncorrelated with factors (as in factor analysis).

• Disturbances are uncorrelated with the explanatory factor (as in regression).

→

22

11

11

2

1

1

000000

,000

ψθ

φN

def

(3.27)

These assumptions make it possible to decompose the variance of observed variables into true score variance (explained by factors) and measurement error variance. R2 is called measurement quality and is represented as κκκκ.

11

111 1

σθκ −=

(3.28)

46

The structural equations become:

+==

+=

221112222

211121

111111

βφψσβφσ

θφσ

(3.29)

Underidentified model: 4 parameters (φφφφ11, θθθθ11, ββββ21, ψψψψ22) and three variances and covariances (only those of observed variables count).

The OLS estimator assumes that θθθθ11=0, which is a specification error and leads to bias. The probability limit of the OLS estimator is:

( )211

11

211111

11

2111

11

21

11

2121

ˆ βκσ

βθσσ

βφσσβ =

−==⇒=

ss

(3.30)

and is thus biased unless κκκκ1=1.

47

The unbiased estimation of the model is only possible if θθθθ11 or κκκκ1 are known. Otherwise θθθθ11 must be estimated. This estimation is possible if two indicators of the variable measured with error are available. EXERCISE 4: Prove that if v2 is measured with error and v1 is error free, then the OLS estimate of ββββ21 is unbiased and the estimate of ψψψψ22 includes both disturbance and measurement error variance.

48

3.3.4. Simple linear regression model with multiple indicators The equation relating the factors is identical to 3.25. The equations relating factors to indicators become:

f2=β21f1+d2 v1=f1+e1 v2=f2 (3.32) v3=λ31f1+e3

β21 f1 f2

d2 v2

1v1 e1

1

v3 e3 λ31

49

The equation includes a loading λ31 which relates the scales of f1 and v3:

The researcher must fix the latent variable scale, usually by anchoring it to the measurement units of an indicator whose λλλλ equals 1.

Standardized instead of raw loadings are usually interpreted. If there is only one factor per indicator, they lie within -1 and +1 and equal the square root of κ.

New assumption of uncorrelated measurement errors of different indicators:

→

22

33

11

11

2

3

1

1

000000000000

,

0000

ψθ

θφ

N

deef

(3.33)

50

Structural equations are the same as in Equation 3.29 with the addition of v3. This is an exactly identified model, all of whose parameters can be solved, even those related to unobservable variables:

(3.34) (3.35)

−=−=−=

===

221112222

231113333

111111

112121

113131

32312111

βφσψλφσθ

φσθφσβφσλ

σσσφ

+===

+==

+=

332311133

31211132

311131

221112222

211121

111111

θλφσλβφσ

λφσβφψσ

βφσθφσ

51

This model could be estimated with data from our example if we explain behavioural intention f2=recom from overall quality f1=quality, measured by its two indicators (v1=glob_sat and v3=per_qua):

recom=β21quality+d2 glob_sat=quality+e1 recomm=recom (3.36) per_qua=λλλλ31quality+e3

And the variances and covariances:

v1 glob_sat v2 recomm v3 per_qua v1 glob_sat 1.706 v2 recomm 1.207 3.058 v3 per_qua 1.225 1.041 1.423

52

=×−=−==×−=−=

=−=−=

===

===

=×==

032.2850.0420.1058.3ˆˆˆ365.0863.0420.1423.1ˆˆˆ

286.0420.1706.1ˆˆ

850.0420.1207.1

ˆˆ

863.0420.1225.1

ˆˆ

420.1041.1225.1207.1ˆ

222111,22

223111_,_33

11_,_11

11

,_21

11

_,_31

_,

_,_,_11

βφψλφθ

φθφ

β

φλ

φ

recommrecomm

quaperquaper

satglobsatglob

recommsatglob

quapersatglob

quaperrecomm

quapersatglobrecommsatglob

sss

s

ss

ss

(3.37)

Measurement quality of indicators can be computed as:

832.0706.1286.01ˆ1ˆ _,_111 =−=−= satglobsatglobsθκ

743.0423.1365.01ˆ1ˆ _,_333 =−=−= quaperquapersθκ

53

Regression model with two indicatorsobserved variablesrecomm per_qua glob_satcovariance matrix3.0581.041 1.4231.207 1.225 1.706sample size: 301latent variablesquality recomrelationshipsper_qua = qualityglob_sat = 1*qualityrecomm = 1*recomrecom = qualitylet the error variance of recomm be 0options me=ml wp rs sc nd=3end of problem

54

Regression model with two indicators

Sample Size = 301

Covariance Matrix to be Analyzedrecomm per_qua glob_sat

-------- -------- --------recomm 3.058per_qua 1.041 1.423

glob_sat 1.207 1.225 1.706

Number of Iterations = 5

55

LISREL Estimates (Maximum Likelihood)Measurement Equations

recomm = 1.000*recom,, R2 = 1.000

per_qua = 0.862*quality, Errorvar.= 0.366 , R2 = 0.742(0.0636) (0.0721)13.551 5.085

glob_sat = 1.000*quality, Errorvar.= 0.286 , R2 = 0.833(0.0912)3.133

Structural Equations

recom = 0.850*quality, Errorvar.= 2.032 , R2 = 0.335(0.0841) (0.178)10.102 11.435

56

Variances of Independent Variables

quality--------

1.420(0.163)

8.704

Covariance Matrix of Latent Variablesrecom quality

-------- --------recom 3.058

quality 1.207 1.420

Goodness of Fit Statistics

Degrees of Freedom = 0Minimum Fit Function Chi-Square = 0.00 (P = 1.000)

Normal Theory Weighted Least Squares Chi-Square = 0.00 (P=1.000)The Model is Saturated, the Fit is Perfect !

57

Regression model with two indicatorsCompletely Standardized Solution

LAMBDA-Yrecom

--------recomm 1.000

LAMBDA-Xquality--------

per_qua 0.862glob_sat 0.912

GAMMAquality

--------recom 0.579

58

Correlation Matrix of ETA and KSI

recom quality-------- --------

recom 1.000quality 0.579 1.000

PSIrecom

--------0.665

THETA-DELTAper_qua glob_sat

-------- --------0.258 0.167

Regression Matrix ETA on KSI (Standardized)

quality--------

recom 0.579

59

3.3.5. Confirmatory factor analysis models. Introduction to reliability and validity assessment This model does not contain equations relating factors to one another but only covariances. At least three indicators are needed for models with one factor and two for models with more factors. For 2 factors and 2 indicators we have the following equations:

v1=f1+e1 v2=λλλλ21f1+e2 v3=f2+e3 v4=λ42f2+e4

φ21

f1 f2 1 v1

e1

v2 e2

v3 e3

v4 e4

1

λ42 λ21

60

with the assumptions:

→

44

33

22

11

1121

2111

4

3

2

1

2

1

0000000000000000000000000000

,

000000

θθ

θθ

φφφφ

N

eeeeff

61

In CFA models it is possible to standardize factors to unit variances instead of fixing a loading to 1. Then φφφφ are factor correlations:

v1=λλλλ11f1+e1 v2=λλλλ21f1+e2

v3=λλλλ32f2+e3 v4=λ42f2+e4

→

44

33

22

11

21

21

4

3

2

1

2

1

000000000000000000000000100001

,

000000

θθ

θθ

φφ

N

eeeeff

62

The model has g=1.

+====

+===

+==

+=

4424244

324243

21422142

11422141

3323233

21322132

11322131

2222122

211121

1121111

θλσλλσ

λλφσλλφσθλσλλφσλλφσθλσ

λλσθλσ

EXERCISE 5: Derive the structural equations if the scale is fixed as λλλλ11=1 and λλλλ32=1.

63

The correlation between two indicators of the same factor depends on κκκκ:

( )( ) ( )( ) 2122

22111

211

221

211

2222111

211

2111

2211

2121 κκ

θλθλλλ

θλθλλλ

σσσρ =

++=

++==

and the correlation between two indicators of different factors is attenuated with respect to the correlation between factors (effect of measurement error):

( )( ) ( )( ) 312133

23211

211

232

211

21

3323211

211

322111

3311

3131 κκφ

θλθλλλφ

θλθλλφλ

σσσρ =

++=

++==

A CFA model is likely to fit the data only if items of the same factor correlate highly and higher than items of different factors. We advise researchers to carefully examine the correlation matrix prior to fitting a CFA model.

64

3.3.6. Random and systematic error. Reliability and validity Reliability: Extent to which a measurement procedure “would” yield the same result upon several independent trials under identical conditions. In other words, absence of random measurement error (any systematic error would replicate).

Validity: Extent to which a measurement procedure measures what it is intended to measure, except for random measurement error. In other words, absence of systematic error.

Assuming the validity of v, its reliability is the percentage κκκκ of variance explained by f.

Always follow this golden rule: Estimate reliability after validity has been diagnosed. Test the specification of measurement equations in a CFA model prior to specifying equations relating factors. Otherwise, relationships among factors might be biased (specification errors) or even meaningless (invalidity).

65

Construct validation: Estimate a CFA model that assumes validity... All items load on the factor they are supposed to measure. No error correlations are specified.

....and diagnose its goodness of fit. You can never be certain of validity, but a CFA model can help detect signs of invalidity such as:

It does not correctly reproduce the covariance matrix (additional loadings or uniqueness correlations are needed, thus revealing mixed items, additional necessary dimensions or method factors). Some variables have a unique variance that is too high to be attributed to solely random error (convergent invalidity). Some factors have correlations very close to unity (discriminant invalidity). Some factors have correlations of unexpected signs or magnitudes (nomological invalidity).

66

3.4. Modelling stages in SEM

1) SPECIFICATION

2) IDENTIFICATION

3) DATA COLLECTION

4) ESTIMATION

5) FIT DIAGNOSTICS

ADEQUATE?

6) UTILIZATION

YES

NO

Model: equations and assumptions

Estimable model

Exploratory data analysis. Computation of S

Methods to fit Σ(p) to S

Discrepancies between Σ(p) and S

Verbal theories

- Theory validation, prediction, reliability assessment ...

MODIFICATION

67

4. Theoretical and statistical grounds 4.1. Specification

Formal establishment of a statistical model: set of statistical and substantive assumptions that structure the data according to a theory.

Equations: one or two of the following systems of equations: Relating factors or error free variables to one another (structural equations). Relating factors to indicators with error (measurement equations).

Parameters: three types:

Free (unknown and freely estimated).

Constrained (constrained to be equal to another parameter or to a function of other parameters).

Fixed (known and constrained to a given value).

Statistical assumptions on the data generation process whose aim is to make the model estimable and testable. Their justification is not theoretical but statistical.

68

The amount of the researchers’ prior knowledge will affect the modelling strategy:

If this knowledge is exhaustive and detailed, it will be easily translated into a model specification. The researchers’ aim will simply be to use the data to estimate and confirm or reject the model (confirmatory strategy).

If this knowledge is less exhaustive and detailed, the fixed or free character of a number of parameters will be dubious. This will lead to a model modification process by repeatedly going through the modelling stages (exploratory strategy).

69

4.1.1 Relationships among factors. Linearity assumption System of simultaneous linear equations of the type: f1 = β 12 f2 + β 13 f3 … + β1m-1 fm-1 + β1m fm+ d 1

f2 = β 21 f1 + β 23 f3… + β2m-1 fm-1 + β2m fm+ d 2 ..... (4.1)

fm = β m1 f1 + β m2 f2 + β m3 f3 …+ βmm-1 fm-1 + d m

Some of the ββββjl coefficients must be fixed or constrained in order to make the model identified. If the jth row contains only zeros, then fj is exogenous and thus dj=fj. Additional parameters and assumptions:

All terms are mean centred. Variances φφφφjj and covariances φφφφjl among exogenous factors are usually free. Variances ψψψψjj of disturbances of endogenous factors are usually free. Covariances ψψψψjl of disturbances may be fixed or free (see identification section). When free they are interpreted as systematic and shared unexplained variance.

In CFA models (see Jöreskog, 1969 and Batista-Foguet and Coenders, 1998) all factors are exogenous, all ββββjl=0, there are no ψψψψ parameters and all φφφφ parameters are free.

70

In matrix notation:

f=Bf+d

+

=

mmmm

m

m

m d

dd

f

ff

f

ff

MM

L

MOMM

L

L

M2

1

2

1

21

221

112

2

1

0

00

ββ

ββββ

The variances of exogenous factors and disturbances, assuming that r of the m factors are exogenous:

Ψ

Φ=

00

)( df,VAR with:

=Φ

rrrr

r

r

φφφ

φφφφφφ

L

MOMM

L

L

21

22221

12111

=Ψ

++

+++++

+++++

mmrmrm

rmrrrr

rmrrrr

ψψψ

ψψψψψψ

L

MOMM

L

L

21

22212

11211

71

4.1.2 Relationships between factors and indicators. Linearity assumption Linear system of equations of the factor analysis type:

v1 = λ 11 f1 + λ 12 f2 +....+ λ 1m fm +e 1

v2 = λ 21 f1 + λ 22 f2 +....+ λ 2m fm +e 2 .... (4.3) vk = λ k1 f1 + λ k2 f2 +....+ λ km fm +e m

Some λλλλ coefficients must be fixed or restricted in order to make identification possible. Additional parameters and assumptions:

All terms are mean centred. Variances θθθθjj of measurement errors are usually free. Covariances θθθθjl among measurement errors are usually fixed. When free they are interpreted as systematic and shared error variance (e.g. a forgotten factor, a common measurement method) and thus invalidity.

If all variables are measured without error:

v1 = f1 ... vk = fk ; with θθθθ11=θθθθ22=…=θθθθkk=0

72

In matrix notation:

v=ΛΛΛΛf+e

+

=

kmkmkk

m

m

k e

ee

f

ff

v

vv

MM

L

MOMM

L

L

M2

1

2

1

21

22221

11211

2

1

λλλ

λλλλλλ

=Θ=

kkkk

k

k

VAR

θθθ

θθθθθθ

L

MOMM

L

L

21

22221

12111

)(e

73

4.1.3. Pseudo-isolation assumptions COV(fj,dl) = 0 (4.6)

If no relevant variables and effects explaining fl have been omitted then it is a realistic assumption. Omitted effects will be in dl and are likely to correlate with other factors.

COV(fj,el) = 0 (4.7)

COV(dj,el) = 0 (4.8)

If measures are valid, then el is pure random error and the assumption is realistic.

4.1.4. Distributional assumptions Multivariate normality of all variation sources (exogenous factors, disturbances and measurement errors).

Unlike other assumptions, violation of this one does not lead to bias, but only to inaccurate standard errors and test statistics.

74

4.1.5 Example quality=ββββ31emplo+ββββ32informa+d3 recom=ββββ41emplo+ββββ42informa+ββββ43quality+d4 (4.9)

per_qua=quality +e1 glob_sat=λλλλ23quality+e2 recomm=recom e_confi=emplo+e4 e_neat=λλλλ51emplo+e5 e_cour=λλλλ61emplo+e6 e_knowl=λλλλ71emplo+e7 e_recogn=λλλλ81emplo+e8 pam_clea=informa+e9 info_ad=λλλλ10 2informa+e10 adv_real=λλλλ11 2informa+e11 off_conv=λλλλ12 2informa+e12 (4.10)

with the additional parameters φφφφ11, φφφφ21, φφφφ22, ψψψψ33, ψψψψ44, θθθθ11, θθθθ22, θθθθ44, θθθθ55, θθθθ66 ,…, θθθθ1212 . The total number of parameters is 29.

75

In matrix form:

+

=

12

11

10

9

8

7

6

5

4

2

1

212

211

210

81

71

61

51

23

0

recomqualityinformaemplo

0000000000010000000000000000110000000100

off_convadv_realinfo_ad

pam_cleae_recogne_knowle_coure_neate_confirecommglob_satper_qua

eeeeeeeee

ee

λλλ

λλλλ

λ

76

=Θ

1212

1111

1010

99

88

77

66

55

44

22

11

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

θθ

θθ

θθ

θθ

θ

θθ

77

+

=

4

3

434241

3231

informaemplo


00000000000


dd

βββββ

=

44

33

2221

2111

0000000000

)(

ψψ

φφφφ

fdVAR

78

λ10 2

1

λ11 2

λ12 2

λ61

λ71

λ81

λ51

e_confi e4

e_neat e5

e_cour e6

e_knowl e7

e_recogn e8

pam_clea e9

info_ad e10

adv_real e11

off_conv e12

emplo quality

informa recom

β43

β31

d3

d4

β42

β32

β41

φ21

1 per_qua

e1

glob_sat

e2

recomm 1

1

λ23

79

4.2 Identification Can model parameters be derived from variances and covariances? Identification must be studied prior to data collection In principle, checks should be made to see if for each free parameter there exists at least one algebraic expression linking it to variances and covariances. A list of necessary and sufficient conditions may help, though not all models can be studied in this way.

Many models will not fulfil the sufficient conditions and will nevertheless be identified. In this case, an empirical check can be carried out:

A random sample is simulated following the model, with reasonable parameter values. The model is estimated using these data. All software programs do an identification test.

If a model is not identified: Seek more restrictive specifications with additional constraints (if theoretically justifiable). Add more indicators or more exogenous factors.

Theoretically identified models may fail to be so for certain data sets. This is an analogous phenomenon to near-perfect collinearity in multiple regression and is called empirical underidentification.

80

4.2.1 Identification conditions Necessary conditions: According to g, models can be classified into:

Never identified (g<0): infinite number of solutions for some parameters that makes S equal ΣΣΣΣ(p).

