# Moral report confidence intervals and not merely

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Moral: report confidence intervals and not merely reject/non-reject when n is large. This also applies for χ 2 -tests. Just as we have little interest in effects which are practically very close to zero, the model may be sufficiently close to the truth to be useful. Unfortunately, the value “Chi-Square” reported by Stata is not comparable across models. In order to get an intuitive feel for what a practically insignificant deviation is, we need to rescale the Chi-Square observation. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 26 / 54

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An empirical example: Mental abilities A common approach (with also a deeper technical motivation) is the Root Mean Square Error of Approximation (RMSEA) RMSEA = r - df ) / ( n - 1 ) df where Δ is the log-likelihood ratio test statistic. The RMSEA is scaled so that it is comparable across models. It’s been suggested that 0.01, 0.05, and 0.08 indicates excellent, good, and mediocre fit respectively. RMSEA should be less than 0 . 10. Stata computes the RMSEA, confidence intervals and a p -value to test RMSEA < 0 . 05 by the command estat gof, stats(rmsea) (directly after the estimation of a CFA). Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 27 / 54
An empirical example: Mental abilities Here, the RMSEA does not indicate an approximate fit. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 28 / 54

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An empirical example: Mental abilities If the model does not pass the Chi-square goodness of fit test, and the RMSEA does not indicate a close fit, we can use the modification indices to identify places where the model can be improved. However: A CFA is built on substantial theoretical knowledge of the data, and any modification must make sense with theoretical knowledge . Adding parameters to the model only on the basis of improving fit, breaks the interpretation of the model as a representation of a substantial theory and is not a valid statistical procedure for CFA. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 29 / 54
An empirical example: Mental abilities We do not observe the factor variables, so we cannot study residuals in the same way as in linear regression. As a CFA is a covariance model, and tries to match up the model implied covariance matrix Σ( θ ) with the empirical covariance matrix, a useful concept of residuals is studying the elements of S - Σ( ˆ θ ) . Problem: Just what is a big residual? Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 30 / 54

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An empirical example: Mental abilities Stata can also compute standardized residuals, equal to S - Σ( ˆ θ ) , but where each element is divided by its estimated standard deviation. The standardized residuals should be between - 1 . 96 and 1 . 96, and not outside [ - 3 , 3 ] . We clearly see that the residual between SCCAPS and VISPERS is large, and SCCAPS is responsible for most large values. Steffen Grønneberg (BI) Lecture 11, GRA6036 17th March 2016 31 / 54
Recall that the Modification indices are the approximate drop in Chi-Square if the suggested parameter is freed.

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An empirical example: Mental abilities Error-terms can be correlated, which corresponds to taking account that unaccounted for factors may be related. This is
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