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General Structural Equation (LISREL) Models

Week #2 Class #2

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Today’s classToday’s class

Latent variable structural equations Latent variable structural equations in matrix form (from yesterday)in matrix form (from yesterday)

Fit measuresFit measures SEM assumptionsSEM assumptions What to write up What to write up LISREL matricesLISREL matrices

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From yesterday’s lab:

Ach1

REDUCE e1

1

1

NEVHAPP e21

NEW_GOAL e31

IMPROVE e41

ACHIEVE e51

CONTENT e61

Reference indicator:

REDUCE

Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- -------

REDUCE <---------- Ach1 1.000 NEVHAPP <--------- Ach1 2.142 0.374 5.721 NEW_GOAL <-------- Ach1 -2.759 0.460 -5.995 IMPROVE <--------- Ach1 -4.226 0.703 -6.009 ACHIEVE <--------- Ach1 -2.642 0.450 -5.874 CONTENT <--------- Ach1 2.657 0.460 5.779

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From yesterday’s lab:

Ach1

REDUCE e1

1

1

NEVHAPP e21

NEW_GOAL e31

IMPROVE e41

ACHIEVE e51

CONTENT e61

Reference indicator:

REDUCE

Standardized Regression Weights: Estimate-------------------------------- --------

REDUCE <---------- Ach1 0.138 NEVHAPP <--------- Ach1 0.332 NEW_GOAL <-------- Ach1 -0.541 IMPROVE <--------- Ach1 -0.682 ACHIEVE <--------- Ach1 -0.410 CONTENT <--------- Ach1 0.357

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From yesterday’s lab:

Reference indicator:

REDUCE

Ach1

REDUCE e1

1

1

NEVHAPP e21

NEW_GOAL e31

IMPROVE e41

ACHIEVE e51

CONTENT e61

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Regression Weights: Estimate S.E. C.R. Label

------------------- -------- ------- ------- -------

REDUCE <---------- Ach1 1.000

NEVHAPP <--------- Ach1 -113.975 1441.597 -0.079

NEW_GOAL <-------- Ach1 215.393 2717.178 0.079

IMPROVE <--------- Ach1 373.497 4711.675 0.079

ACHIEVE <--------- Ach1 211.419 2667.067 0.079

CONTENT <--------- Ach1 -155.262 1961.974 -0.079

Standardized Regression Weights: Estimate-------------------------------- --------

REDUCE <---------- Ach1 0.002 NEVHAPP <--------- Ach1 -0.223 NEW_GOAL <-------- Ach1 0.534 IMPROVE <--------- Ach1 0.762 ACHIEVE <--------- Ach1 0.415 CONTENT <--------- Ach1 -0.264

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Solution:

• Use a different reference indicator

• (Note: REDUCE can be used as a reference indicator in a 2-factor model, though other reference indicators might be better because REDUCE is factorally complex)

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REDUCE

NEVHAPP e21

NEW_GOAL e31

IMPROVE e41

ACHIEVE e51

CONTENT e61

e11

Content

Achieve

1

1

When to add, when not to add parameters

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Modification Indices

Covariances: M.I.Par Change

e1 <--> Ach1 63.668 0.032e1 <--> Cont1 6.692 0.016e6 <--> Ach1 32.540 -0.023e5 <--> Cont1 4.370 0.012e5 <--> e6 13.033 -0.028e4 <--> e1 28.242 0.036e4 <--> e6 24.104 -0.034e3 <--> e1 4.500 0.012e2 <--> e1 5.440 0.016e2 <--> e6 5.290 -0.016e2 <--> e5 14.681 0.025e2 <--> e3 12.410 -0.017