Possibly identified (g=0): there may be a unique solution for all parameters that makes S equal ΣΣΣΣ(p). This type of models is less interesting in that their rejection is not possible (their restrictions are not testable).

Possibly overidentified (g>0): there is no solution for p that makes S equal ΣΣΣΣ(p) but there may be a unique solution that minimizes discrepancies between both matrices. Only these models, more precisely their restrictions, can be tested from the data.

81

Sufficient conditions for models without measurement error: Linear regression models are always identified. Recursive models are those in which variables can be ordered in such a way that effects flow in only one direction. Recursive models with uncorrelated disturbances are always identified. Recursive models without any effect relating two endogenous variables are also identified even if their disturbances are correlated. Non-recursive models have more complicated rules, which can be found in any econometrics book. A useful hint is given by the order condition: let r be the number of exogenous variables and m the total number of variables. If for each endogenous variable there are at most r direct effects pointing to it, then the model is likely to be identified even if disturbances are correlated. Sufficient conditions for models with measurement error: Relationships among factors are identified according to the rules for models with error-free variables. Each factor has at least two pure indicators (i.e. related to no other factor and with uncorrelated errors). Three indicators are however recommended (more precise estimations and more powerful validity tests).

82

Figure 4.2e

v3 v4

v1 v2

Figure 4.2f

v3 v4

v1 v2

Figure 4.2a

v3 v4

v1 v2

v3 v4

v1 v2

Figure 4.2b

Figure 4.2d

v3 v4

v1 v2

Figure 4.2c

v3 v4

v1 v2

83

4.2.2. Example 12 observed variables lead to (12××××13/2)=78 variances and covariances: possibly overidentified model.

The model fulfils enough sufficient conditions:

1) Equations relating factors are recursive

2) Disturbances are uncorrelated

3) All factors have at least two pure indicators except recom. However, recom does not affect any other variable. Thus, ignoring measurement error will not cause bias.

84

4.3. Estimation First estimate the sample variances and covariances (S) and then find the best fitting p parameter values.

Fit function: related to the size of the residuals in S-ΣΣΣΣ(p), which we now arrange as a vector by also dropping duplicated elements. The generic expression of the fit function is:

F= (S-ΣΣΣΣ(p))' W (S-ΣΣΣΣ(p)) (4.11)

(S-ΣΣΣΣ(p))’ is a row vector of residuals, (S-ΣΣΣΣ(p)) a column vector and W the weight matrix.

Estimates obtained by minimizing Equation 4.11 are consistent for general choices of W.

The simplest case is W=I. Equation 4.11 is the sum of squares of residuals.

Estimates are asymptotically efficient if W=ΓΓΓΓ-1, where ΓΓΓΓ is the sampling covariance matrix of the elements in S. This matrix makes less precise residuals have a lower weight.

Under multivariate normality variances and covariances are sufficient parameters and ΓΓΓΓ depends only on ΣΣΣΣ which can be estimated as either S or ΣΣΣΣ(p).

For other distributions, ΓΓΓΓ depends on 4th order moments.

85

Common estimation methods differ only by the choice of W:

Unweighted least squares (ULS) with W=I.

Normal theory weighted least squares (NT-WLS, called Generalized Least Squares –GLS– by LISREL), with W=ΓΓΓΓ-1 and ΓΓΓΓ computed from S. Asymptotically efficient under multivariate normality.

Maximum likelihood (ML), with W=ΓΓΓΓ-1 and ΓΓΓΓ computed from ΣΣΣΣ(p). Asymptotically efficient under multivariate normality.

Asymptotic distribution free method (ADF, called Weighted Least Squares –WLS– by LISREL), with W=ΓΓΓΓ-1 and ΓΓΓΓ computed from 4th order moments. Asymptotically efficient under any distribution. In practice it has been found to be appropriate only for samples larger than 1000 and models with 10 or fewer observed variables.

86

Sample sizes in the 200-500 range are usually enough. Sample requirements increase:

For smaller R2 and κκκκ percentages of explained variance.

When collinearity is greater.

For smaller numbers of indicators per factor.

For larger number of parameters to be estimated.

All methods assume that a covariance matrix is analyzed. Estimations obtained from a correlation matrix are only correct if the model is scale invariant (Cudeck, 1989):

All constraints consist in fixing parameters to zero or in fixing the scale of latent variables (one constrained λλλλ loading or φφφφ variance per latent variable).

The residual matrix contains zeroes in the main diagonal.

Standard errors and test statistics in LISREL The computation of any test statistics requires a realistic ΓΓΓΓ matrix:

Based on ΣΣΣΣ(p) or S under multivariate normality.

Based on 4th order moments otherwise.

87

However, the same W=ΓΓΓΓ-1 does not have to be used for estimation.

Estimation method

ΓΓΓΓ used for W

ΓΓΓΓ used for testing

condition for correct tests

condition for consistent estimation

if 4th order moments are not supplied by the user: ULS I from S normality any distribution NT-LS/GLS from S from S normality any distribution ML from Σ(Σ(Σ(Σ(p)))) from Σ(Σ(Σ(Σ(p)))) normality any distribution

if 4th order moments are supplied by the user: ULS I from moments any distribution NT-LS/GLS from S from moments any distribution ML from Σ(Σ(Σ(Σ(p)))) from moments any distribution ADF/WLS from

moments from moments any distribution, but only for large

samples and small models If 4th order models are supplied, LISREL computes standard errors, t values and two types of χχχχ2 statistic, which are robust to non-normality. The Satorra-Bentler mean scaled χχχχ2 statistic seems to be the best for small to moderate samples (Satorra & Bentler, 1994). Robust standard errors and t-values require larger samples (in the 400-800 range, Boomsma & Hoogland, 2001).

88

6 PRELIS It has the following functions:

Exploratory data analysis: histograms and normality tests.

Computation of S and ΓΓΓΓ. S may be a covariance matrix or a correlation matrix.

Computation of an appropriate ΓΓΓΓ matrix for WLS estimation or, even better, for correct standard errors and test statistics.

Relevant menus

Opening a data file in PRELIS Data (*.psf) format: select file, open from the main menu.

Importing other data types (*.sav of SPSS for Windows, *.xls, ...): select file, import external data in other formats, and choose the file type in list files of type. The program will ask for a name to store the file in *.psf format.

Defining the variable type as numeric or ordinal (covariances assume numeric data): select data, define variables, select some or all the variables in the file, variable type, continuous and OK.

1

90

If data contain missing values, then the same dialog box can be used to:

Indicate the codification of missing values. SPSS’ system missing values are directly recognized by PRELIS. SPSS’s user missing values must be defined. It is advisable to use a common code for missing values on all variables and insert it as the global missing value.

Select the missing value treatment:

Elimination of cases only when computing covariances between pairs of variables that are missing for those cases (pairwise). This can lead to numeric problems and makes it impossible to compute 4th order moments.

Complete elimination of cases with missing cases (listwise). This is the recommended option. It can lead to a large drop in sample size. To prevent this, imputation can be performed first. Available options are:

Hot-deck imputation (statistics, impute missing values).

EM imputation (statistics, multiple imputation).

Maximum likelihood estimation with missing data does not require the elimination or the imputation of any cases. It is done within LISREL, not PRELIS.

91

We select statistics, output options to select the requested matrices and provide file names for them. For numeric variables we select:

Covariances (S) as moment matrix, save to file and provide a file name.

Asymptotic covariance matrix, save to file, and give a name for the file to store ΓΓΓΓ based on 4th order moments.

Perform tests of multivariate normality.

Wide print.

Finally we press OK.

93

The following lines were read from file D:\F\F\LISREL\LLIBRE\BONSHOMS.PR2:!PRELIS SYNTAX: Can be editedSY=D:\F\F\LISREL\LLIBRE\BONSHOMS.PSFOU MA=CM SM=bonshoms.cm AC=bonshoms.ac WP

Total Sample Size = 301Univariate Summary Statistics for Continuous Variables

Variable Mean St. Dev. T-Value Skewness Kurtosis Minimum Freq. Maximum Freq.-------- ---- -------- ------- -------- -------- ------- ----- ------- -----PER_QUA 7.295 1.193 106.108 -0.499 0.753 2.000 1 9.000 57

GLOB_SAT 7.256 1.306 96.387 -0.681 1.202 1.000 1 9.000 61RECOMM 6.641 1.749 65.894 -1.228 2.370 0.000 4 9.000 38E_CONFI 7.511 1.262 103.255 -0.456 -0.677 4.000 3 9.000 85

E_NEAT 7.795 1.050 128.830 -0.670 0.715 3.000 1 9.000 95E_COUR 7.538 1.320 99.069 -0.740 0.532 2.000 1 9.000 94

E_KNOWL 7.427 1.210 106.461 -0.389 -0.580 4.000 2 9.000 69E_RECOGN 6.983 1.881 64.392 -1.041 0.869 1.000 6 9.000 81PAM_CLEA 7.029 1.400 87.091 -0.745 1.045 1.000 1 9.000 46INFO_AD 7.015 1.278 95.223 -0.269 0.176 2.000 1 9.000 46

ADV_REAL 6.911 1.353 88.614 -0.651 1.258 1.000 1 9.000 39OFF_CONV 7.172 1.253 99.319 -0.312 -0.456 4.000 6 9.000 51

Test of Univariate Normality for Continuous VariablesSkewness Kurtosis Skewness and Kurtosis

Variable Z-Score P-Value Z-Score P-Value Chi-Square P-ValuePER_QUA -2.575 0.010 2.252 0.024 11.701 0.003

GLOB_SAT -2.910 0.004 3.075 0.002 17.924 0.000RECOMM -3.553 0.000 4.551 0.000 33.337 0.000E_CONFI -2.478 0.013 -3.486 0.000 18.295 0.000

E_NEAT -2.893 0.004 2.172 0.030 13.086 0.001E_COUR -3.000 0.003 1.760 0.078 12.100 0.002

E_KNOWL -2.308 0.021 -2.729 0.006 12.774 0.002E_RECOGN -3.372 0.001 2.485 0.013 17.548 0.000PAM_CLEA -3.008 0.003 2.811 0.005 16.950 0.000INFO_AD -1.922 0.055 0.790 0.429 4.318 0.115

ADV_REAL -2.862 0.004 3.164 0.002 18.205 0.000OFF_CONV -2.074 0.038 -1.918 0.055 7.978 0.019

94

Relative Multivariate Kurtosis = 1.362

Test of Multivariate Normality for Continuous VariablesSkewness Kurtosis Skewness and Kurtosis

Value Z-Score P-Value Value Z-Score P-Value Chi-Square P-Value------ ------- ------- ------- ------- ------- ---------- -------32.123 26.008 0.000 61.943 14.137 0.000 876.281 0.000

Histograms for Continuous VariablesPER_QUA

Frequency Percentage Lower Class Limit1 0.3 2.0000 0.0 2.7004 1.3 3.400 �0 0.0 4.100

14 4.7 4.800 ��44 14.6 5.500 ��0 0.0 6.200

118 39.2 6.900 ��62 20.6 7.600 ��58 19.3 8.300 ��

.......

OFF_CONVFrequency Percentage Lower Class Limit

6 2.0 4.000 ��0 0.0 4.500

26 8.6 5.000 ��4 1.3 5.500 �

44 14.6 6.000 ��10 3.3 6.500 ��91 30.2 7.000 ��7 2.3 7.500 ��

62 20.6 8.000 ��51 16.9 8.500 ��

95

Covariance MatrixPER_QUA GLOB_SAT RECOMM E_CONFI E_NEAT E_COUR E_KNOWL E_RECOGN PAM_CLEA INFO_AD------- -------- ------- ------- -------- ------ ------- -------- -------- --------

PER_QUA 1.423GLOB_SAT 1.225 1.706RECOMM 1.041 1.207 3.057E_CONFI 0.777 0.885 0.750 1.593

E_NEAT 0.478 0.522 0.371 0.785 1.102E_COUR 0.641 0.847 0.577 1.332 0.776 1.743

E_KNOWL 0.598 0.669 0.559 0.989 0.725 0.956 1.465E_RECOGN 0.933 1.147 0.851 1.372 0.947 1.587 0.949 3.540PAM_CLEA 0.698 0.758 0.573 1.157 0.618 1.047 0.932 1.129 1.961INFO_AD 0.653 0.676 0.560 0.906 0.548 0.863 0.981 1.170 1.050 1.634

ADV_REAL 0.834 0.867 0.690 1.041 0.641 0.971 0.854 1.581 1.290 1.174OFF_CONV 0.589 0.671 0.639 0.947 0.637 0.882 0.983 1.012 1.007 1.027

ADV_REAL OFF_CONV-------- --------

ADV_REAL 1.831OFF_CONV 0.996 1.570

MeansPER_QUA GLOB_SAT RECOMM E_CONFI E_NEAT E_COUR E_KNOWL E_RECOGN PAM_CLEA INFO_AD------- -------- ------- ------- -------- ------ ------- -------- -------- --------

7.295 7.256 6.641 7.511 7.795 7.538 7.427 6.983 7.029 7.015ADV_REAL OFF_CONV-------- --------

6.911 7.172

Standard DeviationsPER_QUA GLOB_SAT RECOMM E_CONFI E_NEAT E_COUR E_KNOWL E_RECOGN PAM_CLEA INFO_AD------- -------- ------- ------- -------- ------ ------- -------- -------- --------

1.193 1.306 1.749 1.262 1.050 1.320 1.210 1.881 1.400 1.278ADV_REAL OFF_CONV-------- --------1.353 1.253

The Problem used 41640 Bytes (= 0.1% of available workspace)

97

LISREL We select file, new, syntax only, and a blank syntax window will appear.

We can save it by selecting file, save as. Syntax files have an *.spl ending.

We can open it by selecting file, open, list files of type: syntax only.

We start with CFA models to test validity.

A syntax file contains:

A freely chosen title line

The observed variables command, followed by a list of all variables in the same order as they appear on the PRELIS data file or the covariance matrix, even the variables that are not part of the model. Variable names can have up to 8 alphabetic or numeric characters without imbedded spaces.

The next command provides file names for S and ΓΓΓΓ (optional, though recommended under non-normality).

The sample size command is also compulsory. If listwise deletion has been applied, then the sample size after deletion is what counts.

98

The latent variables command assigns names to all factors in the model.

The relationships command specifies all equations in the model. Loadings fixed at 1 are indicated as “1*”.

LISREL assumes by default that the θθθθjj measurement error variances, the ψψψψjj disturbance variances, and the φφφφjj variances and φφφφjl covariances of exogenous factors are free parameters. The defaults can be overridden as:

let the error variance of ... be ... to fix a measurement error variance.

let the covariance of ... and ... be ... to fix a covariance among exogenous factors.

let the variance of ... be ... to fix the variance of an exogenous factor or the disturbance of an endogenous factor (rare).

LISREL assumes by default that ψψψψjl disturbance and θθθθjl error covariances are zero. To override the default:

set the covariances of .... and .... free

let the errors of .... and ... correlate

99

The equations in the input file are (note the absence of structural equations for the moment. Measurement validity must be assessed prior to testing relationships): per_qua=quality +e1 glob_sat=λλλλ23quality+e2 recomm=recom e_confi=emplo+e4 e_neat=λλλλ51emplo+e5 e_cour=λλλλ61emplo+e6 e_knowl=λλλλ71emplo+e7 e_recogn=λλλλ81emplo+e8 pam_clea=informa+e9 info_ad=λλλλ10 2informa+e10 adv_real=λλλλ11 2informa+e11 off_conv=λλλλ12 2informa+e12 (6.1)

with the additional parameters φφφφ11, φφφφ21, φφφφ31, φφφφ41, φφφφ22, φφφφ32, φφφφ42, φφφφ33, φφφφ43, φφφφ44,θθθθ11, θθθθ22, θθθθ44, θθθθ55, θθθθ66
,…, θθθθ1212 . The total number of parameters is 29.

100

e1

λ10 2

1

λ11 2

λ12 2

λ61

λ71

λ81

λ51

e_confi e4

e_neat e5

e_cour e6

e_knowl e7

e_recogn e8

pam_clea e9

info_ad e10

adv_real e11

off_conv e12

emplo quality

informa recom

φ43

φ31

φ32

φ41

φ21

1 per_qua

glob_sat

e2

recomm

1

λ23

φ42 1

101

The options command is used to select:

Estimation method. ML, ULS, GLS (i. e. NT-WLS) or WLS (i.e. ADF, not recommended). ML is the default.

130-column output file (wp).

Residual matrix (rs).

Standardized estimates (sc).

Number of decimals (nd=).

The path diagram command (optional) draws a path diagram.

The compulsory final command is end of problem.

To run the file, press the icon or select file, run LISREL.

102

7.1. Results of the LISREL run Path diagram window: The user can:

Modify the layout by dragging variables to the desired position.

Select which parts of the model are shown (view, options, visible).

Select which statistical information is displayed, one at a time (view, estimations):

Estimates.

Standardized solution.

Conceptual diagram: with arrows only, no values.

t-values or Wald tests for included parameters.

Modification indices or Lagrange multiplier tests of omitted parameters.

Expected changes: approximate estimate that would be obtained if the parameter was set free.

A text output file contains more information (select it from the window menu).

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5.2. Goodness of fit assessment 5.2.1. Introduction

Interpretation does not proceed until the goodness of fit has been assessed.

The fit diagnostics attempt to determine if the model is correct and useful.

Correct model: its restrictions are true in the population. Relationships are correctly specified without the omission of relevant parameters or the addition of irrelevant ones.

