Even outside the class of equivalent models to the

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Even outside the class of equivalent models to the given model, there may be many models that fit the data almost as well as the given model. SEM techniques can be used to distinguish sharply between models that fit the data very poorly and those that fit the data reasonably well . To discriminate further between models that fit the data almost equally well requires additional techniques. Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 21 / 37

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Model assessment and modification If the model is rejected, the problem is to determine what is wrong with the model, and if it is possible to modify the model in ways that fit the substantial theory so that the model fit the data better. This process is more complex than for path models and CFAs, as we now have two components of our model. ( I . A ) X = Λ x ξ + δ, ( I . B ) Y = Λ y η + ε, ( II ) η = B η + Γ ξ + ζ. (I) is the measurement model, and must be reasonable in order to get anywhere. This can be tested by CFA-techniques. Suppose (II) is wrong, but (I) is OK. The misspecified path model can in worst case completely distort the estimates of (I). This is called “interpretation confounding”, as the interpretation we think we get from the fitted model may be in part due only to the misspecified path model. Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 22 / 37
Model assessment and modification The path model only makes sense if the latent variables represent what they are supposed to. Interpretational confounding must be avoided. It is therefore recommended to first fit the measurement model of the SEM in a separate step, assess it and possibly modify it, and then fit the full model. The measurement model is ( I . A ) X = Λ x ξ + δ, ( I . B ) Y = Λ y η + ε, where no restrictions is placed on the covariance between ( ξ, η ) other than possibly a scaling. Hence, it is a CFA, where all latent variables are allowed to correlate. Some authors also recommend starting with an EFA, to assess if the discovered structure corresponds to theory. Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 23 / 37

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Model assessment and modification Any model modification must make sense from extra-statistical theory . We do not want to simply find the best fitting model in the sense of best reproducing the empirical covariance matrix. The goal is to quantify and understand the connections between theoretically interpretable latent variables. Theoretically unfounded model modifications may ruin the interpretation of the model. Steffen Grønneberg (BI) Lecture 12, GRA6036 31st March 2016 24 / 37
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 25 / 37

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Direct and indirect effects As for path models, we can use SEM to estimate direct, total and indirect effects amongst the variables. Stata’s syntax to get these estimates is the same as for path models.
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