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Structural Equation Modeling

What is SEM?• Combines measurement models of CFA

with goals of multiple regression analysis to allow the prediction of latent variables from other latent variables.– Simultaneous regression equations– Modeling latent variables from observed variables– Estimate parameters of the measurement model

& structural model– Comparison between implied covariance matrix &

observed covariance matrix

Advantages of SEM• Testing multiple relationships at a time

– Multiple independent and dependent variables can be accommodated (DVs can even related to one another)

• Examining latent variables (but must link them to manifest variables)

• Specifying measurement error in the model– Allows enhanced model fit– No assumption of uncorrelated errors (although by

default errors uncorrelated need to change to allow correlated errors)

Model Testing in SEM• Measurement model

– Also known as confirmatory factor analysis– Tests the relationship between the indicators and

the latent variables they are supposed to measure

• Path model– Tests the relationships between the exogenous

and endogenous variables without the measurement model specified

• Structural model– Tests the relationships between the exogenous

and endogenous variables with the measurement model specified

Measurement Model

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Path Model

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Structural Model

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SEM Lingo• Exogenous variable – Construct that acts only

as a predictor or cause in a model, an IV. Not predicted by anything in the model (A)

• Endogenous variable – Construct that is an outcome variable in at least one causal relationship, a DV, also mediators (B, C, D)

A B

C

D

SEM Lingo: Constructs and Indicators

• Exogenous constructs/latent variables are called Ksis – represented by ξ

• Endogenous constructs/latent variables are called etas –represented by η

• Exogenous indicator/manifest variable - X

• Endogenous indicator/manifest variable - Y

SEM Lingo: Constructs and Indicators

ξ η η

X

X

X

Y

Y Y

Y

Y

Ksis Etas

• Nonrecursive – relationship is reciprocal in path diagram

• Recursive – relationships not reciprocal

• Nested Models – Models that have same constructs but differ in the number and type of causal relationships represented (i.e., parameters estimated)

SEM Lingo

Measurement Model Matrices• Lambda-X (ΛX)

– Loadings of exogenous indicators; tells how you get from the manifest Xs to the latent Xs

• Lambda-Y (ΛY) – Loadings of endogenous indicators; tells how you get

from manifest Ys to latent Ys

• Theta-delta (θ) – Errors of the exogenous indicators, the manifest X

variables

• Theta-epsilon (θε) – Errors of the endogenous indicators, the manifest Y

variables

Example

IntelligenceSchool

Performance

GMAT IQ Test GPA # Pubs

LXLX LY LY

TD TD TE TE

• Beta (B) – Relationships of endogenous constructs to

endogenous constructs; how DVs cause each other

• Gamma (Г) – Relationships of exogenous constructs to endogenous

constructs; how IVs cause DVs

• *Phi (Φ) – Correlations among latent exogenous constructs;

correlations among the IVs

• Psi (Ψ) – Residuals from prediction of latent endogenous

constructs; Tells whether residuals of prediction are correlated

Structural Model Matrices

Example

Intelligence

GMAT IQ Test

SchoolPerformance

GPA # Pubs

WorkPerformance

SupRating

PeerRating

Motivation

SupReport

SelfReport

BetaPhi

Gamma

Gamma

Gamma

Gamma

Psi –Resid

Assumptions in SEM• Observations are independent• Respondents are randomly sampled• In maximum likelihood estimation,

multivariate normality assumption• Continuous variables (using correlations)

except when use:– Polychoric correlation matrix – Tetrachoric correlation matrix– Polyserial correlation matrix – Biserial correlation matrix

SEM Sample Size Requirements• Absolute minimum = number of

covariances or correlations in the matrix• Typical min = 5 respondents/parameter

estimated, 10/parameter preferred– When not MV normal – 15 respondents per

parameter

• ML estimation– Can use as few as 50, but 100-150

recommended– Ideal n = 200

• In SEM, goal of estimation is to minimize error between the observed and reproduced values in VCV matrix choose parameter estimates that increase likelihood of reproducing VCV matrix

Estimation Procedures

Regr: min Σ(y – y’)2

SEM: min (obs VCV – repr VCV)

• Unweighted Least Squares (OLS)– Used in regression, but not in SEM– OLS is scale invariant only if the errors of

measurement are uncorrelated this is an assumption in regression but not in SEM

– Assumes MV normality

Estimation Procedures

• Generalized Least Squares (GLS)– Used in SEM– You take the least squares and weight it with

a VCV matrix yields a scale-free estimation procedure

– Assumes MV Normality

SEM Estimation Procedures

• Maximum Likelihood Estimation (ML)– Weights least squares estimates with VCV

matrix; updates VCV matrix each iteration– Assumes ML normality– As increase sample, GLS = ML

– Finds parameter estimates that maximize the probability of the data

– Most commonly used and default estimation procedure in LISREL

SEM Estimation Procedures

• Weighted Least Squares– Makes no assumptions about distribution– Need huge sample (n = 500+)– No assumption of ML normality required

• In practice, WLS not really used. ML and GLS are robust against assumptions of multivariate normality

SEM Estimation Procedures

Steps in Conducting SEM• Draw a picture of your model including both

your latent and manifest variables• Test the fit of your measurement model.