In a correct model, the differences between S and ΣΣΣΣ(p) are small and random.

Correctness must not be strictly understood. A model must be an approximation of reality, not an exact copy of it.

Thus, a good model will be a compromise between parsimony and approximation.

Diagnostics will usually do well at distinguishing really badly fitting models from fairly well fitting models. Many models will fit fairly well (even exactly equally well if equivalent) and will be hard to distinguish statistically, they can be only distinguished theoretically.

105

5.2.2. The χχχχ2 goodness of fit statistic The minimum value of the discrepancy between S and ΣΣΣΣ(p) –fit function– obtained during estimation is used for the goodness of fit assessment. Under the null hypothesis that the model is correct, a transformation of this fit function called χχχχ2 goodness of fit statistic follows a χχχχ2 distribution with g degrees of freedom and can be used to do a likelihood ratio test. Rejection implies concluding that some relevant parameters have been omitted. The interpretation depends on the power of the test (the probability of rejecting false null hypotheses, i.e. of detecting omitted parameters): Low: rejection implies some large specification errors. Acceptance is inconclusive. High: acceptance means no large specification errors. Rejection is inconclusive. Power increases when: Sample size, R2, κκκκ or the number of indicators per factor increase. Collinearity and the overall number of parameters decrease.

Power is often high. Researchers are usually willing to accept approximately correct models with small misspecifications. Quantifying the degree of misfit is more useful than testing the hypothesis of exact fit.

106

5.2.3 Classification of diagnostics Global or detailed diagnostics

The former summarizes the overall goodness of fit of the model.

The latter seeks parts of the model that are misspecified.

Diagnostics that evaluate whether restrictions are correct or whether free parameters are superfluous or that evaluate the interpretability of the model

The first can lead to adding parameters and improving the fit.

The second can lead to dropping parameters and increasing parsimony.

The third looks at the usefulness of the model to draw conclusions.

Statistical tests or descriptive measures

The χχχχ2 statistic is a statistical test that performs a global diagnostic of the model restrictions.

107

5.2.4 Global diagnostics First look for serious problems:

Lack of convergence of the estimation algorithm.

Underidentification.

Empirical underidentification: very large estimates or standard errors.

Inadmissible estimates (e.g. negative variances, correlations larger than 1). Merge factors with correlations larger than 1. Fix negative unsignificant variances to zero, revise the model if they are significant.

Then look at global descriptive fit indices:

Independent, or less dependent on sample size than the χχχχ2 statistic. However, Hu and Bentler (1999) show that many of these indices tend to reject correct models for small samples. Cut-off criteria must then be flexible.

Some take the number of parameters (parsimony) into account.

It is also recommended to use more that one such index.

108

The simplest global descriptive fit indices evaluate the size of standardized residuals (residual correlations or residuals divided by the product of standard deviations). The standardized root mean squared residual (SRMR) does not take parsimony into account but does well at discriminating between roughly correct and grossly incorrect models (Hu, and Bentler, 1998).

( )[ ]( ) 21

ˆ1

2

1

+

−=∑∑

= =

kk

sssSRMR

p

i

i

jjjiiijij σ

Values below 0.05 are considered acceptable.

It can be improved by using the degrees of freedom instead of the number of variances and covariances (k+1)k/2 (Corten, Saris & Coenders, 2000) to take parsimony into account:

( )[ ]g

kkSRMRg

sssPSRMR

p

i

i

jjjiiijij 2/)1(

ˆ1

2

1 +=−

=∑∑

= =

σ

109

Incremental fit indices (see Bentler, 1990): they compare the χχχχ2 statistics of the researcher’s model and a base model that assumes that all variables are uncorrelated (independence model). They usually lie between 0 and 1 (1 shows a perfect fit).

The base χχχχ2 statistic is usually very large. These indices are often close to unity: in general only values above 0.95 are accepted.

Bentler and Bonnet’s (1980) normed fit index (NFI):

2

22

b

bNFIχ

χχ −=

(5.4)

χχχχ2 corresponds to the researchers’ model, χχχχ2b to the base model.

NFI= 0 if χχχχ2=χχχχ2b and NFI=1 if χχχχ2=0 .

110

The Tucker and Lewis’ (1973) index (TLI) also called non-normed fit index (NNFI) and McDonald and Marsh’s (1990) relative non-centrality index (RNI) introduce the degrees of freedom of the base (gb) and researcher (g) models to account for parsimony. They will increase after adding parameters only if the χχχχ2 statistic decreases more substantially than g. They can be slightly larger than 1.

12

22

−

−=

b

b

b

b

g

ggNNFI

χ

χχ

(5.5)

bb

bb

ggg

RNI−

−−−= 2

22 )()(χ

χχ (5.6)

Bentler’s (1990) comparative fit index (CFI) equals RNI if RNI<1 and equals 1 if RNI≥≥≥≥1.

The TLI seems to be more independent of sample size and to better account for parsimony (Marsh, Balla & Hau, 1996).

111

Other indices do not lie between 0 and 1. Their absolute values are difficult to interpret but they can be used to compare models with the same variables and cases. They take parsimony into account: AIC (Akaike, 1987) y CAIC (Bozdogan, 1987).

AIC = χχχχ2 - 2g (5.7)

CAIC = χχχχ2 - g(ln(N)+1) (5.8)

where ln(N) is the natural logarithm of sample size.

The root mean squared error of approximation (RMSEA) is defined as (Steiger, 1990):

gNNCPRMSEA

×=

(5.9)

Where the noncentrality parameter (NCP) is defined as χχχχ2-g (zero if negative).

Values below 0.05 are considered acceptable (Browne & Cudeck, 1993).

The sampling distribution is known, which makes it possible to do confidence intervals and test the hypothesis of approximate fit. If both extremes of the interval are larger than 0.05, a very bad fit can be concluded. If both extremes are below 0.05, a very good fit can be concluded.

112

5.2.5. Detailed diagnostics Are estimated values reasonable and of the expected sign?

Are there significant residuals that suggest the addition of parameters? (What LISREL calls standardized residuals are actually t-values).

Are there low R2 values suggesting the omission of explanatory variables or low κκκκ values suggesting a lack of validity?

113

The χχχχ2 statistic makes it possible to compare a model with any other that only differs by relaxing one or more restrictions (nested models). The χχχχ2 change or χχχχ2 difference is a likelihood ratio test statistic of these restrictions. Under the null hypothesis (restrictions are correct) the statistic is distributed as a χχχχ2 with degrees of freedom equal to the number of restrictions.

In any case, the less restricted of the models must be correct.

For example, to test the significance of a single parameter (one degree of freedom), two models must be fitted, one with and one without the parameter.

The χχχχ2 change is not robust to non-normality even if the χχχχ2 statistics are. If the Satorra-Bentler χχχχ2 statistic is used, a robust χχχχ2 change can be computed as follows: (Satorra & Bentler, 1999):

)( 10

1100

102

ddcdcd

TTdifferenceRobust

−−−

=χ

where T0 and T1 are the standard minimum fit function χχχχ2 statistics; T*0 and T*

1 the Satorra-Bentler χχχχ2 statistics; c0= T0/T*

0 and c1=T1/T*1 the scaling constants, and d0 and d1 the degrees

of freedom for two nested models, of which Model 0 is more restrictive.

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The Lagrange multiplier test, also known as modification index is asymptotically equivalent to the likelihood ratio test and only requires estimating the model omitting the parameter. Thus, it can be used to test the significance of omitted parameters.

Usually many of these tests are carried out (one for each omitted parameter). In order to prevent risk accumulation, Bonferroni’s inequality can be used by taking ααααindividual = αααα/g, where α α α α is the desired type I risk (e.g. 5%). Reject hypothesis above χχχχ2 value with one d.f. and tail area αααα/g. Always consider the approximate estimated parameter value: if power is high, parameters of a substantially insignificant value can be statistically significant. Robust variants of this test for non-normal data are not available at the moment.

Residuals and Lagrange multiplier tests can suggest the addition of parameters in order to improve fit. A model can also be improved by dropping irrelevant parameters. The Wald test is carried out only with the model including the parameter and thus tests the significance of included parameters.

The test statistic, called t statistic is the ratio of the estimate over its standard error. Under the null hypothesis that the parameter value is zero, it is distributed as a standardized normal variate. Non-significant parameters may be dropped from the model, especially if their theoretical argumentation is weak.

115

5.3. Model modification Frequently models fail to pass the diagnostics. Fortunately, data can not only be used to test models but also to drive their modification.

Model modifications, mainly based on detailed diagnostics and theoretical knowledge, aim at improving either its fit or its parsimony.

Model modification has some undesirable statistical consequences, especially if modifications are blindly done using only statistics, that is, without theory.

Which modifications introduce and in which order?

Introduce modifications one at a time, and carefully examine results before introducing the next.

First improve fit (add parameters). Then improve parsimony (drop parameters).

Use Bonferroni’s adjustment to the Lagrange multiplier tests.

Consider models with good descriptive fit indices, even if the χχχχ2 test rejects them.

Avoid adding theoretically uninterpretable parameters, no matter how significant.

Make few modifications.

116

When must one stop? Which of the modified models will be considered?

It depends on the parsimony-approximation compromise.

The selected model must pass the diagnostics, be theoretically relevant and be useful.

Nested models can be compared by the χχχχ2 change and goodness of fit indices that take parsimony into account: PSRMR, NNFI-TLI, CFI, AIC, CAIC, and RMSEA.

For non-nested models with the same variables: PSRMR, NNFI-TLI, CFI, AIC, CAIC, and RMSEA.

For models with different variables: PSRMR, NNFI-TLI, CFI and RMSEA.

117

5.3.3 Capitalization on chance Even if model modification has been done carefully, modifications are based on a particular sample. Have we reached a model that fits the population?

The introduction of modifications that improve the fit to the sample but not to the population is known as capitalization on chance.

Capitalization on chance is always present. Suggestions in the previous section reduce its extent but do not prevent it.

The results of modified models must be interpreted with caution. Estimates and t-values are inflated because parameters that are relevant to the particular sample have been added or have failed to be eliminated (Luijben, 1989).

The only solution is to check that the model fits well beyond the particular sample used.

Crossvalidation: estimation and goodness of fit test of the model on an independent sample of the same population. If only one sample is available, it can be split: the first half is used for model modification and the second for validation. Crossvalidation is successful if the model fits the second sample reasonably well.

Browne and Cudeck’s (1989) expected crossvalidation index (ECVI) has nothing to do with crossvalidation.

118

8. Example 2. Measurement quality evaluation with the multitrait-multimethod model

Multitrait-multimethod (MTMM) designs (Campbell & Fiske, 1959) consist of multiple measures of a set of factors (traits) with the same set of measurement procedures (methods). So these designs include t××××m measures.

The differences between methods can be any design characteristic, which can be shared by measurements of all traits, such as different response scale lengths or category labels in questionnaires, different data collection modes, different informants, etc.

All measures contain random measurement errors. In addition to these errors, the methods used often produce a systematic error that is called method effect. Method effects are common for all measurements using the same method; the resulting error terms will be correlated.

Random measurement errors tend to attenuate the correlations among observed measurements with respect to the correlations among the trait factors. On the contrary, correlated measurement errors usually increase these correlations.

119

Campbell and Fiske (1959) suggested using MTMM designs by directly examining the elements of the correlation matrix among all t××××m measurements, called MTMM matrix. This approach was cumbersome and often led to confusion (Schmitt & Stults, 1986) so that from the early seventies MTMM matrices began instead to be analyzed by means of SEM. These models are called MTMM models.

Many different MTMM models have been suggested in the literature. Among them are the correlated uniqueness (CU) model (Marsh, 1989; Marsh & Bailey, 1991), the classic confirmatory factor analysis (CFA) model for MTMM data (Althauser, Heberlein & Scott, 1971; Alwin, 1974; Werts & Linn, 1970), the direct product (DP) model (Browne, 1984, 1985), and the true score model for MTMM data (Saris & Andrews, 1991).

120

METHODS USED: Method 1: 101-point (0 to 100) modality with labelled extremes and midpoint (100P) Method 2: 5-point modality with labels on all categories (10P) Method 3: 11-point (0 to 10) modality with labelled extremes (5P) (All scales ranged from “completely satisfied” to “completely dissatisfied”; the midpoint, if any, was “neither satisfied nor dissatisfied”). Method 2 includes two further labels: Satisfied, Dissatisfied TRAITS MEASURED (Satisfaction with...) Trait 1: Life as a whole (GENERAL). Trait 2: Housing (HOUSING). Trait 3: Financial situation (FINANCE). Trait 4: Social contacts (SOCIAL).

The data collection was carried out in November 1989 with CATI (Computer Assisted Telephone Interview).

The population was defined as the residents of Catalonia aged 16 or above. The sampling was proportionate, stratified by town and systematic within each stratum. 406 valid cases were obtained (Batista-Foguet, Coenders and Sureda 1996). The telephone directory was used as sampling frame.

121

Models:

The CU model specifies correlated errors for all pairs of indicators sharing a method.

The CFA model specifies additional uncorrelated factors for all methods.

The CFA model partitions variance into trait, method and error. Frequently some of these variances may be estimated as negative.

For three traits, the CU and CFA models are equivalent. For more than three traits, the CU model is more general.

EXERCISE 6: Write the equations of both models, with and without matrix notation.

Problems:

Some of the methods cannot be assumed to produce continuous data.

122

GEN_10

HOU_10

FIN_10

SOC_10

GEN_100

HOU_100

FIN_100

SOC_100

λ22

λ11

λ33

λ44

GEN_5

HOU_5

FIN_5

SOC_5

λ51

λ62

λ73

λ84

GENERAL

HOUSING

FINANCE

SOCIAL

φ32

φ21

φ42

φ31 φ41

φ43

λ91

λ102

λ113

λ124

100P

e1

e2

e3

e4

λ15

λ25 λ35

λ45

5P

e5

e6

e7

e8

λ56

λ66 λ76

λ86

10P

e9

e10

e11

e12

λ97

λ107 λ117

λ127

123

e3 θ32

θ21

θ42

θ31 θ41

e1

e2

θ43 e4

e7 θ76

θ65

θ86

θ75 θ85

e5

e6

θ87 e8

GEN_10

HOU_10

FIN_10

SOC_10

GEN_100

HOU_100

FIN_100

SOC_100

λ22

λ11

λ33

λ44

GEN_5

HOU_5

FIN_5

SOC_5

λ51

λ62

λ73

λ84

GENERAL

HOUSING

FINANCE

SOCIAL

φ32

φ21

φ42

φ31 φ41

φ43

λ91

λ102

λ113

λ124 e11 θ1110

θ109

θ1210

θ119 θ129

e9

e10

θ1211 e12

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8.2. Extension: ordinal variables If SEM are estimated using covariances, they assume that the data are continuous and have an interval level of measurement.

While the social sciences are often interested in measuring and relating variables that are conceptually continuous, the measurement instruments used in these disciplines often fail to yield continuous and interval-level measures (e.g. a 5-point LIKERT item).

When modelling ordinally measured continuous variables, it is usually assumed that the ordinal measurements v are related to the continuous underlying variables v* through the non-linear step function: y=k iff ττττk-1< v*≤τ≤τ≤τ≤τk , for k=1,2,...,m where ττττk are the thresholds or cutting points of the above-mentioned regions, such that:

ττττ0=-∞∞∞∞, ττττk-1<ττττk , ττττm=∞∞∞∞.

125

Step and linear functions. High transformation errors

Histogram of a symmetric continuous variable

Bar chart of the categorized variable

Types of categorization errors (Johnson and Creech 1983):

Grouping errors derive from discrete measurement, i.e. from collapsing several values of v* into the same y value.

Transformation errors arise from non-interval measurement. The arbitrary values 1,...,m may not be linearly related to the within category expectation of v*, specially if the thresholds ττττi are not equally spaced.

126

Scales for the measurement of satisfaction leading to different degrees of transformation and grouping errors What is your degree of satisfaction concerning ? Scale 1: low transformation errors Scale 2: high transformation errors 1) –Completely dissatisfied 1) –Dissatisfied 2) –Dissatisfied 2) –Neutral 3) –Neither satisfied nor dissatisfied 3) –Fairly satisfied 4) –Satisfied 4) –Very satisfied 5) –Completely satisfied 5) –Completely satisfied Scale 3: high grouping errors Scale 4: “ideal” sale 1) –Completely dissatisfied 2) 1) –Dissatisfied 3) 2) –Neither satisfied nor dissatisfied 4) 3) –Satisfied 5) 6) 7) –Completely satisfied

127

The polychoric correlation ρρρρ(v*i,v*

j) is the correlation coefficient between two v* variables estimated from the scores of the v variables. Therefore, polychoric correlations correct grouping and transformation errors.

The form of the distribution of the vi* variables has to be specified so as to identify the

likelihood function conditional on ρρρρ(v*i,v*

j), ττττi 1, ττττi 2,...,ττττi mi-1, ττττj 1, ττττj 2,...,ττττj mj-1. The bivariate normality assumption is usually made (Olsson 1979).

The polyserial correlation (Olsson et al, 1982) is the correlation between two continuous variables v*

i and v*j, when v*

i is observable whereas only the ordinal scores vj are available. Bivariate normality is also assumed (Jöreskog 1990). Unlike covariances and Pearson correlations, non-normality does affect consistency.