REDUCE

NEVHAPP e21

NEW_GOAL e31

IMPROVE e41

ACHIEVE e51

CONTENT e61

e11

Content

Achieve

1

1

Discrepancy 125.260 0.000Degrees of freedom 8P 0.000 0.000

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REDUCE

NEVHAPP e21

NEW_GOAL e31

IMPROVE e41

ACHIEVE e51

CONTENT e61

e11

Content

Achieve

1

1

Regression Weights: M.I. Par Change

REDUCE <-- Ach1 52.853 0.406REDUCE <-- ACHIEVE 16.291 0.076REDUCE <-- IMPROVE 50.413 0.140REDUCE <-- NEW_GOAL 23.780 0.117CONTENT <-- Ach1 27.051 -0.293CONTENT <-- ACHIEVE 24.336 -0.094CONTENT <-- IMPROVE 31.694 -0.112ACHIEVE <-- REDUCE 4.791 0.033ACHIEVE <-- NEVHAPP 11.086 0.056IMPROVE <-- REDUCE 18.169 0.058IMPROVE <-- CONTENT 16.219 -0.053NEW_GOAL <-- NEVHAPP 6.137 -0.032NEVHAPP <-- REDUCE 4.031 0.029NEVHAPP <-- ACHIEVE 9.687 0.050NEVHAPP <-- NEW_GOAL 9.452 -0.063

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REDUCE

NEVHAPP

.26

e21

NEW_GOAL

.19

e31

IMPROVE

.17

e41

ACHIEVE

.35

e51

CONTENT

.40

e61

.37

e11

.17

Cont1

.07

Ach1

-.05

.99

1.08

1.00

1.03

1.76

1.00

.66

Choice to add or not to add parameter from Ach1 REDUCE

a matter of theoretical judgement. (Note changes in other parameters)

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Goodness of Fit Measures in Structural Equation Models

A Good Reference: Bollen and Long, TESTING STRUCTURAL EQUATION MODELS, Sage, 1993.

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Goodness of Fit Measures in Structural Equation Models

A fit measure expresses the difference between Σ(θ) and S. Using whatever metric it employs, it should register “perfect” whenever Σ(θ) = S exactly.

This occurs trivially when df=0

0 to 1 usually thought of as best metric (see Tanaka in Bollen & Long, 1993)

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Goodness of Fit Measures in Structural Equation Models

Early fit measures: Model Χ2 :

Asks the question, is there a statistically significant difference between S and Σ ?

If the answer to this question is “no”, we should definitely NOT try to add parameters to the model (capitalizing on change)

If the answer to this question is “yes”, we can cautiously add parameters Contemporary thinking is that we need some other

measure that is not sample-size dependent

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Goodness of Fit Measures in Structural Equation Models

Model Χ2 : X2 = (N-1) * Fml

Contemporary thinking is that we need some other measure that is not sample-size dependent

An issue in fit measures: “sample size dependency” (not considered a good thing)

Chi-square is very much sample size dependent (a direct function of N)

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Goodness of Fit Measures in Structural Equation Models

Model Χ2 : X2 = (N-1) * Fml

Contemporary thinking is that we need some other measure that is not sample-size dependent

An issue in fit measures: “sample size dependency” (not considered a good thing)

Chi-square is very much sample size dependent (a direct function of N)

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Goodness of Fit Measures in Structural Equation Models

Problem with Χ2 itself as a measure (aside from the fact that it is a direct function of N):

Logic of trying to “embrace” the null hypothesis. Even if chi-square not used, it IS important

as a “cut off” (never add parameters to a model when chi-square is non-signif.

Many measures are based on Χ2

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Goodness of Fit Measures in Structural Equation Models

The “first generation” fit measures: Jöreskog and Sörbom’s Goodness of Fit

Index (GFI) [LISREL] Bentler’s Normed Fit Index (NFI) [EQS]

These have now been supplemented in most software packages with a wide variety of fit measures

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Fit Measures

GFI = 1 – tr[Σ-1S – I]2

tr (Σ-1S)2

Takes on value from 0 to 1Conventional wisdom: .90 cutoffGFI tends to yield higher values than other

coefficientsGFI is affected by sample size, since in small

samples, we would expect larger differences between Σ and S even if the model is correct (sampling variation is larger)

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Fit Measures

GFI is an “absolute” fit measure

There are “incremental” fit measures that compare the model against some baseline.