Adjust as needed to enhance fit of measurement model

• Once measurement model fits, test fit of structural model

• Modify structural model involves doing exploratory SEM not recommended

Anderson & Gerbing Two Step Approach

• Step 1. Adequacy of Measurement Model– Test measurement model; allow all latent variables

to correlate– Adjust meas model as needed to enhance fit– If fit of measurement model is poor, don’t test

structural model– Fit of structural model is a necessary but not

sufficient condition for the fit of structural model

• Step 2. Adequacy of Structural Model

Hair - Stages in SEM1. Develop a theoretically based model2. Construct a path diagram of causal

relationships3. Convert path diagram into set of structural

and measurement models4. Choose the input matrix type and estimate

the proposed model5. Assess the identification of the model 6. Evaluate goodness-of-fit criteria7. Interpret and modify the model

Develop Model & Path Diagram

productfactors

price-based factors

relationship factors

productusage

satisfactionwith company

Convert path diagram to structural and measurement models

productfactors

price-based factors

relationship factors

productusage

satisfactionw/ company

Y3Y1

X6X5

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X3 X4

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Choose Input Matrix & Estimate Model

• Careful with missing data! – No “Missing data” correlation matrix

• Choice of correlation matrix or VCV matrix– Correlation matrix yields standardized weights

from +1 to -1– VCV matrix is better for validating causal

relationships

Analyzing Correlation versus Covariance Matrices

SEM models are based on the decomposition of covariance matrices, not correlation matrices. The solutions hold, strictly speaking, for the analysis of covariance matrices. To the extent that the solution depends on the scale of the variables, analyses based on covariance matrices and correlation matrices can differ.

• Degrees of freedom (df) are related to the number of parameter estimates

• Model df must be > or = 0– Just-identified model/saturated model: df = 0;

perfect model fit– *Over-identified model: df > 0 because more

information in the matrix than the number of parameters estimates

– Under-identified model: df < 0 because model has more parameters estimated than information available. Can’t run due to infinite solutions

Model Identification

• Often results from a large number of parameters estimated compared to number of correlations provided too few degrees of freedom

• Solution: Estimate fewer parameters

Identification Problems

Model Fit• Fit of model denoted by two things:

– Small residuals– Nonsignificant difference between original

VCV matrix and reconstructed VCV matrix

• To assess model it, SEM provides numerous goodness of fit indices– Different indices assess fit in different ways

Types of Goodness of Fit Indices

• Absolute Fit Measures – Overall model fit, no adjustment for overfitting

• Incremental Fit Measures – Compare proposed model fit to another model

specified by researcher

• Parsimonious Fit Measures – “Adjust” model to provide comparison between

models with differing numbers of estimated coefficients

Specific Goodness of Fit Indices• Absolute Fit Measures

– Chi2 (2)– Goodness-of-fit (GFI)– Root Mean Square Error of Approximation (RMSEA)– Root Mean Square Residual (RMR)

• Incremental Fit Measures– Adjusted-goodness-of-fit (AGFI)– Normed Fit Index (NFI)

• Parsimonious Fit Measures– Parsimony Normed Fit Index (PNFI)– Parsimony Goodness of Fit Index (PGFI)

Chi2 - 2

• A “badness of fit” measure– Represents the extent to which the observed

and reproduced correlation matrices differ

• High power (high n size) increases 2 so that it is significant penalized for large n 2 only look at when n.s.