128

The data are given in the form of a two-way contingency table, where nkl is the number of cases for which vi=k and vj=l and ππππkl the probability that a case has vi=k and vj=l:

****

11

),( jijikl dvvdvvlj

lj

ki

ki

∫∫−−

=τ

τ

τ

τ

φπ

where φφφφ(vi*,vj

*) is the bivariate standardized normal density function. The loglikelihood function of the contingency table is:

∑∑==

=ji m

lklkl

m

knLogL

11logπ

and has to be optimized with respect to ρρρρ(v*i,v*

j), ττττi 1,ττττi 2,..., ττττi mi-1, ττττj 1, ττττj 2,..., ττττj mj-1. The PRELIS program (Jöreskog & Sörbom, 1988, 1993) estimates the polychoric and polyserial correlations by a two-step procedure. The thresholds ττττik and ττττjl are first estimated from the cumulative relative frequency distributions of vi and vj and the inverse of the N(0,1) cumulative distribution function.

129

Standard errors and test statistics for polychoric and polyserial correlations

Regardless of the estimation method used, the ΓΓΓΓ asymptotic sampling covariance matrix of polychoric and polyserial correlations has to be computed in order to obtain correct test statistics.

ΓΓΓΓ is different from that of Pearson correlations or covariances. It can be computed under the normality assumption (it has thus nothing to do with robustness to non-normality). In current versions of PRELIS (up to 2.53) the computation of this matrix seems to be wrong.

PRELIS can also compute ΓΓΓΓ by bootstrapping methods (recommended):

It generates r (e.g. 500) resamples with replacement of size N from the original sample. It computes S from the means of the polychoric and polyserial correlations over the 500 samples.

It computes ΓΓΓΓ from the variances and covariances of the polychoric and polyserial correlations over the 500 samples.

130

A mixed correlation matrix can be produced by PRELIS (it will contain polychoric correlations for pairs of ordinal variables, Pearson correlations for pairs of numeric variables and polyserial correlations for mixed pairs). We must:

Define ordinal variables as such in the data, define variables, and type dialog box.

Select statistics, bootstrapping, and then number of bootstrap samples: 500, sample fraction: 100, output options,

and within the new dialog box select:

Correlations as moment matrix, save to file and provide a file name that will actually contain the mean of the 500 correlation matrices.

Asymptotic covariance matrix, save to file and provide a file name for the bootstrap ΓΓΓΓ matrix.

131

Robustness issues:

Covariances and Pearson correlations: Results of models that do not account for measurement error (e.g. regression) are seriously biased. Point estimates of models that account for measurement error (e.g. factor analysis) are robust unless transformation errors are extreme. Categorization errors seem to be a form of measurement error (Coenders & Saris, 1995; Coenders, Satorra & Saris, 1997). Estimates of parameters relating factors tend to be similar to those obtained with

polychoric correlations. Estimates of loadings and measurement error variances reflect the lower quality of

ordinal measurements. Standard errors are robust if methods robust to non-normality are used (ordinal

data are non-normal), that is, 4th order moments are supplied. If the model is scale dependent (introduces constraints other than fixing to 0 or needed for scaling latent variables), a covariance matrix must be used.

132

Polychoric and polyserial correlations:

Point estimates and standard errors of all models are correct under underlying normality. Estimates of loadings and measurement error variances reflect quality of measurement of the hypothetical underlying continuous variables, not of the actual data. Point estimates of all models are robust under moderate non-normality of the underlying variables. Polyserial correlations are not robust to non-normality of the observed continuous variables. It is advisable to group them into categories and only use Polychoric correlations if this occurs. The model has to be scale invariant.

Conclusion:

Use response scales leading to low grouping and transformation errors, and then use covariances in combination with 4th order moments.

133

Measures of association used for the example:

The models are scale invariant (only factor variances are standardized at 1, so factor covariances are interpretable as factor correlations). Then correlations can be used, and since variances of observed variables are also 1, loadings are standardized.

A Pearson correlation matrix can be estimated by treating all variables as continuous. A 4th order moment-based ΓΓΓΓ is needed.

A Pearson-polyserial-polychoric correlation matrix can be estimated assuming the 5-point scale to be ordinal. A bootstrap-based ΓΓΓΓ is needed.

134

9. Example 3. Parents’ and children’s values With the aim of relating children’s values to their parents’ values, a survey was carried out in Brazil and Catalonia. Children’s questionnaires were distributed in schools, and parents’ questionnaires were taken home by their children. Several dimensions emerged:

Materialistic values (mater): Money (val_mon). Power (val_pow). Own image (val_ima, dropped).

Capacities and knowledge values (abili): Intelligence (val_int). Technological ability (val_htec). Computers knowledge (val_knco, dropped). Profession (val_prof). Knowledge about the world (val_knwo, dropped).

Interpersonal relationships values (perso): Family (val_fam, dropped). Sensitivity (val_sen). Sympathy (val_symp). Social skills (val_soci, dropped).

135

Masurement model. Parents’ variable names start as “p_”.

136

Full model

137

Aims of the study:

Are children’s values related to parents’ values?

Are children of Brazil and Catalonia on average equally attached to values?

Is the relationship between parents and children the same in Brazil and in Catalonia?

Specific problems:

The data set contained a large number of missing values, mainly due to parents’ non-response.

138

9.1. Mean and covariance structure models If means and intercepts are parameters of interest, then the uncentred data must be analyzed, that is, both the covariance matrix and the mean vector must be taken into account (Sörbom, 1974). Thus the model is no longer a covariance structure model but a mean and covariance structure model.

A possible formulation of equation 3.32 including means and intercepts is:

f2=αααα2+β21f1+d2 v1=f1+e1 v2=f2 v3=ττττ3+λ31f1+e3

with E(f1)=αααα1

139

Three new parameter types:

Intercepts of measurement equations (ττττ): expectations of the observed variables for a zero value of the latent variable. The mean of the latent variable can be identified by fixing one ττττ to zero (in the same way as one λλλλ is fixed to one to identify the variance).

Intercepts of equations relating factors (αααα): interpreted as in a regression model.

Expected values of exogenous factors (αααα).

The expected values of the observed variables can be related to the parameter values in another set of structural equations:

E(v1)=E(f1)=αααα1

E(v2)=E(f2)= αααα2+β21 E(f1)=αααα2+β21αααα1 E(v3)=ττττ3+λ31E(f1)= ττττ3+λ31αααα1

This makes it possible to use the means as well as the covariances to estimate the model parameters. In this case, there are three observed variable means and three mean and intercept parameters. The mean structure is exactly identified.

140

In general:

f=αααα+Bf+d

v=ττττ+ΛΛΛΛf+e

where

αααα is a column vector containing the expectations of the r exogenous factors and the intercepts of the m-r endogenous ones: αααα’=(αααα1, αααα2,...,ααααr, ααααr+1,ααααr+2,...,ααααm).

ττττ is a column vector containing the intercepts of measurement equations: ττττ’=(ττττ1,ττττ2,...,ττττk) of which some will usually be constrained to zero.

EXERCISE 7: Write the model equations including intercepts and means, with and without matrix notation. The first indicator of each factor has the loading constrained to one and the intercept to zero.

141

If all possible αααα and all possible ττττ (except the ones used to fix the factor means) are free parameters, the mean structure is exactly identified and the ββββ, φφφφ, λλλλ, θθθθ, ψψψψ and parameter estimates are the same as for a model with only covariances. Of course, more restrictive specifications can be made and constraints on the αααα and ττττ parameters be tested. Then analyses will not be equivalent and the model with means will have more degrees of freedom than a similar model only with covariances.

The same estimation procedures can be used as with covariance structure models. With current versions of LISREL, robust estimation and testing with mean structures is problematic, but this is not so with other software programs.

If there are fewer mean and intercept parameters than means of observed variables, then there will be a residual vector for means. If these residuals are large, then parameters must be added to the mean structure. Modification indices for these parameters will usually reveal this fact.

142

9.2. Missing data treatments Several missing data processes have to be distinguished (see Little and Rubin, 1987):

Data are said to be missing completely at random when the probability that a datum is missing is independent of any characteristic of the individual. Data are said to be missing at random when the probability that a datum is missing depends only on characteristics of the individual that are observed (not missing). Data are said to be missing not at random (also called non-ignorable missing data) when the probability that a datum is missing depends on characteristics of the individual that are missing.

Data missing not at random are the most problematic. They can be brought close to missing at random if the number of variables is large (i.e. a lot of potential predictors of missingness will be observed).

143

Missing data are treated in several alternative ways within the context of SEM. Listwise or pairwise deletion. These procedures are only unbiased if the data are missing completely at random. Even under this unrealistic assumption, the first of the mentioned methods is highly inefficient and the second leads to biased standard errors (Enders, 2001; Enders & Bandalos, 2001). Mean substitution. This method is known to bias variances and covariances even when data are missing completely at random (Graham, Hofer & Piccinin, 1994; Graham, Hofer & MacKinnon, 1996, Enders, 2001) and should thus not be used. Imputation. This is a family of methods including regression imputation, hot deck imputation and EM imputation, in both their simple and multiple variants (Little and Rubin, 1987). This approach has the advantage of providing a complete data set on which standard estimation procedures could in principle be used. However, some of these imputation procedures (simple hot deck imputation and simple regression imputation) lead to biased estimates of factor correlations and residual variances. Multiple imputation (Rubin, 1987) does not have these drawbacks but it is cumbersome to perform unless special software is available. Imputation can be justified both if the data are missing at random or completely at random provided that some variables are correlated with the missing ones.

144

Direct ML assuming that the data are normally distributed and missing at random (Aburckle, 1996; Finkbeiner, 1979, Lee, 1986; Muthén, Kaplan, & Hollis, 1987; Wothke, 2000; Graham, Taylor & Cumsille, 2001). This procedure is currently available in most of the latest commercial software packages for SEM like Mx (Neale et al., 1999), EQS 6.0 (Bentler, 2000), AMOS 4.0 (Aburckle & Wothke, 1999), LISREL 8.51 (Jöreskog et al. 2000; du Toit & du Toit, 2001) and MPLUS 2.1 (Muthén & Muthén, 2001).

This procedure uses all available data to build a case per case likelihood function. For each case, only the measured variables for that case are considered. It thus requires raw data, no covariance matrices or mean vectors. It is consistent, efficient and leads to correct standard errors and test statistics if the data are normal and missing at random (Aburckle, 1996; Enders, 2001; Enders & Bandalos, 2001; Wothke, 2000).

When data are missing not at random (what is also called non-ignorable missing data) none of the procedures are consistent. This is the case when the probability that a datum is missing depends on characteristics of the individual that are missing, for instance on the same variable that is missing for the individual. However, ML is reported to be less biased than the alternative approaches (Muthén et al., 1987). In this example we use ML with missing data.

145

9.3. Equality constraints

Two or more parameters can be constrained to be equal if theoretically reasonable.

This also results in a gain of degrees of freedom and in a more parsimonious model.

These equalities can also be tested by a modification index. The modification index referring to a constrained parameter is the approximate decrease in the χχχχ2 statistic if the parameter is freely estimated instead of constrained.

More complex constraints can also be applied. A parameter can be constrained to a linear or non-linear function of other parameters.

The most frequent case of equality constraints is the fit of the same model to more than one sample. Then it makes sense to constrain certain or all the parameters to be equal in both samples.

146

9.4. Multiple groups If samples of several populations are available, then the same model could be fitted to all of them.

If this is done independently for each population, the estimates cannot be compared because:

It is not easy to test the equality of parameters across populations.

If the model contains latent variables measured by multiple indicators, it must be ensured that latent variables have the same composition and interpretation in the different populations.

The best solution is to fit a model to all the populations constraining certain parameters to be equal across populations.

147

9.5. Factor invariance Factor invariance, also called measurement invariance, measurement equivalence, factor equivalence, and construct comparability, refers to the extent to which items and the dimensions they measure mean the same thing to members of different groups. Factor invariance is needed before the groups can be compared, as otherwise, group differences in means or regression coefficients could be attributable to true differences in group distributions or to a different meaning of variables (Meredith, 1993; Little, 1997). This is especially relevant in cross-cultural research, in which the different groups get translated versions of the questionnaire (e.g. Reise et al. 1993; Steenkamp & Baumgartner, 1998).

A first requisite for factor invariance is the so-called configural invariance, defined as the fact that individuals of different groups conceptualize the constructs in the same way (Riordan & Vandenberg, 1994; Meredith, 1993). Its assessment consists in checking that in all groups there is the same number of factors associated to the same items. Configural invariance may fail due to cultures being so different that the meanings of constructs are different, to translation problems and to many other reasons (Cheung & Rensvold, 2002).

148

A second requisite is metric invariance, which implies that in addition to configural invariance all λλλλ factor loading parameters be equal across groups. Thus, not only the items composing each dimension but also the strength of the relationship between items and factors must be constant. Metric invariance is a requisite for making cross-group comparison of factor variances and of covariances or regression slopes relating different factors. The metric invariance requisite is often not completely satisfied in practice. It is argued that if it holds only for a set of items, it is enough to constrain the loadings of these in order to anchor a common meaning of the factors across groups (Byrne et al., 1989). This is the so-called partial metric invariance. A third requisite is called strong factor invariance (Meredith, 1993). In addition to metric invariance, strong factor invariance requires that ττττ intercepts also be constrained across groups. Strong factor invariance is a prerequisite for comparing factor means. This type of invariance can also hold only partially, that is for a subset of items of each dimension, (Byrne et al. 1989).

149

9.6. LISREL instructions Estimation with missing data by maximum likelihood:

It is done within LISREL, not PRELIS.

A raw data file in free format (with variables separated by blanks, one case per row) must be given.

All missing values must be given a common numeric code.

missing value code 0 raw data from file cataloni.dat

150

Mean structure models:

Either the raw data or both means and covariances must be provided:

means from file:

Intercepts are specified with the keyword const within an equation. Exogenous variables with a mean parameter also require an equation.

relationships: val_din=1*mater val_pod=const mater mater = const

Is equivalent to:

val_din=mater+e1 val_pod=ττττ2+λλλλ21mater+e2

with E(mater)=αααα1 and VAR(mater)=φφφφ11 as free parameters.

Obtaining robust standard errors and test statistics is problematic with current versions of LISREL.

151

Multiple group models:

Two or more model specifications are given in the same input file.

Variable names are common and only provided the first time.

Data (or covariances and means) must be in separate files for each group.

The title line of each input should include the group name.

If no relationships, variances and covariances are given for the second group, then the same pattern of free and fixed parameters is assumed as in the first group and all parameters are assumed to be constrained to be equal across groups.

In order to release these constraints for certain parameters, the appropriate relationships, variances and covariances have to be added in the specification of the second group. This requirement includes variances and covariances that are free by default in the first group, so the second group input can be much longer than the first.

153 FIRST MEASUREMENT MODELobserved variablesper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recognpam_clea info_ad adv_real off_convcovariance matrix from file bonshoms.cmasymptoric covariances from file bonshoms.acsample size: 301latent variablesquality recom emplo informarelationshipsper_qua = 1*qualityglob_sat = qualityrecomm = 1*recome_confi = 1*emploe_neat = emploe_cour = emploe_knowl = emploe_recogn = emplopam_clea = 1*informainfo_ad = informaadv_real = informaoff_conv = informalet the error variance of recomm be 0options me=ml wp rs sc nd=3path diagramend of problemSample Size = 301

Covariance Matrix to be Analyzedper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recogn pam_clea info_ad

per_qua 1.423glob_sat 1.225 1.706recomm 1.041 1.207 3.058e_confi 0.777 0.885 0.750 1.593

e_neat 0.478 0.522 0.371 0.785 1.102e_cour 0.641 0.847 0.577 1.332 0.776 1.743

e_knowl 0.599 0.669 0.559 0.989 0.725 0.956 1.465e_recogn 0.933 1.147 0.851 1.372 0.947 1.587 0.949 3.540pam_clea 0.698 0.758 0.573 1.157 0.618 1.047 0.932 1.129 1.961info_ad 0.653 0.676 0.560 0.906 0.548 0.863 0.981 1.170 1.050 1.634

adv_real 0.834 0.867 0.690 1.041 0.641 0.971 0.854 1.581 1.290 1.174off_conv 0.589 0.671 0.639 0.947 0.637 0.882 0.983 1.012 1.007 1.027

adv_real off_convadv_real 1.831off_conv 0.996 1.570

154

Number of Iterations = 7LISREL Estimates (Maximum Likelihood)

per_qua = 1.000*quality, Errorvar.= 0.356 , R² = 0.750(0.0623)5.716

glob_sat = 1.148*quality, Errorvar.= 0.300 , R² = 0.824(0.0658) (0.0720)17.449 4.170

recomm = 1.000*recom,, R² = 1.000

e_confi = 1.000*emplo, Errorvar.= 0.302 , R² = 0.810(0.0457)6.616

e_neat = 0.620*emplo, Errorvar.= 0.605 , R² = 0.451(0.0565) (0.0971)10.973 6.231

e_cour = 0.995*emplo, Errorvar.= 0.464 , R² = 0.734(0.0456) (0.0622)21.831 7.460

e_knowl = 0.784*emplo, Errorvar.= 0.671 , R² = 0.542(0.0616) (0.0958)12.741 7.003

e_recogn = 1.135*emplo, Errorvar.= 1.878 , R² = 0.469(0.0825) (0.297)13.756 6.331

pam_clea = 1.000*informa, Errorvar.= 0.750 , R² = 0.618(0.100)7.473

info_ad = 0.918*informa, Errorvar.= 0.613 , R² = 0.625(0.0758) (0.0894)12.110 6.858