- one such baseline is the “Independence Model

- Independence Model: models only the variances of manifest variables (no covariances) [=assumpt. all MVs independent] “Independence Model chi-square” (usually very large)

- Σ will have 0’s in the off-diagonals

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Fit Measures

NFI = (Χ2b-Χ2

m)/ Χ2b Normed Fit Index (Bentler)

(subscript b = baseline m=model)

Both NFI and GFI will increase as the number of model parameters increases and are affected by N (though not as a simple *N or *N-1 function).

GFI = widely used in earlier literature since it was the only measure (along with AGFI) available in LISREL

NFI (along with NNFI) only measure available in early versions of EQs

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Fit MeasuresThinking about fit indices:

Desirable properties:

1. Normed (esp. to 0 1)Some measures only approx: TLI

Arbitrary metric: AIC (Tanaka: AIC could be normed)

2. Not affected by sample size (GFI, NFI are)

3. “Penalty function” for extra parameters (no inherent advantage to complex models) – “Parsimony” indices deal with this

4. Consistent across estimation techniques (ML, GLS, other methods)

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Fit MeasuresBollens delta-2

(Χ2b – Χ2

m )/ Χ2b – dfm

RMR – root mean residual (only works with standardized residuals)

SRMR - standardized RMRParsimony GFI 2df/p * (p+1) * GFIAGFI = 1 – [1(q+1) / 2df ] [1 – GFI]RNI (Relative Noncentrality Index)

= [(Χ2b – dfb) – (X2

m- dfm)] / (Χ2b – dfb)

CFI = 1 – max[(X2m- dfm),0] / max[(X2

m- dfm), (X2 b- dfb),0]

RMSEA = sqrt (MAX[(X2m- dfm),(n-1),0) / dfm

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Fit Measures

Some debate on conventional .90 criterion for most of these measures

Hu & Bentler, SEM 6(1), 1999 suggest: • Use at least 2 measures• Use criterion of >.95 for 0-1 measure, <.06 for

RMSEA or SRMR

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SEM Assumptions

Fml estimator:

1. No Kurtosis

2. Covariance matrix analysed *

3. Large sample

4. H0: S = Σ(θ) holds exactly

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SEM Assumptions

Fml estimator:

1. Consistent

2. Asymptotically efficient

3. Scale invariant

4. Distribution approximately normal as N increases

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SEM Assumptions

Fml estimator:

Small Samples

1980s simulations:

- Not accurate N<50

- 100 + highly recommended

- “large sample” usually 200+

- in small samples, chi-square tends ot be too large

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Writing up results from Writing up results from Structural Equation ModelsStructural Equation Models

What to Report, What to OmitWhat to Report, What to Omit

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Writing up results from Structural Equation Writing up results from Structural Equation ModelsModels

Reference: Hoyle and Panter Reference: Hoyle and Panter chapter in Hoyle.chapter in Hoyle.

Important to note that there is a Important to note that there is a wide variety of reporting styles (no wide variety of reporting styles (no one “standard”).one “standard”).

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Writing up results from Structural Equation Writing up results from Structural Equation ModelsModels

A DiagramA Diagram Construct Equation ModelConstruct Equation Model Measurement Equation modelMeasurement Equation model

Some simplification may be required.Some simplification may be required.

Adding parameter estimates may clutter Adding parameter estimates may clutter (but for simple models helps with (but for simple models helps with reporting).reporting).

Alternatives exist (present matrices).Alternatives exist (present matrices).