2 difference test – compares nested models– Most practical use of chi2

SEM - Degrees of Freedom

• df = number of known pieces of information – unknowns to be estimated

• Important in model fit if estimate fewer paths, fit will reduce just by chance (almost all paths not 0 just due to chance)

• Best possible model fit saturated model in which all the links are estimated by default

)2/2(1 NullModelGFI

• The quality of the original model and its ability to reproduce the actual variance-covariance matrix is more easily gauged by GFI

• This index is similar to R2 in multiple regression

• This index tells us how much better our model compared to the null model

• Want higher values, > or = .90

Goodness of fit index (GFI)

SEM - Degrees of Freedom

Latent A

Latent B

Latent C

Manifest B1

Manifest B2

Manifest C1

Manifest C2

Manifest A1

Manifest A2Latent A

Latent B

Latent C

Manifest B1

Manifest B2

Manifest C1

Manifest C2

Manifest A1

Manifest A2

Larger 2 - Worse Fit Smaller 2 - Better Fit

Root Mean Square Error of Approximation (RMSEA)

• A normed index with rules of thumb

• Prefer a RMSEA < or = .05

Model

NModelModel

df

dfSQRTRMSEA

/)2(

Root Mean Squared Residual (RMR)

• Want this value to be small, < or = .05 is ideal, < or = .1 is probably good

• Not a normed fit index; size of residuals influenced by variance of variables involved

22 )()( qpreprVCVobsVCVsumSQRTRMR

.)]/#(#1/)1[(1 dataptsarametersestimatedpGFIAGFI

• Adjusted goodness-of-fit index (AGFI) was created to account for increases in fit due to chance

• Similar to the adjusted R2 in multiple regression

• Can be negative

• Less sensitive to changes in df than is PNFI/PGFI

Adjusted Goodness of fit index (AGFI)

Normed Fit Index (NFI)

• NFI compares fit of the null model to fit of the theoretical model– Large values of NFI are best (ideally > or = .9)

• Criticism of NFI: Comparing the model fit to a model of nothing. Is this meaningful?

null

HypModelnull

NFI2

22

Parsimony Normed Fit Index (PNFI)

• Problem: Most fit indices increase just by estimating more parameters (free more links to be estimated)– Just identified model perfect fit

• PNFI penalizes you for lack of parsimony• No clear benchmark for what is “good” –

best to compare between models

HypModelnull

HypModelNFI

df

dfPNFI

• Gets smaller as you increase the number of paths in the model

• No clear benchmark for what is “good” – best to compare between models

Parsimony Goodness of Fit Index (PGFI)

GFIdataptsarametersestimatedpPGFI .)]/#(#1[

Some Common Rules of Thumb for Model Fit

Test or Index Good Fit Acceptable Fit

Chi-Square Goodness of Fit

p > .20 p > .05

GFI .95 .90

AGFI .90 .80

RMRDepends on scale.

Closer to 0 is betterDepends on scale.

Closer to 0 is better

RMSEA < .05 < .08

Model Testing Strategies with SEM

• Model Confirmation– Single model tested to fit or not fit– Problem with “confirmation bias” – many

possible models fit

• *Competing Models Strategy– Compares competing models for best fit– Nested models should be used

• Exploratory SEM/Model development– Capitalizes on Chance

Testing Competing Nested Models

• Comparing the fit of hypothesized and alternative models that have the same constructs but differ in number of parameters estimated

• Models must be nested within one another to compare them use 2 difference test to compare

• Typical nested models involve deleting or adding a single path

Testing Nested Models

A CBMODEL 1

A CBMODEL 2

These two models are nested within one another because they differ only in the addition of a single link in the second model

Likelihood Ratio Test

• Problem: sensitive to sample size

• Solution: CFI – changes in CFI less than -.01– Cheung & Rensvold (2002) Structural Equation

Modeling, 9, 233-255

222

nedunconstraidconstraine

nedunconstraidconstraine dfdfdf

Model Modification: Exploratory SEM

• Common management practice not to do this don’t edit structural models a confirmatory technique (but edit meas model OK)

• t values– Tells us where deleting a path would enhance

model fit (paths with n.s. t values)

• Modification Indices– Suggest where paths could be added to

increase fit– Represents the degree to which 2 would

decrease if you added a path

• To justify change due to MI– Fairly large– Justifiable– Impact model fit– Not central to theory

Model Modification: Exploratory SEM

In LISREL, the modification indices are the changes in the goodness-of-fit 2 that would result from setting that parameter free.

In LISREL, the modification indices are the changes in the goodness-of-fit 2 that would result from setting that parameter free.

