Data was collected from 932 persons in two rural

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Data was collected from 932 persons in two rural regions in Illinois both in 1967 and 1971. The social scientists wanted to assess how socioeconomic status affects “attitude of alienation” in the two time periods, and how feeling of alienation in ’67 affects the feeling of alienation in ’71. We cannot observe “attitude of alienation”, but two psychological tests (the Anomia subscale and the Powerlessness subscale) were taken as indicators of alienation. Education (years of schooling completed) and Duncan’s Socioeconomic Index (SEI) were taken as indicators of the respondent’s socioeconomic status (Ses). Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 8 / 37
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An empirical example The researches’ theory was formulated through the following path-diagram Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 9 / 37
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An empirical example The researches’ theory was formulated through the following path-diagram Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 10 / 37
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An empirical example The researches’ theory was formulated through the following path-diagram Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 11 / 37
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An empirical example The researches’ theory was formulated through the following path-diagram EDUC & SEI measures SES. The two ANOMIA and POWERL measures felling of alienation in ’67 or ’71. For the latent path-model: SES is exogenous, and affects both (endogenous) alienation-variables. The feeling of the alienation in 67 affects the feeling of alienation in 71. Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 12 / 37
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SEM-theory Theory Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 13 / 37
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Contents 1 Finishing the R 2 -discussion 2 General SEM General SEMs: Introduction and diagram conventions An empirical example SEM-theory Statistical estimation of SEMs Model assessment and modification Direct and indirect effects An empirical example Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 14 / 37
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Statistical estimation of SEMs As in CFA, we work with unobservable variables, and must have substantial extra-statistical knowledge of the data to justify this. As in path models, SEM researchers wants to work with causal connections, again substantial extra-statistical knowledge must justify this. Once a model has been carefully formulated on the basis of substantial extra-statistical theory, it can be confronted with empirical data. How should we do this? Just as we did for CFA and path models. Suppose our data is mean zero multivariate Normally distributed. We observe independent copies of ( Y , X ) t , where ( I . A ) X = Λ x ξ + δ, ( I . B ) Y = Λ y η + ε, ( II ) η = B η + Γ ξ + ζ. Hence, ( Y , X ) t N ( 0 , Σ) . Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 15 / 37
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Statistical estimation of SEMs To use the MLE, we need the model implied covariance matrix Σ( θ ) , where θ = ( B , Γ , Λ X , Λ Y , Ψ , Θ Y , Θ X , Φ) Matrix calculations show that Σ( θ ) = Λ Y [ A - 1 (ΓΦΓ t + Ψ) A t t Y + Θ ε Λ Y A ΓΦΛ t X Λ X ΦΓ t A Λ t Y Λ X ΦΛ t X + Θ δ where A = ( I - B ) - 1 .
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