Linear Structural Equation Models

Linear Structural Equation Models Program: 18.04.05 25.04.05 02.05.05 09.05.05 23.05.05 30.05.05 06.06.05 13.06.05 20.06.05 27.06.05 04.07.05 11.07.05...
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Linear Structural Equation Models Program: 18.04.05 25.04.05 02.05.05 09.05.05 23.05.05 30.05.05 06.06.05 13.06.05 20.06.05 27.06.05 04.07.05 11.07.05 12.07.05

Introduction and Organization of the Course The Logic of Linear Modeling Before LISREL: Preparing data with PRELIS Using LISREL: SIMPLIS and LISREL-Language Simulation and Bootstrapping with LISREL a) Causality and Effect Decomposition b) Reliability, Validity and the Consequences of Measurement Errors Identification and Estimation Statistical Tests and Evaluation of Structural Equation Models Multi-Sample Analysis and Structured Means Coping with Ordinal Indicators Analyzing Panel Data Interaction and Non-Linearity Summary

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Introduction: Basic Ideas of LISREL-Models Starting Point: Multiple Regression Equation: Y1 = β0 + β1⋅X1+ β2⋅X2+ β3⋅X3+E1

Path diagram: X2 X1

Equation after subtracting means: (Y − y) = β1 ⋅ (X1 − x1 ) + β 2 ⋅ (X 2 − x 2 ) + β3 ⋅ (X 3 − x 3 ) + E1 Y1

LISREL notation: y1 = γ1 ⋅ x1 + γ 2 ⋅ x 2 + γ 3 ⋅ x 3 + ζ1

X3 What is different compared with OLS-regression: • Usually all variables are measured in deviations from their means • Most often all (dependent, independet and residual) variables are treated as random variables • Analysis is based on sufficient statistics (variances and covariances) and not on raw data

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Path Analysis: Simultaneous Analysis of a Set of Linear Relations: ζ2 (σ2 =ψ22) Y2 γ21 (σ2 = ϕ11)

X1

γ31

LISREL equations: y1 = γ11 ⋅ x1 + β12 ⋅ y 2 + β13 ⋅ y3 + ζ1

β12 γ11 β13

Y1

ζ1 (σ2 =ψ11)

y 2 = γ 21 ⋅ x1 + ζ 2 y3 = γ 31 ⋅ x1 + ζ 3

Y3 ζ3 (σ2 =ψ33) New Terms: An exogenous variable is not dependent in any equation; an endogenous variable is dependent in one equation but may be an independent variable in any other equation. Advantages of simultaneous analysis of a set of regression equations: • Effect decomposition: direct and indirect effects, • Evaluation of goodness-of fit between model and data if the model is over-identified. 3

Confirmatory Factor Analysis Linear Dependencies of Indicators (Observed Variables) on Factors (Latent Variables)

θδ32

δ1 (σ2 =θδ11)

x1

δ2 (σ2 =θδ22)

x2

λx21

x3

λx

δ3 (σ2 =θδ33) δ4 (σ2 =θδ44) δ5 (σ2 =θδ55) δ6

x4 x5

=θδ66)

x6

δ7 (σ2 =θδ77)

x7

(σ2

δ8 (σ2 =θδ88)

λx11

ξ2

λx42 λx53

ϕ11 ϕ21

32

ϕ22

ϕ21

ϕ32

λx

43

ξ3

ϕ33

ϕ41 ϕ42

ϕ43

λx63 λx74

x8

ξ1

ξ4

ϕ44

λx84 4

The Full LISREL-Model: Path Analysis of Latent Variables Controlling for Measurement Errors in Path Analysis

δ1(θδ11) δ2(θδ22)

x1

x2

λx11 λx

21

(ϕ11) ξ1

γ11 γ21 γ31

ζ1(ψ11) η1 β21 η2

x x 2 = λ 21 ⋅ ξ1 + δ2

λy

21

ζ2(ψ22) β23 η3 ζ3(ψ33)

Measurement Model for X: x x1 = λ11 ⋅ ξ1 + δ1

λy11

λy32 λy42 λy43 λy53

λy63

y1

ε1(θε11)

y2

ε2(θε22)

y3

ε3(θε33)

y4

ε4(θε44)

y5

ε5(θε55)

y6

ε6(θε66)