SEM with LISREL• Traditional LISREL language

– Uses matrix language– Specify whether aspects of matrix are free (FR)

or fixed (FI)

• SIMPLIS language*– More recent LISREL language– More user-friendly syntax

• PRELIS– Prepares raw data for use in LISREL - generates

correlation matrix or VCV matrix to input

Running LISREL Program

• Title line

• Input Specification

• Model Specification

• Path Diagram

• Output Specification

Exercise: CFA Model

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Exercise: CFA Model

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Draw the diagram

Title line: ExerciseObserved variables: X1 X2 X3 X4 X5 X6 X7 X8Covariance matrix:1.650.45 1.140.35 0.30 1.010.51 0.49 0.43 1.580.07 0.20 0.20 0.23 0.750.17 0.14 0.17 0.27 0.23 0.650.41 0.20 0.07 0.21 0.11 0.25 0.850.22 0.23 0.26 0.36 0.24 0.25 0.16 0.75Sample size: 200Latent variables: LAT1 LAT2

Input Specification: SIMPLIS

LISREL/SIMPLIS Code (continued)RELATIONSHIPSX1 = 1*LAT1X2 = LAT1X3 = LAT1X4 = LAT1X5 = 1*LAT2X6 = LAT2X7 = LAT2X8 = LAT2LAT1 = LAT2LISREL OUTPUT: SS SC EF AD = OFFPRINT RESIDUALSPATH DIAGRAMEND OF PROBLEM

LISREL/SIMPLIS Code:Estimation Techniques

• Default estimation technique is ML

• Other Techniques available:– Generalized least squares (GLS)– Unweighted least squares (ULS)– Weighted least squares (WLS)

• Other techniques with this syntax

Method of estimation: GLS

LISREL/SIMPLIS Code:LISREL Output

SS Print standardized solutionSC Print completely standardized solutionEF Print total & indirect effects, their standard errors

& t valuesVA Print variances and covariancesFS Print factor scores regressionPC Print correlations of parameter estimatesPT Print technical information

Example: Confirmatory Factor Analysis

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Tests of significance for parameter estimates: t values.

Parameter estimates

CHI-SQUARE WITH 51 DEGREES OF FREEDOM = 55.50 (P = 0.31)ESTIMATED NON-CENTRALITY PARAMETER (NCP) = 4.5090 PERCENT CONFIDENCE INTERVAL FOR NCP = (0.0 ; 26.77)MINIMUM FIT FUNCTION VALUE = 0.11POPULATION DISCREPANCY FUNCTION VALUE (F0) = 0.009090 PERCENT CONFIDENCE INTERVAL FOR F0 = (0.0 ; 0.054)ROOT MEAN SQUARE ERROR OF APPROXIMATION (RMSEA) = 0.01390 PERCENT CONFIDENCE INTERVAL FOR RMSEA = (0.0 ; 0.032)P-VALUE FOR TEST OF CLOSE FIT (RMSEA < 0.05) = 1.00EXPECTED CROSS-VALIDATION INDEX (ECVI) = 0.2290 PERCENT CONFIDENCE INTERVAL FOR ECVI = (0.21 ; 0.26)ECVI FOR SATURATED MODEL = 0.31ECVI FOR INDEPENDENCE MODEL = 3.98CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES OF FREEDOM = 1962.12INDEPENDENCE AIC = 1986.12MODEL AIC = 109.50SATURATED AIC = 156.00INDEPENDENCE CAIC = 2048.69MODEL CAIC = 250.29SATURATED CAIC = 562.74ROOT MEAN SQUARE RESIDUAL (RMR) = 0.028STANDARDIZED RMR = 0.028GOODNESS OF FIT INDEX (GFI) = 0.98ADJUSTED GOODNESS OF FIT INDEX (AGFI) = 0.97PARSIMONY GOODNESS OF FIT INDEX (PGFI) = 0.64NORMED FIT INDEX (NFI) = 0.97NON-NORMED FIT INDEX (NNFI) = 1.00PARSIMONY NORMED FIT INDEX (PNFI) = 0.75COMPARATIVE FIT INDEX (CFI) = 1.00INCREMENTAL FIT INDEX (IFI) = 1.00RELATIVE FIT INDEX (RFI) = 0.96CRITICAL N (CN) = 696.82

CHI-SQUARE WITH 51 DEGREES OF FREEDOM = 55.50 (P = 0.31) (This test models the variances and covariances as implied by the parameter expectations)

CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES OF FREEDOM = 1962.12 (This test only models the variances of the variables and assumes all covariances are 0)