155 adv_real = 1.024*informa, Errorvar.= 0.561 , R² = 0.693

(0.0695) (0.0895)14.736 6.269

off_conv = 0.861*informa, Errorvar.= 0.671 , R² = 0.573(0.0839) (0.0938)10.261 7.154

Covariance Matrix of Independent Variablesquality recom emplo informa

-------- -------- -------- --------quality 1.067

(0.142)7.492

recom 1.047 3.058(0.144) (0.367)

7.264 8.328

emplo 0.756 0.686 1.291(0.100) (0.129) (0.111)

7.598 5.304 11.590

informa 0.705 0.648 1.067 1.211(0.102) (0.134) (0.115) (0.187)

6.884 4.830 9.261 6.486

156 Goodness of Fit Statistics

Degrees of Freedom = 49Minimum Fit Function Chi-Square = 200.993 (P = 0.0)

Normal Theory Weighted Least Squares Chi-Square = 197.928 (P = 0.0)Satorra-Bentler Scaled Chi-Square = 151.510 (P = 0.00)

Chi-Square Corrected for Non-Normality = 154.078 (P = 0.00)Estimated Non-centrality Parameter (NCP) = 102.510

90 Percent Confidence Interval for NCP = (69.182 ; 143.462)

Minimum Fit Function Value = 0.670Population Discrepancy Function Value (F0) = 0.342

90 Percent Confidence Interval for F0 = (0.231 ; 0.478)Root Mean Square Error of Approximation (RMSEA) = 0.0835

90 Percent Confidence Interval for RMSEA = (0.0686 ; 0.0988)P-Value for Test of Close Fit (RMSEA < 0.05) = 0.000189

Expected Cross-Validation Index (ECVI) = 0.69890 Percent Confidence Interval for ECVI = (0.587 ; 0.835)

ECVI for Saturated Model = 0.520ECVI for Independence Model = 7.893

Chi-Square for Independence Model with 66 Degrees of Freedom = 2344.001Independence AIC = 2368.001

Model AIC = 209.510Saturated AIC = 156.000

Independence CAIC = 2424.486Model CAIC = 346.016

Saturated CAIC = 523.155

Root Mean Square Residual (RMR) = 0.0810Standardized RMR = 0.0432

Goodness of Fit Index (GFI) = 0.901Adjusted Goodness of Fit Index (AGFI) = 0.842Parsimony Goodness of Fit Index (PGFI) = 0.566

Normed Fit Index (NFI) = 0.914Non-Normed Fit Index (NNFI) = 0.910

Parsimony Normed Fit Index (PNFI) = 0.679Comparative Fit Index (CFI) = 0.933Incremental Fit Index (IFI) = 0.934

Relative Fit Index (RFI) = 0.885

Critical N (CN) = 112.824

157 Fitted Covariance Matrix

per_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recogn pam_clea info_ad-------- -------- ------- ------- -------- ------ ------- -------- -------- --------

per_qua 1.423glob_sat 1.225 1.706recomm 1.047 1.202 3.058e_confi 0.756 0.868 0.686 1.593

e_neat 0.469 0.539 0.425 0.801 1.102e_cour 0.753 0.864 0.683 1.285 0.797 1.743

e_knowl 0.593 0.681 0.538 1.012 0.628 1.007 1.465e_recogn 0.858 0.985 0.778 1.465 0.908 1.458 1.149 3.540pam_clea 0.705 0.810 0.648 1.067 0.662 1.062 0.837 1.211 1.961info_ad 0.647 0.743 0.595 0.980 0.608 0.975 0.768 1.112 1.112 1.634

adv_real 0.722 0.829 0.663 1.092 0.678 1.087 0.857 1.240 1.240 1.138off_conv 0.607 0.697 0.558 0.919 0.570 0.915 0.721 1.043 1.043 0.958

adv_real off_conv-------- --------

adv_real 1.831off_conv 1.068 1.570

Fitted Residualsper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recogn pam_clea info_ad

-------- -------- ------- ------- -------- ------ ------- -------- -------- --------per_qua 0.000

glob_sat 0.000 0.000recomm -0.007 0.005 0.000e_confi 0.020 0.016 0.064 0.000

e_neat 0.008 -0.017 -0.054 -0.016 0.000e_cour -0.112 -0.017 -0.106 0.047 -0.020 0.000

e_knowl 0.005 -0.013 0.022 -0.023 0.097 -0.052 0.000e_recogn 0.074 0.161 0.073 -0.092 0.038 0.129 -0.199 0.000pam_clea -0.007 -0.051 -0.075 0.089 -0.044 -0.016 0.095 -0.082 0.000info_ad 0.006 -0.067 -0.035 -0.073 -0.060 -0.112 0.213 0.058 -0.062 0.000

adv_real 0.113 0.038 0.027 -0.051 -0.037 -0.117 -0.003 0.341 0.050 0.036off_conv -0.018 -0.027 0.081 0.027 0.066 -0.033 0.262 -0.031 -0.037 0.070


adv_real 0.000off_conv -0.072 0.000

158 Summary Statistics for Fitted ResidualsSmallest Fitted Residual = -0.199

Median Fitted Residual = 0.000Largest Fitted Residual = 0.341

Stemleaf Plot- 2|0- 1|2111- 0|98777766555544433332222222111000000000000000

0|11122233444556677778991|01362|163|4

Standardized Residualsper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recogn pam_clea info_ad

-------- -------- ------- ------- -------- ------ ------- -------- -------- --------per_qua - -

glob_sat - - - -recomm -0.229 0.229 - -e_confi 0.706 0.595 1.659 - -

e_neat 0.194 -0.378 -0.774 -0.904 - -e_cour -3.051 -0.476 -1.947 4.243 -0.829 - -

e_knowl 0.116 -0.268 0.299 -1.293 2.916 -2.080 - -e_recogn 0.982 2.033 0.592 -2.987 0.681 2.988 -3.424 - -pam_clea -0.144 -1.081 -1.053 2.672 -0.977 -0.388 1.979 -1.022 - -info_ad 0.138 -1.570 -0.544 -2.429 -1.453 -3.053 4.893 0.801 -2.132 - -

adv_real 2.830 0.976 0.463 -1.831 -0.914 -3.375 -0.063 4.785 1.986 1.585off_conv -0.405 -0.586 1.166 0.852 1.565 -0.858 5.824 -0.418 -1.156 2.440


adv_real - -off_conv -2.856 - -

159 Summary Statistics for Standardized ResidualsSmallest Standardized Residual = -3.424

Median Standardized Residual = 0.000Largest Standardized Residual = 5.824

Stemleaf Plot- 3|44110- 2|9411- 1|9865321100- 0|99988655444432110000000000000

0|1122356677891|0026672|00047893|04|2895|8

Largest Negative Standardized ResidualsResidual for e_cour and per_qua -3.051Residual for e_recogn and e_confi -2.987Residual for e_recogn and e_knowl -3.424Residual for info_ad and e_cour -3.053Residual for adv_real and e_cour -3.375Residual for off_conv and adv_real -2.856

Largest Positive Standardized ResidualsResidual for e_cour and e_confi 4.243Residual for e_knowl and e_neat 2.916Residual for e_recogn and e_cour 2.988Residual for pam_clea and e_confi 2.672Residual for info_ad and e_knowl 4.893Residual for adv_real and per_qua 2.830Residual for adv_real and e_recogn 4.785Residual for off_conv and e_knowl 5.824

160 Qplot of Standardized Residuals

3.5........................................................................... . .. . x. . .. . x. . x. . x. . xx .

N . . x x .o . . xx x .r . . xx x .m . . x x x .a . . x* x .l . x *x .

. x*x .Q . xxx.x .u . xxx . .a . xx . .n . x x . .t . ** . .i . xxx . .l . xx x . .e . xxx . .s . x x . .

. x x . .

. x . .

. x . .

.x . .

. . .

.x . .

. . .-3.5..........................................................................

-3.5 3.5Standardized Residuals

161 The Modification Indices Suggest to Add the

Path to from Decrease in Chi-Square New Estimateinfo_ad emplo 11.9 -0.94

The Modification Indices Suggest to Add an Error CovarianceBetween and Decrease in Chi-Square New Estimate

e_cour per_qua 11.0 -0.11e_cour e_confi 24.5 0.24e_knowl e_neat 9.7 0.14e_recogn e_confi 13.5 -0.28e_recogn e_cour 10.5 0.24e_recogn e_knowl 12.6 -0.26pam_clea e_confi 12.3 0.13pam_clea e_recogn 8.7 -0.23info_ad e_knowl 22.8 0.21adv_real e_recogn 35.3 0.42off_conv e_knowl 22.0 0.21off_conv info_ad 8.7 0.17off_conv adv_real 13.6 -0.24

Standardized SolutionLAMBDA-X

quality recom emplo informa-------- -------- -------- --------

per_qua 1.033 - - - - - -glob_sat 1.186 - - - - - -recomm - - 1.749 - - - -e_confi - - - - 1.136 - -

e_neat - - - - 0.705 - -e_cour - - - - 1.131 - -

e_knowl - - - - 0.891 - -e_recogn - - - - 1.289 - -pam_clea - - - - - - 1.101info_ad - - - - - - 1.010

adv_real - - - - - - 1.127off_conv - - - - - - 0.948

PHIquality recom emplo informa

-------- -------- -------- --------quality 1.000

recom 0.580 1.000emplo 0.645 0.345 1.000

informa 0.620 0.337 0.854 1.000

162 Completely Standardized Solution

LAMBDA-Xquality recom emplo informa

-------- -------- -------- --------per_qua 0.866 - - - - - -

glob_sat 0.908 - - - - - -recomm - - 1.000 - - - -e_confi - - - - 0.900 - -

e_neat - - - - 0.671 - -e_cour - - - - 0.857 - -

e_knowl - - - - 0.736 - -e_recogn - - - - 0.685 - -pam_clea - - - - - - 0.786info_ad - - - - - - 0.790

adv_real - - - - - - 0.833off_conv - - - - - - 0.757


-------- -------- -------- --------quality 1.000

recom 0.580 1.000emplo 0.645 0.345 1.000

informa 0.620 0.337 0.854 1.000

THETA-DELTAper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recogn pam_clea info_ad

-------- -------- ------- ------- -------- ------ ------- -------- -------- --------0.250 0.176 - - 0.190 0.549 0.266 0.458 0.531 0.382 0.375


0.307 0.427

The Problem used 49752 Bytes (= 0.1% of Available Workspace)Time used: 0.902 Seconds

163 MODIFIED MEASUREMENT MODELobserved variablesper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recognpam_clea info_ad adv_real off_convcovariance matrix from file bonshoms.cmasymptoric covariances from file bonshoms.acsample size: 301latent variablesquality recom emplo informarelationshipsper_qua = 1*qualityglob_sat = qualityrecomm = 1*recome_confi = 1*emploe_neat = emploe_cour = emploe_knowl = emplo informae_recogn = emplopam_clea = 1*informainfo_ad = informaadv_real = informaoff_conv = informalet the error variance of recomm be 0let the errors of adv_real and e_recogn correlateoptions me=ml wp rs sc nd=3path diagramend of problem

164 LISREL Estimates (Maximum Likelihood)



recomm = 1.000*recom,, R² = 1.000




e_knowl = 0.332*emplo + 0.532*informa, Errorvar.= 0.613 , R² = 0.582(0.105) (0.117) (0.0935)3.174 4.537 6.558

e_recogn = 1.122*emplo, Errorvar.= 1.908 , R² = 0.464(0.0798) (0.304)14.049 6.277



adv_real = 1.00*informa, Errorvar.= 0.624 , R² = 0.659(0.0642) (0.0986)15.571 6.332

165 off_conv = 0.880*informa, Errorvar.= 0.632 , R² = 0.597

(0.0825) (0.0889)10.668 7.114

Error Covariance for adv_real and e_recogn = 0.444(0.0968)

4.585

Covariance Matrix of Independent Variablesquality recom emplo informa

-------- -------- -------- --------quality 1.064

(0.142)7.484

recom 1.046 3.058(0.144) (0.367)

7.261 8.328

emplo 0.749 0.680 1.316(0.102) (0.130) (0.113)

7.373 5.234 11.679

informa 0.691 0.643 1.032 1.210(0.100) (0.134) (0.114) (0.185)

6.900 4.800 9.054 6.554

Goodness of Fit StatisticsDegrees of Freedom = 47

Normal Theory Weighted Least Squares Chi-Square = 138.054 (P = 0.00)Satorra-Bentler Scaled Chi-Square = 105.861 (P = 0.000)

Root Mean Square Error of Approximation (RMSEA) = 0.064690 Percent Confidence Interval for RMSEA = (0.0482 ; 0.0811)

Non-Normed Fit Index (NNFI) = 0.944Comparative Fit Index (CFI) = 0.960

166 Standardized Residuals

per_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recogn pam_clea info_ad-------- -------- ------- ------- -------- ------ ------- -------- -------- --------

per_qua - -glob_sat - - - -recomm -0.180 0.180 - -e_confi 1.002 0.894 1.989 - -

e_neat 0.474 -0.089 -0.614 -1.151 - -e_cour -3.103 -0.474 -2.060 1.797 -1.239 - -

e_knowl -0.405 -0.895 -0.115 0.212 3.538 -1.394 - -e_recogn 1.198 2.221 0.714 -3.277 0.806 2.582 -2.532 -0.776pam_clea 0.154 -0.758 -0.958 3.473 -0.254 0.293 -1.696 -0.341 - -info_ad 0.314 -1.396 -0.546 -1.561 -0.817 -2.515 2.411 1.299 -2.415 - -

adv_real 3.289 1.648 0.741 0.304 0.250 -1.706 -4.561 -0.450 2.644 2.025off_conv -0.429 -0.655 1.086 1.146 1.879 -0.737 3.836 -0.097 -1.893 1.539


adv_real -0.118off_conv -2.396 - -

The Modification Indices Suggest to Add thePath to from Decrease in Chi-Square New Estimate

e_cour quality 8.2 -0.25info_ad emplo 12.1 -0.65


e_cour per_qua 10.4 -0.10e_knowl e_neat 13.3 0.15e_recogn e_confi 27.0 -0.43e_recogn e_cour 14.7 0.27pam_clea e_confi 11.9 0.13info_ad e_knowl 9.8 0.15adv_real e_knowl 19.0 -0.22adv_real pam_clea 21.2 0.28off_conv e_knowl 15.6 0.19off_conv adv_real 9.5 -0.17

167 FINAL MEASUREMENT MODELobserved variablesper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recognpam_clea info_ad adv_real off_convcovariance matrix from file bonshoms.cmasymptoric covariances from file bonshoms.acsample size: 301latent variablesquality recom emplo informarelationshipsper_qua = 1*qualityglob_sat = qualityrecomm = 1*recome_confi = 1*emploe_neat = emploe_cour = emplopam_clea = 1*informainfo_ad = informaadv_real = informaoff_conv = informalet the error variance of recomm be 0options me=ml wp rs sc nd=3path diagramend of problem




recomm = 1.000*recom,, R² = 1.000e_confi = 1.000*emplo, Errorvar.= 0.193 , R² = 0.879

(0.0457)4.226







169 Covariance Matrix of Independent Variables

quality recom emplo informa-------- -------- -------- --------

quality 1.074(0.142)

7.552

recom 1.051 3.058(0.144) (0.367)

7.303 8.328

emplo 0.770 0.708 1.400(0.104) (0.133) (0.115)

7.403 5.310 12.209

informa 0.715 0.653 1.067 1.233(0.103) (0.135) (0.116) (0.189)

6.945 4.831 9.231 6.535


Normal Theory Weighted Least Squares Chi-Square = 59.963 (P = 0.000930)Satorra-Bentler Scaled Chi-Square = 47.026 (P = 0.0247)Root Mean Square Error of Approximation (RMSEA) = 0.0435

90 Percent Confidence Interval for RMSEA = (0.0158 ; 0.0664)Non-Normed Fit Index (NNFI) = 0.973

Comparative Fit Index (CFI) = 0.982


info_ad emplo 12.5 -0.60adv_real emplo 10.9 -0.66


e_cour per_qua 9.6 -0.10e_cour glob_sat 7.9 0.10off_conv info_ad 11.8 0.21off_conv adv_real 12.6 -0.24


LAMBDA-Xquality recom emplo informa

-------- -------- -------- --------per_qua 0.869 - - - - - -

glob_sat 0.905 - - - - - -recomm - - 1.000 - - - -e_confi - - - - 0.937 - -

e_neat - - - - 0.645 - -e_cour - - - - 0.850 - -

pam_clea - - - - - - 0.793info_ad - - - - - - 0.786

adv_real - - - - - - 0.834off_conv - - - - - - 0.751


-------- -------- -------- --------quality 1.000

recom 0.580 1.000emplo 0.628 0.342 1.000

informa 0.622 0.336 0.813 1.000

THETA-DELTAper_qua glob_sat recomm e_confi e_neat e_cour pam_clea info_ad adv_real off_conv------- -------- ------- ------- -------- ------- -------- ------- -------- -------

0.245 0.182 - - 0.121 0.584 0.278 0.371 0.383 0.304 0.436

171 FIRST COMPLETE MODELobserved variablesper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recognpam_clea info_ad adv_real off_convcovariance matrix from file bonshoms.cmasymptoric covariances from file bonshoms.acsample size: 301latent variablesquality recom emplo informarelationshipsper_qua = 1*qualityglob_sat = qualityrecomm = 1*recome_confi = 1*emploe_neat = emploe_cour = emplopam_clea = 1*informainfo_ad = informaadv_real = informaoff_conv = informaquality = emplo informarecom = quality emplo informalet the error variance of recomm be 0options me=ml wp rs sc nd=3path diagramend of problem

172 quality = 0.317*emplo + 0.306*informa, Errorvar.= 0.611 , R² = 0.431

(0.105) (0.100) (0.0836)3.028 3.048 7.309

recom = 1.026*quality - 0.0240*emplo - 0.0450*informa, Errorvar.= 2.025 , R² = 0.338(0.110) (0.143) (0.164) (0.345)9.332 -0.168 -0.274 5.876

Covariance Matrix of Independent Variablesemplo informa

-------- --------emplo 1.400

(0.115)12.209

informa 1.067 1.233(0.116) (0.189)

9.231 6.535

Completely Standardized SolutionBETA

quality recom-------- --------

quality - - - -recom 0.608 - -

GAMMAemplo informa

-------- --------quality 0.361 0.328

recom -0.016 -0.029

PSINote: This matrix is diagonal.