θε32

Measurement Model for Y: Structural Model: y1 = λ11y ⋅ η1 + ε1 η1 = γ11 ⋅ ξ1 + ζ1 y y 2 = λ 21 ⋅ η1 + ε 2 η2 = γ 21 ⋅ ξ1 + β21 ⋅ η1 + β23 ⋅ η3 + ζ 2 y y3 = λ 32 ⋅ η2 + ε3

η3 = γ 31 ⋅ ξ1 + ζ 3

y y y 4 = λ 42 ⋅ η2 + λ 43 ⋅ η3 + ε 4 y y5 = λ 53 ⋅ η3 + ε5 y y 6 = λ 63 ⋅ η3 + ε6

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The Process of Linear Causal Modeling

Starting Hypotheses

Data Collection & Preprocessing

model modification

Reality

Model Specification

Model Estimation

model does not fit

Model Evaluation

model fits the data 6

An example: Causal Chain from „Political Interest“ over „Trust in Politicians“ to „Impact on Politics“ Hypotheses: H1: The higher political interest, the higher is trust in politicians H2: The higher trust in politicians, the higher is the perceived impact on politics Formulation as a Path Diagram Impact Polint Politicn

Note: From the path diagram it follows that political interest affect perceived impact on politics only indirectly via trust in politicians, but has no direct effect. The indirect effect follows from the two hypotheses but the exclusion of the direct effect is not stated expliceitely. Very often hypotheses from theory give not enough information to draw a path diagram. Then additional assumptions/hypotheses have to be added in the process of formulation a linear structural equation model. 7

Empirical data in this example are the answers to three questions in a survey where each question is intended to measure one of the three variables of the model: Variables: Polint: Political Interest How strongly do you be interested in politics: very strongly (5), strongly (4), mean (3), less (2), not at all (1). Politicn: Trust in Politicians Politicians elected try to keep their promises from the campaign: agree strongly (5), agree (4), neither/nor (3), disagree (2), disagree strongly (1). Impact: Impact of Citizen In the average citizen have strong impact on politic: agree strongly (5), agree (4), neither/nor (3), disagree (2), disagree strongly (1). From the answers the variances and covariances of the the three variables are computed: Data: (Co-) Variances, n=2021 Impact Politicn Polint Impact 1.067 Politicn 0.290 1.108 Polint 0.106 0.034 0.975

Note that in LISREL names of variables have no more than 8 character

(Source: German General Social Survey (ALLBUS) 1996) 8

The SIMPLIS Command Language Specification of LISREL-Models in nearly natural Language Title line(s):

Example of the Logic of Linear Modeling

Observed variables Definition of Observed Variables IMPACT POLITICN POLINT Input data & Sample Size

Model Specification

Output Options

Last Command

Sample Size: 2021 Covariance matrix: 1.067 0.290 1.108 0.106 0.034 0.975 Relationships POLITICN = POLINT IMPACT = POLITICN Number of Decimals = 3 Print Residuals Path Diagram End of Problem

Data: (Co-) Variances, n=2021

Impact Politicn Polint

Impact 1.067 0.290 0.106

Politicn

Polint

1.108 0.034

0.975

Impact Polint Politicn

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Graphical Output (Path Diagram): IMPACT 0.97

0.99

POLINT 0.26 0.03

POLI TICN

1.11

Chi-Square=19.73, df=1, P-value=0.00001, RMSEA=0.096

SIMPLIS Output: IMPACT = 0.262*POLITICN, Errorvar.= 0.991 , R² = 0.0711 (0.0210) (0.0312) 12.435 31.773 POLITICN = 0.0349*POLINT, Errorvar.= 1.107 , R² = 0.00107 (0.0237) (0.0348) 1.471 31.773 10

Model Evaluation: The model does not fit: From the chi-square goodness-of-fit-test it follows that there is a probability of only 0.00001 that a chi-square value of 19.73 or greater with 1 degree of freedom will result if the postulated model fits perfect to the population variances and covariances and a simple random sample of 2021 cases is drawn from this population. The RMSEA of 0.096 suggests that even the close fit statistic is not well (>0.05). The bad fit is caused by the strong underestimation of the covariance between Polint and Impact. This can be seen by a comparison of the observed and implied (fitted) moments: Observed Covariance matrix Impact Politicn Polint Impact 1.067 Politicn 0.290 1.108 Polint 0.106 0.034 0.975