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Reporting Structural Equation ModelsReporting Structural Equation Models

““Written explanation justifying each Written explanation justifying each path and each absence of a path” path and each absence of a path” (Hoyle and Panter)(Hoyle and Panter)

(just how much journal space is (just how much journal space is available here? )available here? )

It might make more sense to try to It might make more sense to try to identify potential controversies identify potential controversies (with respect to inclusion, (with respect to inclusion, exclusion).exclusion).

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Controversial paths?Controversial paths?

LV1

y11

1

y2

1

y3

1

LV21

111

e1

e2

LV320

y4

1

y5

1

y6

1

LV411

1

1

3333

What to report and what not to report…..What to report and what not to report…..

Present the details of the statistical Present the details of the statistical modelmodel

Clear indication of all free parametersClear indication of all free parameters Clear indication of all fixed parametersClear indication of all fixed parameters It should be possible for the It should be possible for the

reader to reproduce the model reader to reproduce the model

4.4. Describe the dataDescribe the data1.1. Correlations and standard errors (or Correlations and standard errors (or

covariances) for all variables ??covariances) for all variables ??

Round to 3-4 digits and not just 2 if you do Round to 3-4 digits and not just 2 if you do thisthis

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What to report and what not to report…What to report and what not to report…

4. Describing the data (continued)4. Describing the data (continued) Distributions of the dataDistributions of the data

Any variable highly skewed?Any variable highly skewed? Any variable only nominally continuous Any variable only nominally continuous

(i.e., 5-6 discrete values or less)?(i.e., 5-6 discrete values or less)? Report Mardia’s Kurtosis coefficient Report Mardia’s Kurtosis coefficient

(multivariate statistic)(multivariate statistic) Dummy exogenous variables, if anyDummy exogenous variables, if any

5. Estimation Method5. Estimation MethodIf the estimation method is not ML, report If the estimation method is not ML, report

ML results.ML results.

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What to report and what not to report…What to report and what not to report…

6. Treatment of Missing Data6. Treatment of Missing Data How big is the problem?How big is the problem? Treatment method used?Treatment method used?

Pretend there are no missing dataPretend there are no missing data Listwise deletionListwise deletion Pairwise deletionPairwise deletion FIML estimation (AMOS, LISREL >=8.5)FIML estimation (AMOS, LISREL >=8.5) Nearest neighbor imputation (LISREL Nearest neighbor imputation (LISREL

>=8.1)>=8.1) EM algorithm (covariance matrix EM algorithm (covariance matrix

imputation ) (LISREL >=8.5)imputation ) (LISREL >=8.5)

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What to report and what not to report…What to report and what not to report…

7. Fit criterion7. Fit criterion Hoyle and Panter suggest “.90; justify Hoyle and Panter suggest “.90; justify

if lower”.if lower”. Choice of indices also an issue.Choice of indices also an issue.

There appears to be “little consensus There appears to be “little consensus on the best index” (H & P recommend on the best index” (H & P recommend using multiple indices in presentations)using multiple indices in presentations)

Standards:Standards:

Bollen’s delta 2 (IFI)Bollen’s delta 2 (IFI)

Comparative Fit IndexComparative Fit Index

RMSEARMSEA

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Fit indicesFit indices

Older measures:Older measures: GFI (Joreskog & Sorbom)GFI (Joreskog & Sorbom) Bentler’s Normed Fit indexBentler’s Normed Fit index Model Chi-SquareModel Chi-Square

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What to report & what not to report….What to report & what not to report….

8. Alternative Models used for Nested Comparisons 8. Alternative Models used for Nested Comparisons (if (if appropriate)appropriate)

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1

1

US West

U.S. Midwest

U.S. Rust Belt

US South

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9. Plausible explanation for correlated errors9. Plausible explanation for correlated errors[“these things were just too darned big to ignore”][“these things were just too darned big to ignore”]

Generally assumed when working with panel model with Generally assumed when working with panel model with equivalent indicators across time:equivalent indicators across time:

1

1 1 1

1

1 1 1

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What to reportWhat to report

10. Interpretation of regression-based 10. Interpretation of regression-based modelmodel

Present standardized and Present standardized and unstandardized coefficients (usually)unstandardized coefficients (usually)

Standard errors? (* significance test Standard errors? (* significance test indicators?)indicators?)