0.569 0.662

173 FINAL COMPLETE MODELobserved variablesper_qua glob_sat recomm e_confi e_neat e_cour e_knowl e_recognpam_clea info_ad adv_real off_convcovariance matrix from file bonshoms.cmasymptoric covariances from file bonshoms.acsample size: 301latent variablesquality recom emplo informarelationshipsper_qua = 1*qualityglob_sat = qualityrecomm = 1*recome_confi = 1*emploe_neat = emploe_cour = emplopam_clea = 1*informainfo_ad = informaadv_real = informaoff_conv = informaquality = emplo informarecom = qualitylet the error variance of recomm be 0options me=ml wp rs sc nd=3path diagramend of problem




recomm = 1.000*recom,, R² = 1.000








175 quality = 0.316*emplo + 0.303*informa, Errorvar.= 0.615 , R² = 0.427

(0.104) (0.1000) (0.0834)3.027 3.036 7.377

recom = 0.974*quality, Errorvar.= 2.039 , R² = 0.333(0.0936) (0.349)10.407 5.850

Covariance Matrix of Independent Variablesemplo informa

-------- --------emplo 1.400

(0.115)12.211

informa 1.067 1.232(0.116) (0.189)

9.232 6.535


Degrees of Freedom = 32Normal Theory Weighted Least Squares Chi-Square = 60.335 (P = 0.00178)

Satorra-Bentler Scaled Chi-Square = 48.276 (P = 0.0324)Root Mean Square Error of Approximation (RMSEA) = 0.0412

90 Percent Confidence Interval for RMSEA = (0.0123 ; 0.0638)Non-Normed Fit Index (NNFI) = 0.976

Comparative Fit Index (CFI) = 0.983

176 Standardized Residuals

per_qua glob_sat recomm e_confi e_neat e_cour pam_clea info_ad adv_real off_conv------- -------- ------- ------- -------- ------- -------- ------- -------- -------

per_qua - -glob_sat -0.588 - -recomm -0.161 0.563 - -e_confi 0.427 0.434 0.043 - -

e_neat 0.869 0.439 -0.662 -1.619 - -e_cour -2.291 0.455 -1.441 1.167 0.699 - -

pam_clea -0.284 -1.163 -1.180 2.901 0.152 0.834 - -info_ad 0.220 -1.363 -0.718 -2.069 -0.097 -1.343 -2.312 - -

adv_real 2.807 1.036 -0.153 -1.758 0.458 -1.633 1.572 1.815 - -off_conv -0.316 -0.407 0.544 1.319 2.597 0.576 -1.257 2.817 -2.698 - -


info_ad emplo 12.3 -0.59adv_real emplo 10.6 -0.65


e_cour per_qua 9.6 -0.10e_cour glob_sat 8.0 0.10off_conv info_ad 11.7 0.20off_conv adv_real 12.7 -0.24


LAMBDA-Yquality recom

-------- --------per_qua 0.869 - -

glob_sat 0.906 - -recomm - - 1.000

LAMBDA-Xemplo informa

-------- --------e_confi 0.938 - -

e_neat 0.645 - -e_cour 0.850 - -

pam_clea - - 0.793info_ad - - 0.786

adv_real - - 0.834off_conv - - 0.752

BETAquality recom

-------- --------quality - - - -

recom 0.577 - -

GAMMAemplo informa

-------- --------quality 0.361 0.325

recom - - - -

PSINote: This matrix is diagonal.


0.573 0.667

THETA-EPSper_qua glob_sat recomm

-------- -------- --------0.245 0.179 - -

THETA-DELTAe_confi e_neat e_cour pam_clea info_ad adv_real off_conv

-------- -------- -------- -------- -------- -------- --------0.121 0.584 0.278 0.372 0.383 0.304 0.435

178 Classic multitrait-multimethod model. Pearson correlationsobserved variablesGEN_100 HOU_100 FIN_100 S0C_100GEN_5 HOU_5 FIN_5 S0C_5GEN_10 HOU_10 FIN_10 S0C_10correlation matrix from file: sat.pemasymptotic covariances from file: sat.peasample size: 406latent variablesGENERAL HOUSING FINANCE SOCIAL 100P 5P 10PrelationshipsGEN_100 = GENERAL 100PHOU_100 = HOUSING 100PFIN_100 = FINANCE 100PS0C_100 = SOCIAL 100PGEN_5 = GENERAL 5PHOU_5 = HOUSING 5PFIN_5 = FINANCE 5PS0C_5 = SOCIAL 5PGEN_10 = GENERAL 10PHOU_10 = HOUSING 10PFIN_10 = FINANCE 10PS0C_10 = SOCIAL 10Plet the covariance of GENERAL and 100P be zerolet the covariance of GENERAL and 5P be zerolet the covariance of GENERAL and 10P be zerolet the covariance of HOUSING and 100P be zerolet the covariance of HOUSING and 5P be zerolet the covariance of HOUSING and 10P be zerolet the covariance of FINANCE and 100P be zerolet the covariance of FINANCE and 5P be zerolet the covariance of FINANCE and 10P be zerolet the covariance of SOCIAL and 100P be zerolet the covariance of SOCIAL and 5P be zerolet the covariance of SOCIAL and 10P be zerolet the covariance of 100P and 5P be zerolet the covariance of 100P and 10P be zerolet the covariance of 5P and 10P be zerooptions me=uls rs wp nd=3 AD=OFF IT=300end of problem

179 Sample Size = 406Classic multitrait-multimethod model. Pearson correlations

Correlation MatrixGEN_100 HOU_100 FIN_100 S0C_100 GEN_5 HOU_5 FIN_5 S0C_5 GEN_10 HOU_10

-------- -------- -------- -------- -------- -------- -------- -------- -------- --------GEN_100 1.000HOU_100 0.453 1.000FIN_100 0.464 0.430 1.000S0C_100 0.340 0.229 0.223 1.000

GEN_5 0.574 0.207 0.241 0.330 1.000HOU_5 0.261 0.662 0.276 0.211 0.334 1.000FIN_5 0.349 0.307 0.762 0.195 0.300 0.292 1.000S0C_5 0.175 0.039 0.019 0.646 0.390 0.163 0.119 1.000

GEN_10 0.639 0.334 0.346 0.309 0.630 0.322 0.326 0.281 1.000HOU_10 0.370 0.733 0.386 0.280 0.313 0.739 0.312 0.166 0.518 1.000FIN_10 0.399 0.360 0.788 0.166 0.251 0.275 0.791 0.052 0.420 0.462S0C_10 0.276 0.121 0.121 0.657 0.282 0.161 0.143 0.692 0.445 0.339

Correlation Matrix

FIN_10 S0C_10-------- --------

FIN_10 1.000S0C_10 0.237 1.000

180 Classic multitrait-multimethod model. Pearson correlationsNumber of Iterations = 36

LISREL Estimates (Unweighted Least Squares)

Measurement Equations

GEN_100 = 0.784*GENERAL + 0.402*100P, Errorvar.= 0.224 , R² = 0.776(0.0330) (0.0636) (0.0772)23.784 6.329 2.901

HOU_100 = 0.790*HOUSING + 0.383*100P, Errorvar.= 0.228 , R² = 0.772(0.0314) (0.0568) (0.0636)25.164 6.745 3.592

FIN_100 = 0.858*FINANCE + 0.391*100P, Errorvar.= 0.110 , R² = 0.890(0.0221) (0.0484) (0.0471)38.778 8.080 2.335

S0C_100 = 0.869*SOCIAL + 0.172*100P, Errorvar.= 0.215 , R² = 0.785(0.0357) (0.0735) (0.0639)24.353 2.336 3.364

GEN_5 = 0.718*GENERAL + 0.733*5P, Errorvar.= -0.0517, R² = 1.052(0.0391) (0.524) (0.759)18.360 1.397 -0.0680

W_A_R_N_I_N_G : Error variance is negative.

HOU_5 = 0.792*HOUSING + 0.0957*5P, Errorvar.= 0.364 , R² = 0.636(0.0366) (0.0841) (0.0708)21.606 1.137 5.144



GEN_10 = 0.851*GENERAL + 0.348*10P, Errorvar.= 0.155 , R² = 0.845(0.0291) (0.0605) (0.0679)29.237 5.748 2.286

181 HOU_10 = 0.950*HOUSING + 0.388*10P, Errorvar.= -0.0525 , R² = 1.053

(0.0249) (0.0602) (0.0553)38.206 6.443 -0.949

W_A_R_N_I_N_G : Error variance is negative.



Correlation Matrix of Independent Variables

GENERAL HOUSING FINANCE SOCIAL 100P 5P 10P-------- -------- -------- -------- -------- -------- --------

GENERAL 1.000

HOUSING 0.465 1.000(0.063)

7.321

FINANCE 0.458 0.426 1.000(0.052) (0.054)

8.774 7.922

SOCIAL 0.428 0.242 0.175 1.000(0.065) (0.059) (0.062)

6.538 4.088 2.816

100P - - - - - - - - 1.000

5P - - - - - - - - - - 1.000

10P - - - - - - - - - - - - 1.000

182 Goodness of Fit Statistics


Satorra-Bentler Scaled Chi-Square = 68.785 (P = 0.000810)Chi-Square Corrected for Non-Normality = 95.473 (P = 0.000)

Estimated Non-centrality Parameter (NCP) = 32.78590 Percent Confidence Interval for NCP = (13.149 ; 60.222)




Expected Cross-Validation Index (ECVI) = 0.37790 Percent Confidence Interval for ECVI = (0.329 ; 0.445)

ECVI for Saturated Model = 0.385ECVI for Independence Model = 10.780

Chi-Square for Independence Model with 66 Degrees of Freedom = 4341.927Independence AIC = 4365.927

Model AIC = 152.785Saturated AIC = 156.000

Independence CAIC = 4426.003Model CAIC = 363.052

Saturated CAIC = 546.496




Critical N (CN) = 674.570


Goodness of Fit Index (GFI) = 0.995Adjusted Goodness of Fit Index (AGFI) = 0.989Parsimony Goodness of Fit Index (PGFI) = 0.459

183 Correlated Uniqueness MTMM model. Pearson correlationsobserved variablesGEN_100 HOU_100 FIN_100 SOC_100GEN_5 HOU_5 FIN_5 SOC_5GEN_10 HOU_10 FIN_10 SOC_10correlation matrix from file: sat.pemasymptotic covariances from file: sat.peasample size: 406latent variablesGENERAL HOUSING FINANCE SOCIALrelationshipsGEN_100 = GENERALHOU_100 = HOUSINGFIN_100 = FINANCESOC_100 = SOCIALGEN_5 = GENERALHOU_5 = HOUSINGFIN_5 = FINANCESOC_5 = SOCIALGEN_10 = GENERALHOU_10 = HOUSINGFIN_10 = FINANCESOC_10 = SOCIALlet the errors of GEN_100 and HOU_100 correlatelet the errors of GEN_5 and HOU_5 correlatelet the errors of GEN_10 and HOU_10 correlatelet the errors of GEN_100 and FIN_100 correlatelet the errors of GEN_5 and FIN_5 correlatelet the errors of GEN_10 and FIN_10 correlatelet the errors of GEN_100 and SOC_100 correlatelet the errors of GEN_5 and SOC_5 correlatelet the errors of GEN_10 and SOC_10 correlatelet the errors of HOU_100 and FIN_100 correlatelet the errors of HOU_5 and FIN_5 correlatelet the errors of HOU_10 and FIN_10 correlatelet the errors of HOU_100 and SOC_100 correlatelet the errors of HOU_5 and SOC_5 correlatelet the errors of HOU_10 and SOC_10 correlatelet the errors of FIN_100 and SOC_100 correlatelet the errors of FIN_5 and SOC_5 correlatelet the errors of FIN_10 and SOC_10 correlateoptions me=uls wp nd=3 AD=OFF IT=300end of problem

184 Correlated Uniequeness MTMM model. Pearson correlationsNumber of Iterations = 5LISREL Estimates (Unweighted Least Squares)


GEN_100 = 0.785*GENERAL, Errorvar.= 0.383 , R² = 0.617(0.0412) (0.0791)19.075 4.840

HOU_100 = 0.791*HOUSING, Errorvar.= 0.374 , R² = 0.626(0.0343) (0.0727)23.083 5.144

FIN_100 = 0.858*FINANCE , Errorvar.= 0.265 , R² = 0.735(0.0239) (0.0646)35.806 4.098

SOC_100 = 0.872*SOCIAL , Errorvar.= 0.239 , R² = 0.761(0.0435) (0.0889)20.043 2.685





185 GEN_10 = 0.850*GENERAL, Errorvar.= 0.277 , R² = 0.723

(0.0394) (0.0862)21.562 3.215




Error Covariance for HOU_100 and GEN_100 = 0.167(0.0399)

4.185

Error Covariance for FIN_100 and GEN_100 = 0.152(0.0316)

4.823

Error Covariance for FIN_100 and HOU_100 = 0.139(0.0325)

4.269

Error Covariance for SOC_100 and GEN_100 = 0.0430(0.0380)

1.129

Error Covariance for SOC_100 and HOU_100 = 0.0592(0.0389)

1.523

Error Covariance for SOC_100 and FIN_100 = 0.0984(0.0318)

3.090


1.946

186 Error Covariance for FIN_5 and GEN_5 = 0.00958

(0.0267)0.360

Error Covariance for FIN_5 and HOU_5 = -0.006(0.0287)-0.219


4.076


0.588


0.371


3.369


1.598


2.518


2.701


3.579


2.886

187 Correlation Matrix of Independent Variables

GENERAL HOUSING FINANCE SOCIAL-------- -------- -------- --------

GENERAL 1.000

HOUSING 0.460 1.000(0.061)

7.586

FINANCE 0.462 0.429 1.000(0.050) (0.051)

9.272 8.356

SOCIAL 0.433 0.245 0.166 1.000(0.063) (0.058) (0.058)

6.860 4.199 2.876










Relative Fit Index (RFI) = 0.983Root Mean Square Residual (RMR) = 0.0329

Standardized RMR = 0.0329

188 Correlated Uniequeness MTMM model. Polychoric correlationsobserved variablesGEN_100 HOU_100 FIN_100 SOC_100GEN_5 HOU_5 FIN_5 SOC_5GEN_10 HOU_10 FIN_10 SOC_10correlation matrix from file: sat.pomasymptotic covariances from file: sat.pobsample size: 406latent variablesGENERAL HOUSING FINANCE SOCIALrelationshipsGEN_100 = GENERALHOU_100 = HOUSINGFIN_100 = FINANCESOC_100 = SOCIALGEN_5 = GENERALHOU_5 = HOUSINGFIN_5 = FINANCESOC_5 = SOCIALGEN_10 = GENERALHOU_10 = HOUSINGFIN_10 = FINANCESOC_10 = SOCIALlet the errors of GEN_100 and HOU_100 correlatelet the errors of GEN_5 and HOU_5 correlatelet the errors of GEN_10 and HOU_10 correlatelet the errors of GEN_100 and FIN_100 correlatelet the errors of GEN_5 and FIN_5 correlatelet the errors of GEN_10 and FIN_10 correlatelet the errors of GEN_100 and SOC_100 correlatelet the errors of GEN_5 and SOC_5 correlatelet the errors of GEN_10 and SOC_10 correlatelet the errors of HOU_100 and FIN_100 correlatelet the errors of HOU_5 and FIN_5 correlatelet the errors of HOU_10 and FIN_10 correlatelet the errors of HOU_100 and SOC_100 correlatelet the errors of HOU_5 and SOC_5 correlatelet the errors of HOU_10 and SOC_10 correlatelet the errors of FIN_100 and SOC_100 correlatelet the errors of FIN_5 and SOC_5 correlatelet the errors of FIN_10 and SOC_10 correlateoptions me=uls wp nd=3 AD=OFF IT=300end of problem

189 Sample Size = 406

Correlated Uniequeness MTMM model. Polychoric correlations

Correlation Matrix

GEN_100 HOU_100 FIN_100 SOC_100 GEN_5 HOU_5 FIN_5 SOC_5 GEN_10 HOU_10-------- -------- -------- -------- -------- -------- -------- -------- -------- --------

GEN_100 1.000HOU_100 0.453 1.000FIN_100 0.464 0.430 1.000SOC_100 0.340 0.229 0.223 1.000

GEN_5 0.602 0.212 0.233 0.352 1.000HOU_5 0.274 0.700 0.286 0.219 0.387 1.000FIN_5 0.366 0.326 0.793 0.207 0.332 0.323 1.000SOC_5 0.189 0.036 0.011 0.677 0.449 0.198 0.140 1.000