GOODNESS OF FIT INDEX (GFI) = 0.98

ADJUSTED GOODNESS OF FIT INDEX (AGFI) = 0.97

Hypothesized Model Fit Statistics

CORRELATION MATRIX TO BE ANALYZED V1 V2 V3 V4 M1 M2 -------- -------- -------- -------- -------- -------- V1 1.00 V2 0.52 1.00 V3 0.52 0.48 1.00 V4 0.54 0.54 0.49 1.00 M1 0.16 0.22 0.19 0.23 1.00 M2 0.22 0.28 0.23 0.23 0.48 1.00 M3 0.19 0.21 0.13 0.17 0.47 0.46 M4 0.22 0.23 0.23 0.17 0.48 0.49 R1 0.23 0.25 0.29 0.23 0.14 0.22 R2 0.22 0.17 0.21 0.17 0.17 0.23 R3 0.28 0.22 0.26 0.22 0.18 0.23 R4 0.27 0.25 0.24 0.26 0.21 0.23 CORRELATION MATRIX TO BE ANALYZED M3 M4 R1 R2 R3 R4 -------- -------- -------- -------- -------- -------- M3 1.00 M4 0.50 1.00 R1 0.15 0.28 1.00 R2 0.11 0.19 0.47 1.00 R3 0.17 0.25 0.50 0.51 1.00 R4 0.15 0.23 0.51 0.52 0.52 1.00

FITTED COVARIANCE MATRIX V1 V2 V3 V4 M1 M2 -------- -------- -------- -------- -------- -------- V1 1.00 V2 0.53 1.00 V3 0.51 0.49 1.00 V4 0.54 0.52 0.50 1.00 M1 0.21 0.20 0.20 0.21 1.00 M2 0.21 0.21 0.20 0.21 0.48 1.00 M3 0.21 0.20 0.19 0.20 0.46 0.47 M4 0.22 0.21 0.21 0.22 0.49 0.50 R1 0.24 0.23 0.22 0.23 0.19 0.20 R2 0.23 0.23 0.22 0.23 0.19 0.19 R3 0.25 0.24 0.23 0.24 0.20 0.20 R4 0.25 0.24 0.23 0.25 0.20 0.21 FITTED COVARIANCE MATRIX M3 M4 R1 R2 R3 R4 -------- -------- -------- -------- -------- -------- M3 1.00 M4 0.48 1.00 R1 0.19 0.20 1.00 R2 0.19 0.20 0.48 1.00 R3 0.20 0.21 0.50 0.50 1.00 R4 0.20 0.21 0.51 0.51 0.53 1.00

FITTED RESIDUALS V1 V2 V3 V4 M1 M2 -------- -------- -------- -------- -------- -------- V1 0.00 V2 -0.01 0.00 V3 0.01 -0.01 0.00 V4 0.00 0.02 -0.01 0.00 M1 -0.05 0.01 -0.01 0.02 0.00 M2 0.01 0.07 0.03 0.02 0.00 0.00 M3 -0.02 0.01 -0.06 -0.04 0.01 -0.01 M4 0.00 0.01 0.02 -0.04 -0.01 -0.01 R1 -0.01 0.02 0.07 -0.01 -0.05 0.02 R2 -0.01 -0.05 -0.01 -0.06 -0.03 0.04 R3 0.03 -0.02 0.03 -0.03 -0.02 0.03 R4 0.02 0.00 0.01 0.02 0.01 0.03 FITTED RESIDUALS M3 M4 R1 R2 R3 R4 -------- -------- -------- -------- -------- -------- M3 0.00 M4 0.01 0.00 R1 -0.04 0.08 0.00 R2 -0.08 -0.01 -0.01 0.00 R3 -0.02 0.04 0.00 0.01 0.00 R4 -0.05 0.01 0.00 0.01 -0.01 0.00

STANDARDIZED RESIDUALS V1 V2 V3 V4 M1 M2 -------- -------- -------- -------- -------- -------- V1 0.00 V2 -0.87 0.00 V3 0.77 -0.64 0.00 V4 0.17 1.20 -0.66 0.00 M1 -1.43 0.45 -0.21 0.63 0.00 M2 0.16 2.27 0.79 0.66 0.19 0.00 M3 -0.51 0.37 -1.84 -1.12 0.81 -0.40 M4 0.05 0.38 0.63 -1.40 -0.49 -1.08 R1 -0.30 0.59 2.23 -0.24 -1.44 0.68 R2 -0.35 -1.68 -0.27 -1.83 -0.76 1.11 R3 1.02 -0.62 0.92 -0.92 -0.59 0.81 R4 0.60 0.12 0.21 0.57 0.36 0.83 STANDARDIZED RESIDUALS M3 M4 R1 R2 R3 R4 -------- -------- -------- -------- -------- -------- M3 0.00 M4 1.00 0.00 R1 -1.10 2.45 0.00 R2 -2.38 -0.34 -0.32 0.00 R3 -0.70 1.15 -0.03 0.74 0.00 R4 -1.44 0.45 -0.21 0.74 -0.91 0.00

The chosen model fits the data quite well. How would other models do? A complete confirmatory analysis would not only test the preferred model but also examine alternative models to assess how easily they could account for the data. To the extent that reasonable alternatives exist, the preferred model must be considered with more caution.