R-square for equationsR-square for equations Measurement model too?Measurement model too? (expect higher R-squares) (expect higher R-squares)

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What to report.What to report.

Problems and issuesProblems and issues Negative error variances or other Negative error variances or other

reasons for non-singular parameter reasons for non-singular parameter covariance matricescovariance matrices

How dealt with? Does the final model How dealt with? Does the final model entail any “improper estimates”?entail any “improper estimates”?

Convergence difficulties, if anyConvergence difficulties, if any LISREL: can look at FLISREL: can look at Fmlml across values of across values of

given parameter, holding other parameters given parameter, holding other parameters constantconstant

Collinearity among exogenous Collinearity among exogenous variablesvariables

Factorially complex itemsFactorially complex items

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What to report & what not to report….What to report & what not to report….

General Model Limitations, Future General Model Limitations, Future Research issues:Research issues:

Where the number of available Where the number of available indicators compromised the modelindicators compromised the model

2-indicator variables? (any constraints 2-indicator variables? (any constraints required?)required?)

Single-indicator variables? (what Single-indicator variables? (what assumptions made about error variances?)assumptions made about error variances?)

Indicators not broadly representative of the Indicators not broadly representative of the construct being measured?construct being measured?

Where the distribution of data Where the distribution of data presented problems presented problems

Larger sample sizes can helpLarger sample sizes can help

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What to report & what not to report….What to report & what not to report….

General Model Limitations, Future General Model Limitations, Future Research issues:Research issues:

Missing data (extent of, etc.)Missing data (extent of, etc.) Cause-effect issues, if any (what Cause-effect issues, if any (what

constraints went into non-recursive constraints went into non-recursive model? How reasonable are these?)model? How reasonable are these?)

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

Manifest variables: X’s

Measurement errors: DELTA ( δ)

Coefficients in measurement equations: LAMBDA ( λ )

Sample equation:

X1 = λ1 ξ1+ δ1

MATRICES:

LAMBDA-x THETA-DELTA PHI

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

A slightly more complex example:

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

Labeling shown here applies ONLY if this matrix is specified as “diagonal”

Otherwise, the elements would be: Theta-delta 1, 2, 5, 9, 15.

OR, using double-subscript notation:

Theta-delta 1,1

Theta-delta 2,2

Theta-delta 3,3

Etc.

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

While this numbering is common in some journal articles, the LISREL program itself does not use it. Two subscript notations possible:

Single subscript Double subscript

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

Models with correlated measurement errors:

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

Measurement models for endogenous latent variables (ETA) are similar:

Manifest variables are Ys

Measurement error terms: EPSILON ( ε )

Coefficients in measurement equations: LAMBDA (λ)

• same as KSI/X side

•to differentiate, will sometimes refer to LAMBDAs as Lambda-Y (vs. Lambda-X)

Equations

Y1 = λ1 η 1+ ε1

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Matrix form: LISREL MEASUREMENT MODEL MATRICES

Measurement models for endogenous latent variables (ETA) are similar:

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LISREL MATRIX FORM

An Example:

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LISREL MATRIX FORM

An Example:

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LISREL MATRIX FORM

An Example:

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LISREL MATRIX FORM

An Example:

+ theta-epsilon, 8 x 8 matrix with parameters in diagonal and 0s in off diagonals (a “diagonal” matrix)

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Class Exercise

1

1

1

1

1

#1

Provide labels for each of the variables

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#2

1

1

1

1

1

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#1

delta

epsilon

ksieta

zeta

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#2

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Lisrel Matrices for examples.

No Beta Matrix in this model

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Lisrel Matrices for examples.

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Lisrel Matrices for examples (example #2)

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Lisrel Matrices for examples (example #2)