GEN_10 0.639 0.334 0.346 0.309 0.668 0.339 0.343 0.310 1.000HOU_10 0.370 0.733 0.386 0.280 0.326 0.783 0.330 0.176 0.518 1.000FIN_10 0.399 0.360 0.788 0.166 0.247 0.289 0.826 0.047 0.420 0.462SOC_10 0.276 0.121 0.121 0.657 0.308 0.172 0.151 0.745 0.445 0.339

Correlation Matrix

FIN_10 SOC_10-------- --------

FIN_10 1.000SOC_10 0.237 1.000

190 Correlated Uniequeness MTMM model. Polychoric correlationsNumber of Iterations = 5LISREL Estimates (Unweighted Least Squares)




FIN_100 = 0.852*FINANCE, Errorvar.= 0.274 , R² = 0.726(0.0215) (0.0620)39.581 4.424

SOC_100 = 0.862*SOCIAL, Errorvar.= 0.257 , R² = 0.743(0.0311) (0.0716)27.753 3.590





191 GEN_10 = 0.860*GENERAL, Errorvar.= 0.260 , R² = 0.740

(0.0274) (0.0699)31.365 3.714





4.275


4.815


3.893


1.126


1.549


3.001


2.212

192 Error Covariance for FIN_5 and GEN_5 = 0.0189

(0.0442)0.428

Error Covariance for FIN_5 and HOU_5 = -0.007(0.0473)-0.147


3.881


0.771


0.513


3.708


1.873


2.925


2.767


3.560


2.801

193 Correlation Matrix of Independent Variables

GENERAL HOUSING FINANCE SOCIAL-------- -------- -------- --------

GENERAL 1.000

HOUSING 0.455 1.000(0.031)14.678

FINANCE 0.453 0.427 1.000(0.027) (0.028)16.550 15.293

SOCIAL 0.439 0.241 0.162 1.000(0.030) (0.029) (0.030)14.455 8.399 5.453












194 Only Catalonia children and parents. Listwise deletionobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imasample size: 1313raw data from file catalist.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem


Covariance Matrix

val_int val_htec val_prof val_sen val_symp val_mon val_pow p_v_int p_v_htec p_v_prof-------- -------- -------- -------- -------- -------- -------- -------- -------- --------

val_int 0.785val_htec 0.461 0.806val_prof 0.403 0.382 0.954val_sen 0.298 0.260 0.344 1.010

val_symp 0.298 0.236 0.320 0.443 0.724val_mon 0.158 0.259 0.291 0.218 0.098 1.614val_pow 0.166 0.282 0.294 0.186 0.099 1.255 1.576p_v_int 0.094 0.027 0.066 0.041 0.031 0.014 -0.009 0.504

p_v_htec 0.059 0.081 0.052 0.030 0.018 0.033 0.028 0.257 0.497p_v_prof 0.030 0.024 0.069 0.081 0.021 0.095 0.075 0.155 0.097 0.558p_v_sen 0.044 0.028 0.049 0.108 0.036 -0.011 0.003 0.098 0.086 0.228

p_v_symp 0.032 0.022 0.030 0.065 0.049 -0.003 0.000 0.145 0.118 0.176p_v_mon 0.013 0.006 0.063 0.013 -0.008 0.243 0.188 0.101 0.142 0.155p_v_pow 0.023 0.023 0.077 0.021 0.006 0.243 0.246 0.128 0.179 0.148

Covariance Matrix

p_v_sen p_v_symp p_v_mon p_v_pow-------- -------- -------- --------

p_v_sen 0.514p_v_symp 0.263 0.458p_v_mon 0.055 0.089 1.238p_v_pow 0.059 0.110 1.048 1.308

Means


3.770 3.641 3.893 3.755 4.182 2.619 2.604 4.200 3.938 4.418

Means


4.389 4.335 2.519 2.475

196 Only Catalonia children and parents. Listwise deletionLISREL Estimates (Maximum Likelihood)

val_int = 1.000*abili, Errorvar.= 0.451 , R² = 0.426(0.0260)17.332

val_htec = 0.0956 + 0.941*abili, Errorvar.= 0.511 , R² = 0.367(0.180) (0.0474) (0.0272)0.531 19.838 18.765

val_prof = - 0.667 + 1.210*abili, Errorvar.= 0.465 , R² = 0.513(0.285) (0.0752) (0.0311)-2.341 16.084 14.943

val_sen = 1.000*perso, Errorvar.= 0.526 , R² = 0.479(0.0339)15.530

val_symp = 0.744 + 0.916*perso, Errorvar.= 0.318 , R² = 0.561(0.215) (0.0569) (0.0258)3.461 16.081 12.344

val_mon = 1.000*mater, Errorvar.= 0.333 , R² = 0.793(0.0726)4.593

val_pow = 0.0380 + 0.980*mater, Errorvar.= 0.347 , R² = 0.780(0.153) (0.0577) (0.0699)0.249 16.981 4.969

197 p_v_int = 1.000*p_abili, Errorvar.= 0.409 , R² = 0.189

(0.0182)22.425

p_v_htec = 0.450 + 0.831*p_abili, Errorvar.= 0.431 , R² = 0.132(0.301) (0.0714) (0.0183)1.498 11.628 23.524

p_v_prof = - 1.892 + 1.503*p_abili, Errorvar.= 0.343 , R² = 0.385(0.561) (0.133) (0.0230)-3.376 11.274 14.931

p_v_sen = 1.000*p_perso, Errorvar.= 0.230 , R² = 0.553(0.0181)12.697

p_v_symp = 0.273 + 0.925*p_perso, Errorvar.= 0.215 , R² = 0.531(0.251) (0.0571) (0.0159)1.087 16.198 13.530

p_v_mon = 1.000*p_mater, Errorvar.= 0.269 , R² = 0.783(0.0568)4.733

p_v_pow = - 0.251 + 1.082*p_mater, Errorvar.= 0.173 , R² = 0.868(0.165) (0.0651) (0.0658)-1.517 16.618 2.631

Error Covariance for val_htec and val_int = 0.146(0.0214)

6.838

Error Covariance for p_v_htec and p_v_int = 0.178(0.0144)12.348


mater abili perso p_mater p_abili p_perso-------- -------- -------- -------- -------- --------

mater 1.281(0.094)13.573

abili 0.230 0.334(0.027) (0.032)

8.688 10.584

perso 0.148 0.292 0.484(0.029) (0.024) (0.043)

5.109 12.399 11.233

p_mater 0.225 0.037 0.008 0.969(0.037) (0.020) (0.024) (0.073)

6.149 1.857 0.315 13.249

p_abili 0.044 0.039 0.032 0.108 0.095(0.015) (0.009) (0.011) (0.015) (0.014)

2.886 4.225 3.018 6.991 6.561

p_perso -0.003 0.034 0.066 0.079 0.136 0.284(0.021) (0.012) (0.015) (0.019) (0.013) (0.024)-0.141 2.753 4.338 4.230 10.434 11.903

Mean Vector of Independent Variables


2.619 3.770 3.755 2.519 4.200 4.389(0.035) (0.024) (0.028) (0.031) (0.020) (0.020)74.676 154.079 135.333 82.005 214.190 221.793


Minimum Fit Function Chi-Square = 234.762 (P = 0.0)Root Mean Square Error of Approximation (RMSEA) = 0.047690 Percent Confidence Interval for RMSEA = (0.0413 ; 0.0540)

P-Value for Test of Close Fit (RMSEA < 0.05) = 0.726

199 Only Catalonia children and parents. Maximum Likelihood with missing dataobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imamissing value code 0sample size: 3118raw data from file cataloni.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem


Only Catalonia children and parents. Maximum Likelihood with missing dataCovariance Matrix



val_symp 0.336 0.291 0.318 0.450 0.767val_mon 0.183 0.281 0.284 0.224 0.096 1.735val_pow 0.181 0.308 0.281 0.198 0.101 1.356 1.708p_v_int 0.106 0.037 0.065 0.047 0.036 0.014 -0.007 0.522

p_v_htec 0.069 0.097 0.059 0.032 0.021 0.034 0.035 0.275 0.510p_v_prof 0.021 0.025 0.066 0.062 0.016 0.099 0.086 0.155 0.105 0.565p_v_sen 0.053 0.038 0.048 0.109 0.046 0.008 0.030 0.109 0.100 0.225

p_v_symp 0.032 0.031 0.030 0.058 0.049 -0.007 0.007 0.154 0.127 0.193p_v_mon 0.024 0.014 0.050 0.016 -0.013 0.284 0.225 0.113 0.162 0.152p_v_pow 0.040 0.040 0.067 0.023 -0.003 0.276 0.284 0.143 0.198 0.147

Covariance Matrix



Means


3.759 3.672 3.885 3.732 4.167 2.678 2.664 4.188 3.934 4.418

Means


4.377 4.308 2.531 2.485

201 Only Catalonia children and parents. Maximum Likelihood with missing data


val_htec = - 0.0565 + 0.992*abili, Errorvar.= 0.503 , R² = 0.418(0.123) (0.0324) (0.0193)-0.460 30.599 26.027





val_pow = - 0.0123 + 0.999*mater, Errorvar.= 0.353 , R² = 0.793(0.112) (0.0416) (0.0546)-0.109 24.034 6.466


(0.0172)24.418




p_v_symp = - 0.0649 + 0.999*p_perso, Errorvar.= 0.220 , R² = 0.551(0.252) (0.0574) (0.0160)-0.258 17.411 13.751




16.234

Error Covariance for p_v_htec and p_v_int = 0.185(0.00683)

27.068



mater 1.357(0.069)19.661

abili 0.247 0.368(0.019) (0.023)13.112 15.913

perso 0.146 0.324 0.477(0.020) (0.017) (0.029)

7.464 19.462 16.718

p_mater 0.257 0.040 0.005 0.976(0.034) (0.019) (0.022) (0.065)

7.585 2.051 0.234 15.050

p_abili 0.048 0.044 0.030 0.121 0.102(0.015) (0.009) (0.010) (0.015) (0.014)

3.240 4.757 2.918 8.141 7.300

p_perso 0.008 0.037 0.062 0.086 0.143 0.271(0.019) (0.012) (0.013) (0.017) (0.012) (0.022)

0.424 3.127 4.617 5.072 11.718 12.431



2.678 3.759 3.733 2.533 4.192 4.377(0.024) (0.017) (0.019) (0.028) (0.018) (0.018)113.162 225.494 201.036 90.951 232.456 237.568

Global Goodness of Fit Statistics, Missing Data CaseDegrees of Freedom = 60

Full Information ML Chi-Square = 300.854 (P = 0.0)Root Mean Square Error of Approximation (RMSEA) = 0.0359


204 Only Catalonia children and parents. Maximum Likelihood with missing data

Modification Indices and Expected Change


val_int mater 46.4 -0.10val_htec mater 31.2 0.08val_htec p_perso 91.5 -2.20val_sen mater 29.2 0.10val_symp mater 34.8 -0.11val_symp p_mater 29.1 -0.30


val_symp val_int 21.8 0.05val_mon val_sen 14.2 0.05val_pow val_int 8.8 -0.03val_pow val_htec 19.6 0.05p_v_int val_int 66.5 0.06p_v_int val_htec 52.6 -0.06p_v_int val_pow 13.1 -0.03p_v_htec val_int 9.5 -0.02p_v_htec val_htec 67.0 0.06p_v_prof val_int 15.9 -0.04p_v_prof val_sen 8.1 0.03p_v_prof val_mon 9.1 0.03p_v_prof p_v_int 107.6 0.19p_v_prof p_v_htec 43.8 -0.07p_v_sen val_sen 23.5 0.05p_v_sen val_symp 12.7 -0.03p_v_sen p_v_int 42.7 -0.06p_v_symp val_sen 8.6 -0.03p_v_symp p_v_int 10.8 0.03p_v_mon val_mon 43.3 0.06p_v_mon val_pow 42.0 -0.06p_v_pow val_mon 35.1 -0.06p_v_pow val_pow 33.9 0.06p_v_pow p_v_htec 21.1 0.03p_v_pow p_v_prof 33.2 -0.07p_v_pow p_v_sen 10.0 -0.02p_v_pow p_v_symp 8.4 0.02


LAMBDA-X


val_int - - 0.653 - - - - - - - -val_htec - - 0.647 - - - - - - - -val_prof - - 0.658 - - - - - - - -val_sen - - - - 0.669 - - - - - -

val_symp - - - - 0.744 - - - - - -val_mon 0.884 - - - - - - - - - -val_pow 0.891 - - - - - - - - - -p_v_int - - - - - - - - 0.442 - -

p_v_htec - - - - - - - - 0.398 - -p_v_prof - - - - - - - - 0.589 - -p_v_sen - - - - - - - - - - 0.707

p_v_symp - - - - - - - - - - 0.742p_v_mon - - - - - - 0.882 - - - -p_v_pow - - - - - - 0.932 - - - -

PHI


mater 1.000abili 0.350 1.000perso 0.182 0.774 1.000

p_mater 0.223 0.066 0.007 1.000p_abili 0.130 0.227 0.134 0.384 1.000p_perso 0.014 0.117 0.173 0.167 0.865 1.000

206 THETA-DELTA


val_int 0.573val_htec 0.146 0.582val_prof - - - - 0.567val_sen - - - - - - 0.552

val_symp - - - - - - - - 0.446val_mon - - - - - - - - - - 0.218val_pow - - - - - - - - - - - - 0.207p_v_int - - - - - - - - - - - - - - 0.805

p_v_htec - - - - - - - - - - - - - - 0.359 0.842p_v_prof - - - - - - - - - - - - - - - - - - 0.653p_v_sen - - - - - - - - - - - - - - - - - - - -

p_v_symp - - - - - - - - - - - - - - - - - - - -p_v_mon - - - - - - - - - - - - - - - - - - - -p_v_pow - - - - - - - - - - - - - - - - - - - -

THETA-DELTA


p_v_sen 0.501p_v_symp - - 0.449p_v_mon - - - - 0.222p_v_pow - - - - - - 0.131

Time used: 2.764 Seconds

207 Only Brazil children and parents. Maximum Likelihood with missing dataobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imamissing value code 0sample size: 860raw data from file brazil.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem


Only Brazil children and parents. Maximum Likelihood with missing data

Covariance Matrix



val_symp 0.280 0.264 0.269 0.529 0.947val_mon 0.350 0.363 0.414 0.408 0.325 2.012val_pow 0.392 0.325 0.406 0.355 0.232 1.444 1.942p_v_int 0.078 0.054 0.119 0.087 0.063 0.096 0.034 0.442

p_v_htec 0.014 0.096 0.118 0.059 0.080 0.110 -0.006 0.287 0.777p_v_prof 0.090 0.081 0.136 0.123 0.145 0.104 0.084 0.182 0.268 0.530p_v_sen 0.049 0.040 0.029 0.232 0.105 0.032 0.025 0.130 0.191 0.206

p_v_symp 0.051 0.065 0.020 0.103 0.075 0.001 -0.067 0.143 0.170 0.186p_v_mon 0.076 0.046 0.121 -0.040 0.095 0.516 0.313 0.170 0.261 0.262p_v_pow 0.105 0.108 0.087 -0.038 0.055 0.397 0.280 0.184 0.394 0.266

Covariance Matrix



Means


4.057 4.061 4.370 3.821 4.207 3.218 3.137 4.465 4.102 4.424

Means


4.269 4.459 2.897 2.925

209

Only Brazil children and parents. Maximum Likelihood with missing data

Number of Iterations = 12

LISREL Estimates (Maximum Likelihood)










(0.0355)8.856

p_v_htec = - 2.426 + 1.459*p_abili, Errorvar.= 0.511 , R² = 0.343(0.958) (0.214) (0.0617)-2.532 6.824 8.280



p_v_symp = 0.781 + 0.861*p_perso, Errorvar.= 0.252 , R² = 0.522(0.597) (0.139) (0.0469)1.309 6.176 5.359




9.081


5.605



mater 1.557(0.133)11.716

abili 0.381 0.274(0.041) (0.038)

9.251 7.247

perso 0.370 0.290 0.606(0.051) (0.031) (0.070)

7.305 9.283 8.667

p_mater 0.383 0.073 -0.004 1.312(0.113) (0.054) (0.075) (0.216)

3.390 1.352 -0.054 6.081

p_abili 0.059 0.066 0.075 0.189 0.125(0.039) (0.021) (0.029) (0.046) (0.035)

1.484 3.148 2.613 4.110 3.590

p_perso -0.011 0.043 0.146 0.159 0.144 0.371(0.065) (0.032) (0.048) (0.063) (0.032) (0.082)-0.174 1.313 3.048 2.527 4.553 4.534



3.216 4.057 3.821 2.900 4.472 4.264(0.050) (0.031) (0.038) (0.087) (0.042) (0.056)64.482 132.021 100.632 33.392 106.375 76.150





LAMBDA-Xmater abili perso p_mater p_abili p_perso

-------- -------- -------- -------- -------- --------val_int - - 0.590 - - - - - - - -

val_htec - - 0.553 - - - - - - - -val_prof - - 0.642 - - - - - - - -val_sen - - - - 0.720 - - - - - -

val_symp - - - - 0.700 - - - - - -val_mon 0.879 - - - - - - - - - -val_pow 0.833 - - - - - - - - - -p_v_int - - - - - - - - 0.534 - -

p_v_htec - - - - - - - - 0.586 - -p_v_prof - - - - - - - - 0.704 - -p_v_sen - - - - - - - - - - 0.693

p_v_symp - - - - - - - - - - 0.723p_v_mon - - - - - - 0.841 - - - -p_v_pow - - - - - - 0.900 - - - -

