# For example t ˆ φ 1 2 φ 1 2 se ˆ φ 1 2 is

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For example, t = ˆ φ 1 , 2 - φ 1 , 2 SE ( ˆ φ 1 , 2 ) is approximately N ( 0 , 1 ) when n is large. The default standard error computed by Stata assumes that the data is Normal (but this can be adjusted) When using the MLE, normality needs to be checked. Although MLE is consistent, inference procedures are invalid for data with distributions far away from the multivariate Normal. In these cases, Quasi Maximum Likelihood procedures are needed, and are supported by Stata. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 9 / 54

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Inference, goodness of fit and modification indices Are the two ability factors positively correlated as expected? Is their correlation statistically significantly different from zero? Yes, the correlation (recall: covariance between standardized variables equals their correlation) is estimated to 56 % and is highly statistically significant. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 10 / 54
Inference, goodness of fit and modification indices As in logistic regression, we can use the log likelihood ratio-test to compare two nested models. Let L 1 be the maximized log likelihood of the largest model Let L 2 be the maximized log likelihood of the smaller model We know that L 1 L 2 and so Δ = 2 ( L 1 - L 2 ) is positive, and signifies how much the larger model manages to further increase the log likelihood. If Δ 0, it seems that we can be content with the smaller model. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 11 / 54

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Inference, goodness of fit and modification indices We can formally test H 0 : small model is correct versus H A : Large model is correct through the result that if H 0 is true, then Δ χ 2 df where df equals how many more parameters the bigger model has, compared to the small. The test has p-value = P ( χ 2 df > Δ) . Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 12 / 54
Inference, goodness of fit and modification indices For CFA, the most important log likelihood ratio test is H 0 : Postulated model is correct versus H A : A model with completely free covariance matrix is correct Both models assume X N ( 0 , Σ) , but with different assumptions on the structure of Σ . QML-tests also exist (i.e. they do not assume Normality but test the structural covariance assumptions). Clearly the postulated model implies a covariance matrix, so that the two models are nested. This means we can use the log-likelihood ratio as before, with Δ = 2 ( L 1 - L 2 ) and p-value = P ( χ 2 df > Δ) . Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 13 / 54

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Inference, goodness of fit and modification indices Because the model under H A places no restrictions on the covariance structure, this test is seen as a goodness-of-fit test. If our model passes the goodness-of-fit (GOF) test, it means there is no statistical difference between S and Σ( ˆ θ ) , indicating a correctly specified covariance structure. Note we simply assume Normality.
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