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Alternative Measurement Model 1

ParameterEstimates

CHI-SQUARE WITH 54 DEGREES OF FREEDOM = 214.02 (P = 0.0)

CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES OF FREEDOM = 1962.12

GOODNESS OF FIT INDEX (GFI) = 0.93

ADJUSTED GOODNESS OF FIT INDEX (AGFI) = 0.90

Alternative Model 1 Fit Indices

F1

Alternative Measurement Model 2

Parameter Estimates

CHI-SQUARE WITH 54 DEGREES OF FREEDOM = 770.57 (P = 0.0)

CHI-SQUARE FOR INDEPENDENCE MODEL WITH 66 DEGREES OF FREEDOM = 1962.12

GOODNESS OF FIT INDEX (GFI) = 0.73

ADJUSTED GOODNESS OF FIT INDEX (AGFI) = 0.61

Alternative Model 2 Fit Indices

Summary Fit StatisticsModel Chi2 df GFI AGFIHyp 55.50 51 .98 .97Alt 1 214.02 54 .93 .90Alt 2 770.57 54 .73 .61

Note: Our hypothesized measurement model is the best fit!!

Example Structural Model Testing

Hypothesized Model

Home

AchieveAbility

Aspire

Family Income

Father Education

Mother Education

Verbal Ability

Quant. Ability

Ed. Aspirations

Occ. Aspirations

Verbal Achieve

Quant. Achieve

e

e

e

e

ee

e

e

e

e

e

Syntax to Specify Hypothesized

Structural Model

RELATIONSHIPSfaminc = 1*homefaed = homemoed = homeverbab = 1*abilityquantab = abilityedasp = 1*aspireocasp = aspireverach = 1*achievequantach = achievehome = abilityaspire = home abilityachieve = home ability

COVARIANCE MATRIX TO BE ANALYZED edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 1.02 ocasp 0.79 1.08 verach 1.03 0.92 1.84 quantach 0.76 0.70 1.24 1.29 faminc 0.57 0.54 0.88 0.63 0.85 faed 0.44 0.42 0.68 0.53 0.52 0.67 moed 0.43 0.39 0.64 0.50 0.48 0.55 verbab 0.58 0.56 0.89 0.72 0.55 0.42 quantab 0.49 0.50 0.89 0.65 0.51 0.39 COVARIANCE MATRIX TO BE ANALYZED moed verbab quantab -------- -------- -------- moed 0.72 verbab 0.37 0.85 quantab 0.34 0.63 0.87

GOODNESS OF FIT STATISTICSCHI-SQUARE WITH 21 DEGREES OF FREEDOM = 57.17 (P = 0.000034)ROOT MEAN SQUARE ERROR OF APPROXIMATION (RMSEA) = 0.093CHI-SQUARE FOR INDEPENDENCE MODEL WITH 36 DEGREES OF FREEDOM = 1407.10ROOT MEAN SQUARE RESIDUAL (RMR) = 0.047STANDARDIZED RMR = 0.048GOODNESS OF FIT INDEX (GFI) = 0.94ADJUSTED GOODNESS OF FIT INDEX (AGFI) = 0.87

STANDARDIZED SOLUTION LAMBDA-Y aspire achieve -------- -------- edasp 0.93 - - ocasp 0.85 - - verach - - 1.28 quantach - - 0.97 LAMBDA-X home ability -------- -------- faminc 0.73 - - faed 0.73 - - moed 0.70 - - verbab - - 0.81 quantab - - 0.77

STANDARDIZED SOLUTION BETA aspire achieve -------- -------- aspire - - - - achieve 0.40 - - GAMMA home ability -------- -------- aspire 0.32 0.52 achieve 0.14 0.48 CORRELATION MATRIX OF ETA AND KSI aspire achieve home ability -------- -------- -------- -------- aspire 1.00 achieve 0.85 1.00 home 0.70 0.76 1.00 ability 0.75 0.88 0.73 1.00 PSI aspire achieve -------- -------- 0.39 0.14

FITTED COVARIANCE MATRIX edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 1.02 ocasp 0.79 1.08 verach 1.01 0.93 1.84 quantach 0.77 0.71 1.24 1.29 faminc 0.47 0.43 0.71 0.54 0.85 faed 0.48 0.44 0.72 0.54 0.54 0.67 moed 0.46 0.42 0.69 0.52 0.51 0.52 verbab 0.57 0.52 0.91 0.69 0.43 0.43 quantab 0.54 0.49 0.87 0.66 0.41 0.41 FITTED COVARIANCE MATRIX moed verbab quantab -------- -------- -------- moed 0.72 verbab 0.42 0.85 quantab 0.39 0.63 0.87