Fitted Covariance matrix Impact Politicn Polint Impact 1.067 Politicn 0.290 1.108 Polint 0.009 0.034 0.975

Residuals (Observed – Implied) Moments Impact Politicn Polint Impact 0.000 Politicn 0.000 0.000 Polint 0.097 0.000 0.000 11

The Process of Linear Causal Modeling: Model Modification

Starting Hypotheses

Data Collection & Preprocessing

model modification

Reality

Model Specification

Model Estimation

model does not fit

Model Evaluation

model fits the data 12

Model modification: Modified example model Observed variables: IMPACT POLITICN POLINT Sample Size: 2021 Covariance matrix: 1.067 0.290 1.108 0.106 0.034 0.975 POLITICN = POLINT IMPACT = POLITICN POLINT Path diagram Options: ND=3 RS End of problem Alternative specification: Alternatively modified model Observed variables: MPACT POLINT POLITICN Sample Size: 2021 Covariance matrix: 1.067 0.106 0.975 0.290 0.034 1.108 IMPACT = POLITICN POLINT Path diagram Options: ND=3 RS End of problem

Results (Standard Errors in Brackets)

0. 100 (0.022) 0.975 (0.031)

Polint 0. 035 (0.024)

0. 981 (0.031) 0. 259 (0.021) 1.107 Politicn (0.035) Impact

χ02 = 0 (P = 1.000)

0.975 (0.031)

Polint

0. 100 (0.022)

0.034 (0.023)

0. 981 (0.031)

Impact 1.108 (0.035)

Politicn

0. 259 (0.021)

χ02 = 0 (P = 1.000) 13

Question: What are the differences between the models from a substantive point of view ? Starting model 0. 991 (0.031) 0. 262 (0.021) 1.107 Politicn (0.035) Impact

0.975 (0.031)

Polint 0. 035 (0.024)

χ12 = 19.731 (P ≤ 0.001)

Model modification 0. 100 (0.022) 0.975 (0.031)

Polint 0. 035 (0.024)

Alternative model modification

0. 981 Impact (0.031) 0. 259 (0.021) 1.107 Politicn (0.035)

χ02 = 0 (P = 1.000)

0.975 (0.031)

Polint

0. 100 (0.022) Impact

0.034 (0.023) 1.108 (0.035)

Politicn

0. 259 (0.021)

χ02 = 0 (P = 1.000)

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0. 981 (0.031)

Interactive LISREL (LISREL 8.70) A free student version can be downloaded from „http://www.ssicentral.com“. In the student version the number of variables are limited. Additionally, reading data from spreat sheet programs and statistical software (via DBMS-module) is not feasible.

components: PRELIS LISREL-Preprocessor: • Reading ASCII-Data, SPSS system files and files from many software systems • Data screening, recoding, computations of means, variances and covariances, raw moments, different types of correlations including polychoric correlations and covariances based on unrestricted or restricted thresholds • estimation of variances and covariances of moments (asymptotic covariances) for ADF GLS-estimators • Different methods to cope with missing values • Exploratory data analysis, PC and EFA • OLS-, 2-Step-Least-Square, Logit- and Probit-regression and GLIM-models • Linear and non-linear multi-level analysis • Simulation und Bootstrapping

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Interactive LISREL (LISREL 8.70) LISREL

Program for specifying and estimating SEMs with latent variables • Confirmatory factor analysis • recursive and non-recursive path models (with latent variables) • Multi-level SEMs • Simultaneous group comparisions • SEMs with structured means (e.g. latent growth curves) • Different estimation methods (Normal Theory ML, ADF-GLS, robust ML, FIML with missing data, ...) • specification of non-linear restrictions on parameters • auxiliary parameters to specify arbitrary moment models • Export of path diagrams • Specification of SEMs using SIMPLIS, LISREL-Language, menues or by drawing a path diagram

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