PHImater abili perso p_mater p_abili p_perso

-------- -------- -------- -------- -------- --------mater 1.000abili 0.584 1.000perso 0.381 0.711 1.000

p_mater 0.268 0.121 -0.005 1.000p_abili 0.133 0.358 0.272 0.466 1.000p_perso -0.015 0.134 0.307 0.228 0.666 1.000

213 THETA-DELTA



val_symp - - - - - - - - 0.510val_mon - - - - - - - - - - 0.228val_pow - - - - - - - - - - - - 0.306p_v_int - - - - - - - - - - - - - - 0.715

p_v_htec - - - - - - - - - - - - - - 0.174 0.657p_v_prof - - - - - - - - - - - - - - - - - - 0.504p_v_sen - - - - - - - - - - - - - - - - - - - -

p_v_symp - - - - - - - - - - - - - - - - - - - -p_v_mon - - - - - - - - - - - - - - - - - - - -p_v_pow - - - - - - - - - - - - - - - - - - - -

THETA-DELTA


p_v_sen 0.520p_v_symp - - 0.478p_v_mon - - - - 0.293p_v_pow - - - - - - 0.189

214 Multiple group 1. All free. ml. group = Cataloniaobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imamissing value code 0sample size: 3118raw data from file cataloni.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlategroup = Brazilmissing value code 0sample size: 860raw data from file brazil.dat

215 relationships:val_pow=const materval_htec=const abilival_prof=const abilival_symp=const persop_v_pow=const p_materp_v_htec=const p_abilip_v_prof=const p_abilip_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constset the error variances of val_mon freeset the error variances of val_pow freeset the error variances of val_int freeset the error variances of val_htec freeset the error variances of val_prof freeset the error variances of val_sen freeset the error variances of val_symp freeset the error variances of p_v_mon freeset the error variances of p_v_pow freeset the error variances of p_v_int freeset the error variances of p_v_htec freeset the error variances of p_v_prof freeset the error variances of p_v_sen freeset the error variances of p_v_symp freeset the variances of mater freeset the variances of abili freeset the variances of perso freeset the variances of p_mater freeset the variances of p_abili freeset the variances of p_perso freeset the covariances of mater abili perso p_mater p_abili p_perso freeset the covariances of abili perso p_mater p_abili p_perso freeset the covariances of perso p_mater p_abili p_perso freeset the covariances of p_mater p_abili p_perso freeset the covariances of p_abili p_perso freelet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem

216 SOME ESTIMATES OF THE CATALONIA GROUP




SOME ESTIMATES OF THE BRAZIL GROUP







217 Multiple group 2. Scale invariance. ml group = Cataloniaobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imamissing value code 0sample size: 3118raw data from file cataloni.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlate

218 group = Brazilmissing value code 0sample size: 860raw data from file brazil.datrelationships:mater = constabili = constperso = constp_mater = constp_abili = constp_perso = constset the error variances of val_mon freeset the error variances of val_pow freeset the error variances of val_int freeset the error variances of val_htec freeset the error variances of val_prof freeset the error variances of val_sen freeset the error variances of val_symp freeset the error variances of p_v_mon freeset the error variances of p_v_pow freeset the error variances of p_v_int freeset the error variances of p_v_htec freeset the error variances of p_v_prof freeset the error variances of p_v_sen freeset the error variances of p_v_symp freeset the variances of mater freeset the variances of abili freeset the variances of perso freeset the variances of p_mater freeset the variances of p_abili freeset the variances of p_perso freeset the covariances of mater abili perso p_mater p_abili p_perso freeset the covariances of abili perso p_mater p_abili p_perso freeset the covariances of perso p_mater p_abili p_perso freeset the covariances of p_mater p_abili p_perso freeset the covariances of p_abili p_perso freelet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem

219 Multiple group 2. Scale invariance. ml group = CataloniaModification Indices and Expected Change


val_int abili 13.3 1.03 IN GROUP 1p_v_prof p_abili 32.7 1.29 IN GROUP 1p_v_symp p_perso 26.9 1.02 IN GROUP 1


val_int CONST 13.7 0.12 IN GROUP 1p_v_prof CONST 32.4 -1.18 IN GROUP 1p_v_symp CONST 28.7 -0.04 IN GROUP 1




220 Multiple group 3. Partial Scale invariance. ml group = Cataloniaobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imamissing value code 0sample size: 3118raw data from file cataloni.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlate

221 group = Brazilmissing value code 0sample size: 860raw data from file brazil.datrelationships:p_v_prof=const p_abilimater = constabili = constperso = constp_mater = constp_abili = constp_perso = constset the error variances of val_mon freeset the error variances of val_pow freeset the error variances of val_int freeset the error variances of val_htec freeset the error variances of val_prof freeset the error variances of val_sen freeset the error variances of val_symp freeset the error variances of p_v_mon freeset the error variances of p_v_pow freeset the error variances of p_v_int freeset the error variances of p_v_htec freeset the error variances of p_v_prof freeset the error variances of p_v_sen freeset the error variances of p_v_symp freeset the variances of mater freeset the variances of abili freeset the variances of perso freeset the variances of p_mater freeset the variances of p_abili freeset the variances of p_perso freeset the covariances of mater abili perso p_mater p_abili p_perso freeset the covariances of abili perso p_mater p_abili p_perso freeset the covariances of perso p_mater p_abili p_perso freeset the covariances of p_mater p_abili p_perso freeset the covariances of p_abili p_perso freelet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem

222 Multiple group 3. Partial Scale invariance. ml group = CataloniaNumber of Iterations = 10LISREL Estimates (Maximum Likelihood)










(0.0171)24.648








18.338


27.069



mater 1.412(0.060)23.351

abili 0.245 0.346(0.018) (0.020)13.549 16.975

perso 0.152 0.318 0.486(0.020) (0.016) (0.027)

7.618 20.314 17.732

p_mater 0.264 0.039 0.005 0.984(0.034) (0.019) (0.022) (0.060)

7.739 2.064 0.231 16.345

p_abili 0.049 0.042 0.029 0.120 0.097(0.015) (0.009) (0.010) (0.015) (0.013)

3.312 4.798 2.926 8.284 7.292

p_perso 0.007 0.036 0.063 0.086 0.141 0.270(0.020) (0.011) (0.014) (0.017) (0.012) (0.021)

0.355 3.115 4.641 5.099 11.824 12.767



2.681 3.742 3.737 2.531 4.196 4.363(0.024) (0.016) (0.018) (0.028) (0.018) (0.018)113.716 234.098 207.222 91.304 235.213 240.069

225 group = BrazilNumber of Iterations = 10LISREL Estimates (Maximum Likelihood)









p_v_int = 1.000*p_abili, Errorvar.= 0.300 , R² = 0.354(0.0363)8.281

226 p_v_htec = 0.00528 + 0.934*p_abili, Errorvar.= 0.572 , R² = 0.201

(0.248) (0.0586) (0.0587)0.0213 15.955 9.742






Error Covariance for val_htec and val_int = 0.131(0.0131)10.015


6.016



mater 1.511(0.102)14.836

abili 0.365 0.255(0.035) (0.025)10.551 10.052

perso 0.350 0.272 0.571(0.046) (0.025) (0.050)

7.581 11.114 11.490

p_mater 0.355 0.068 -0.002 1.244(0.105) (0.050) (0.071) (0.147)

3.382 1.345 -0.027 8.466

p_abili 0.072 0.080 0.091 0.205 0.165(0.045) (0.023) (0.032) (0.046) (0.038)

1.604 3.494 2.884 4.408 4.317

p_perso -0.015 0.037 0.121 0.145 0.152 0.299(0.059) (0.028) (0.041) (0.054) (0.029) (0.049)-0.262 1.299 2.972 2.711 5.283 6.068



3.198 4.122 3.802 2.927 4.449 4.421(0.048) (0.025) (0.033) (0.077) (0.041) (0.044)67.143 162.707 114.015 37.868 108.277 100.001




228 Multiple group 4. Partial Sc. inv. Constrained factor means. ml. g = Cataloniaobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imamissing value code 0sample size: 3118raw data from file cataloni.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = constabili = constperso = constp_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlate

229 group = Brazilmissing value code 0sample size: 860raw data from file brazil.datrelationships:p_v_prof=const p_abiliset the error variances of val_mon freeset the error variances of val_pow freeset the error variances of val_int freeset the error variances of val_htec freeset the error variances of val_prof freeset the error variances of val_sen freeset the error variances of val_symp freeset the error variances of p_v_mon freeset the error variances of p_v_pow freeset the error variances of p_v_int freeset the error variances of p_v_htec freeset the error variances of p_v_prof freeset the error variances of p_v_sen freeset the error variances of p_v_symp freeset the variances of mater freeset the variances of abili freeset the variances of perso freeset the variances of p_mater freeset the variances of p_abili freeset the variances of p_perso freeset the covariances of mater abili perso p_mater p_abili p_perso freeset the covariances of abili perso p_mater p_abili p_perso freeset the covariances of perso p_mater p_abili p_perso freeset the covariances of p_mater p_abili p_perso freeset the covariances of p_abili p_perso freelet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem

230

Multiple group 4. Partial Sc. inv. Constrained factor means. ml. g = Catalonia


mater CONST 13.1 2.72 IN GROUP 1abili CONST 113.9 3.76 IN GROUP 1perso CONST 20.3 3.78 IN GROUP 1p_abili CONST 21.8 4.26 IN GROUP 1




group = Brazil


mater CONST 11.3 2.88 IN GROUP 2abili CONST 95.0 4.00 IN GROUP 2perso CONST 21.7 3.64 IN GROUP 2p_mater CONST 13.4 2.71 IN GROUP 2p_abili CONST -109.9 3.92 IN GROUP 2

231 Multiple group 5. Partial Scale invariance. Regressions. ml. group = Cataloniaobserved variables:val_int val_htec val_soci val_knco val_prof val_fam val_sen val_sympval_mon val_pow val_knwo val_imaglob_satp_v_int p_v_htec p_v_soci p_v_knco p_v_prof p_v_fam p_v_sen p_v_sympp_v_mon p_v_pow p_v_knwo p_v_imamissing value code 0sample size: 3118raw data from file cataloni.datlatent variablesmater abili perso p_mater p_abili p_persorelationships:val_mon=1*materval_pow=const materval_int=1*abilival_htec=const abilival_prof=const abilival_sen=1*persoval_symp=const persop_v_mon=1*p_materp_v_pow=const p_materp_v_int=1*p_abilip_v_htec=const p_abilip_v_prof=const p_abilip_v_sen=1*p_persop_v_symp=const p_persomater = const p_materabili = const p_abiliperso = const p_persop_mater = constp_abili = constp_perso = constlet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateset the covariances of mater abili perso freeset the covariances of abili perso free

232 group = Brazilmissing value code 0sample size: 860raw data from file brazil.datrelationships:p_v_prof=const p_abilimater = const p_materabili = const p_abiliperso = const p_persop_mater = constp_abili = constp_perso = constset the error variances of val_mon freeset the error variances of val_pow freeset the error variances of val_int freeset the error variances of val_htec freeset the error variances of val_prof freeset the error variances of val_sen freeset the error variances of val_symp freeset the error variances of p_v_mon freeset the error variances of p_v_pow freeset the error variances of p_v_int freeset the error variances of p_v_htec freeset the error variances of p_v_prof freeset the error variances of p_v_sen freeset the error variances of p_v_symp freeset the variances of mater freeset the variances of abili freeset the variances of perso freeset the variances of p_mater freeset the variances of p_abili freeset the variances of p_perso freeset the covariances of mater abili perso freeset the covariances of abili perso freeset the covariances of p_mater p_abili p_perso freeset the covariances of p_abili p_perso freelet the errors of val_htec and val_int correlatelet the errors of p_v_htec and p_v_int correlateoptions: mi nd=3 wp ad=off scend of problem

233 Multiple group 5. Partial Scale invariance. Regressions. ml. group = CataloniaLISREL Estimates (Maximum Likelihood)












234 p_v_sen = 1.000*p_perso, Errorvar.= 0.268 , R² = 0.505

(0.0168)15.999





18.105


26.907


mater = 2.006 + 0.268*p_mater, Errorvar.= 1.339 , R² = 0.0508(0.0837) (0.0318) (0.0596)23.975 8.417 22.477

abili = 2.308 + 0.343*p_abili, Errorvar.= 0.336 , R² = 0.0345(0.287) (0.0685) (0.0205)8.032 4.999 16.355

perso = 2.667 + 0.245*p_perso, Errorvar.= 0.471 , R² = 0.0338(0.205) (0.0469) (0.0271)13.002 5.238 17.364

235 Error Covariance for abili and mater = 0.235

(0.0179)13.116

Error Covariance for perso and mater = 0.149(0.0197)

7.574

Error Covariance for perso and abili = 0.308(0.0156)19.722

Covariance Matrix of Independent Variables

p_mater p_abili p_perso-------- -------- --------

p_mater 1.000(0.058)17.115

p_abili 0.122 0.102(0.015) (0.014)

8.356 7.524

p_perso 0.086 0.140 0.274(0.017) (0.012) (0.021)

5.060 11.847 12.937



2.515 4.184 4.357(0.027) (0.018) (0.018)92.195 236.044 240.206

236 Multiple group 5. Partial Scale invariance. Regressions. ml. group = CataloniaWithin Group Completely Standardized Solution

LAMBDA-Y

mater abili perso-------- -------- --------

val_int - - 0.638 - -val_htec - - 0.642 - -val_prof - - 0.670 - -val_sen - - - - 0.675

val_symp - - - - 0.739val_mon 0.900 - - - -val_pow 0.875 - - - -

LAMBDA-X


p_v_int - - 0.443 - -p_v_htec - - 0.406 - -p_v_prof - - 0.601 - -p_v_sen - - - - 0.710

p_v_symp - - - - 0.741p_v_mon 0.892 - - - -p_v_pow 0.922 - - - -

GAMMA


mater 0.225 - - - -abili - - 0.186 - -perso - - - - 0.184

237 THETA-EPS

val_int val_htec val_prof val_sen val_symp val_mon val_pow-------- -------- -------- -------- -------- -------- --------


val_symp - - - - - - - - 0.454val_mon - - - - - - - - - - 0.190val_pow - - - - - - - - - - - - 0.235

THETA-DELTA

p_v_int p_v_htec p_v_prof p_v_sen p_v_symp p_v_mon p_v_pow-------- -------- -------- -------- -------- -------- --------

p_v_int 0.804p_v_htec 0.355 0.835p_v_prof - - - - 0.638p_v_sen - - - - - - 0.495

p_v_symp - - - - - - - - 0.451p_v_mon - - - - - - - - - - 0.205p_v_pow - - - - - - - - - - - - 0.150

238 group = Brazil












239 p_v_sen = 1.000*p_perso, Errorvar.= 0.498 , R² = 0.393

(0.0581)8.566





9.915


5.738


mater = 2.242 + 0.312*p_mater, Errorvar.= 1.386 , R² = 0.0842(0.231) (0.0725) (0.108)9.717 4.302 12.802

abili = 2.397 + 0.381*p_abili, Errorvar.= 0.230 , R² = 0.101(0.510) (0.112) (0.0272)4.702 3.402 8.446

perso = 1.830 + 0.435*p_perso, Errorvar.= 0.513 , R² = 0.106(0.536) (0.117) (0.0543)3.414 3.710 9.454

240 Error Covariance for abili and mater = 0.341

(0.0350)9.733

Error Covariance for perso and mater = 0.337(0.0462)

7.304

Error Covariance for perso and abili = 0.247(0.0257)

9.608

Covariance Matrix of Independent Variables


p_mater 1.310(0.152)

8.621

p_abili 0.225 0.178(0.049) (0.040)

4.618 4.427

p_perso 0.167 0.165 0.322(0.056) (0.030) (0.052)

2.991 5.494 6.249



3.119 4.553 4.568(0.056) (0.037) (0.037)55.382 122.465 124.015




241 group = Brazil

Within Group Completely Standardized Solution

LAMBDA-Y


val_int - - 0.573 - -val_htec - - 0.561 - -val_prof - - 0.644 - -val_sen - - - - 0.702

val_symp - - - - 0.718val_mon 0.868 - - - -val_pow 0.845 - - - -

LAMBDA-X


p_v_int - - 0.613 - -p_v_htec - - 0.452 - -p_v_prof - - 0.699 - -p_v_sen - - - - 0.627

p_v_symp - - - - 0.763p_v_mon 0.833 - - - -p_v_pow 0.914 - - - -

GAMMA


mater 0.290 - - - -abili - - 0.318 - -perso - - - - 0.325

PSI


mater 0.916abili 0.548 0.899perso 0.362 0.645 0.894

242 THETA-EPS

val_int val_htec val_prof val_sen val_symp val_mon val_pow-------- -------- -------- -------- -------- -------- --------


val_symp - - - - - - - - 0.485val_mon - - - - - - - - - - 0.247val_pow - - - - - - - - - - - - 0.287

THETA-DELTA

p_v_int p_v_htec p_v_prof p_v_sen p_v_symp p_v_mon p_v_pow-------- -------- -------- -------- -------- -------- --------

p_v_int 0.624p_v_htec 0.177 0.795p_v_prof - - - - 0.512p_v_sen - - - - - - 0.607

p_v_symp - - - - - - - - 0.417p_v_mon - - - - - - - - - - 0.306p_v_pow - - - - - - - - - - - - 0.165

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