FITTED RESIDUALS edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 0.00 ocasp 0.00 0.00 verach 0.01 -0.01 0.00 quantach -0.01 -0.01 0.00 0.00 faminc 0.09 0.10 0.16 0.09 0.00 faed -0.03 -0.01 -0.04 -0.02 -0.02 0.00 moed -0.02 -0.03 -0.05 -0.02 -0.04 0.03 verbab 0.01 0.04 -0.02 0.02 0.11 -0.01 quantab -0.05 0.00 0.02 -0.01 0.10 -0.02 FITTED RESIDUALS moed verbab quantab -------- -------- -------- moed 0.00 verbab -0.04 0.00 quantab -0.06 0.00 0.00 SUMMARY STATISTICS FOR FITTED RESIDUALS SMALLEST FITTED RESIDUAL = -0.06 MEDIAN FITTED RESIDUAL = 0.00 LARGEST FITTED RESIDUAL = 0.16

STANDARDIZED RESIDUALS edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 0.00 ocasp 0.00 0.00 verach 1.42 -0.80 0.00 quantach -0.78 -0.36 0.00 0.00 faminc 3.54 3.11 5.35 2.80 0.00 faed -2.25 -0.58 -2.63 -0.86 -2.81 0.00 moed -1.03 -1.03 -2.15 -0.84 -3.24 6.34 verbab 0.88 1.96 -2.28 1.31 4.59 -0.90 quantab -2.56 0.18 1.82 -0.57 3.47 -1.29 STANDARDIZED RESIDUALS moed verbab quantab -------- -------- -------- moed 0.00 verbab -2.14 0.00 quantab -2.37 0.00 0.00 SUMMARY STATISTICS FOR STANDARDIZED RESIDUALS SMALLEST STANDARDIZED RESIDUAL = -3.24 MEDIAN STANDARDIZED RESIDUAL = 0.00 LARGEST STANDARDIZED RESIDUAL = 6.34

QPLOT OF STANDARDIZED RESIDUALS 3.5.......................................................................... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . . . x . . x N . . x O . . x R . . x x . M . . x x . A . . xx . L . x . x . . x x . . Q . x x . . U . x . . A . xx . . N . xx . . T . x x . . I . * . . L . * . . E . x . . S . x . . . x . . . x . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -3.5.......................................................................... -3.5 3.5

STANDARDIZED RESIDUALS

Home

AchieveAbility

Aspire

Family Income

Father Education

Mother Education

Verbal Ability

Quant. Ability

Ed. Aspirations

Occ. Aspirations

Verbal Achieve

Quant. Achieve

e

e

e

e

ee

e

e

e

e

e

Alternative Model 1

Syntax to Specify

Alternative Structural

Model

RELATIONSHIPSfaminc = 1*homefaed = homemoed = homeverbab = 1*abilityquantab = abilityedasp = 1*aspireocasp = aspireverach = 1*achievequantach = achievehome = ability

aspire = home ability

achieve = home abilityLet the errors for faed and moed correlate

CHI-SQUARE WITH 20 DEGREES OF FREEDOM = 19.17 (P = 0.51)ROOT MEAN SQUARE ERROR OF APPROXIMATION (RMSEA) = 0.090 PERCENT CONFIDENCE INTERVAL FOR RMSEA = (0.0 ; 0.058)CHI-SQUARE FOR INDEPENDENCE MODEL WITH 36 DEGREES OF FREEDOM = 1407.10ROOT MEAN SQUARE RESIDUAL (RMR) = 0.015STANDARDIZED RMR = 0.015GOODNESS OF FIT INDEX (GFI) = 0.98ADJUSTED GOODNESS OF FIT INDEX (AGFI) = 0.95

STANDARDIZED SOLUTION LAMBDA-Y aspire achieve -------- -------- edasp 0.93 - - ocasp 0.85 - - verach - - 1.29 quantach - - 0.97 LAMBDA-X home ability -------- -------- faminc 0.81 - - faed 0.64 - - moed 0.59 - - verbab - - 0.81 quantab - - 0.77

BETA aspire achieve -------- -------- aspire - - - - achieve 0.38 - -GAMMA home ability -------- -------- aspire 0.44 0.39 achieve 0.19 0.43CORRELATION MATRIX OF ETA AND KSI aspire achieve home ability -------- -------- -------- -------- aspire 1.00 achieve 0.85 1.00 home 0.76 0.83 1.00 ability 0.75 0.87 0.81 1.00PSI aspire achieve -------- -------- 0.37 0.14

COVARIANCE MATRIX TO BE ANALYZED

edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 1.02 ocasp 0.79 1.08 verach 1.03 0.92 1.84 quantach 0.76 0.70 1.24 1.29 faminc 0.57 0.54 0.88 0.63 0.85 faed 0.44 0.42 0.68 0.53 0.52 0.67 moed 0.43 0.39 0.64 0.50 0.48 0.55 verbab 0.58 0.56 0.89 0.72 0.55 0.42 quantab 0.49 0.50 0.89 0.65 0.51 0.39

COVARIANCE MATRIX TO BE ANALYZED

moed verbab quantab -------- -------- -------- moed 0.72 verbab 0.37 0.85 quantab 0.34 0.63 0.87

FITTED COVARIANCE MATRIX

edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 1.02 ocasp 0.79 1.08 verach 1.02 0.93 1.84 quantach 0.77 0.70 1.24 1.29 faminc 0.57 0.53 0.87 0.66 0.85 faed 0.45 0.41 0.68 0.51 0.52 0.67 moed 0.41 0.38 0.63 0.47 0.48 0.54 verbab 0.57 0.52 0.91 0.69 0.54 0.42 quantab 0.54 0.49 0.87 0.65 0.51 0.40

FITTED COVARIANCE MATRIX

moed verbab quantab -------- -------- -------- moed 0.72 verbab 0.39 0.85 quantab 0.37 0.63 0.87

FITTED RESIDUALS

edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 0.00 ocasp 0.00 0.00 verach 0.01 -0.01 0.00 quantach -0.01 -0.01 0.00 0.00 faminc -0.01 0.01 0.01 -0.02 0.00 faed 0.00 0.01 0.00 0.01 0.00 0.00 moed 0.02 0.01 0.01 0.03 0.00 0.00 verbab 0.01 0.04 -0.02 0.03 0.01 0.00 quantab -0.05 0.00 0.02 -0.01 0.00 -0.01

FITTED RESIDUALS

moed verbab quantab -------- -------- -------- moed 0.00 verbab -0.01 0.00 quantab -0.03 0.00 0.00

STANDARDIZED RESIDUALS

edasp ocasp verach quantach faminc faed -------- -------- -------- -------- -------- -------- edasp 0.00 ocasp 0.00 0.00 verach 1.26 -1.01 0.00 quantach -0.52 -0.23 0.00 0.00 faminc -0.64 0.45 0.55 -1.17 0.00 faed -0.25 0.45 -0.23 0.58 0.15 0.00 moed 0.82 0.30 0.36 0.91 -0.15 0.00 verbab 0.88 1.93 -2.34 1.50 0.72 0.10 quantab -2.53 0.16 1.59 -0.38 -0.13 -0.50

STANDARDIZED RESIDUALS

moed verbab quantab -------- -------- -------- moed 0.00 verbab -0.63 0.00 quantab -1.10 0.00 0.00

SUMMARY STATISTICS FOR STANDARDIZED RESIDUALS SMALLEST STANDARDIZED RESIDUAL = -2.53 MEDIAN STANDARDIZED RESIDUAL = 0.00 LARGEST STANDARDIZED RESIDUAL = 1.93

QPLOT OF STANDARDIZED RESIDUALS 3.5.......................................................................... . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .x . . . . . .x . . .x . N . .x . O . x. . R . xx . M . x . A . xx . L . * . . .* . Q . .* . U . x. x . A . xx . N . . * . T . x x . I . . x . L . . * . E . x . S . .x . . . x . . x . . . . . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -3.5.......................................................................... -3.5 3.5

STANDARDIZED RESIDUALS

BETA aspire achieve -------- -------- aspire - - - - achieve 0.40 - - GAMMA home ability -------- -------- aspire 0.32 0.52 achieve 0.14 0.48

BETA aspire achieve -------- -------- aspire - - - - achieve 0.38 - -GAMMA home ability -------- -------- aspire 0.44 0.39 achieve 0.19 0.43

Original Model Modified Model

Original Model Modified Model

Chi2 57.17 (df = 21) 19.17 (df = 20)

RMSEA 0.093 0.0

RMR 0.047 0.015

SRMR 0.048 0.015

GFI 0.94 0.98

AGFI 0.87 0.95

Chi2 difference (1) = 38.00