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First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

level.The simulations

level.The simulations were carried out exactly as for Table 4, so that the errorterms in the latent regression are multiples of standardized variates fromthe base distribution. A sample size of n = 250 was employed for the mildparameter configuration, while n = 1000 was necessary for all three teststo have an approximate 0.05 Type I error rate with the severe parameterconfiguration. For each curve in Figures 3, ten thousand simulated data setswere generated for each of eleven equally spaced γ 2 values, ranging from -0.5to +0.5, and a cubic spline was fit to the points to produce smooth curves.Note that the likelihood ratio tests and Wald tests are based on a normalmodel for all the curves, even though only the data in the top panel arenormal.In Figure 3, we see that the shapes of the power curves depend substantiallyupon the parameter configuration, but very little upon the basedistribution. The power curves of the three tests coincide almost exactly forthe mild parameter configuration, and it is noteworthy that the distributionfreetest based on weighted least squares does about as well as the likelihoodratio test with normal data.For the severe parameter configuration, with its strongly correlated latentindependent variables and substantial measurement error, the Wald andweighted least squares tests are biased; that is, the minimum probability ofrejecting the null hypothesis occurs at a parameter value for which the nullhypothesis is false. Compared to the normal Wald test, the weighted leastsquares test is clearly more powerful for the skewed and heavy tailed dataarising from a Pareto base distribution.The likelihood ratio test is unbiased, but still the power curve is notsymmetrical. There is a better chance of detecting the incorrectness of thenull hypothesis for parameter values that are negative. Apart inadmissiblylow power for small negative values of γ 2 , Figure 3 provides little basis forchoosing among the three tests. Their performance is equivalent for themild parameter configuration, while for the severe parameter configurationthe likelihood ratio test is more powerful against some alternatives, whilethe Wald and weighted least squares tests are more powerful against others.Recall from Tables 3 and 4, though, that the likelihood ratio test protectsmuch better against Type I error. And protection against Type I error isprimary, from both a theoretical and an applied point of view.Thus, a likelihood ratio test based upon the assumption of a multivariatenormal distribution appears to be practically superior to both the Wald and34

weighted least squares test for the case we are examining, regardless of thedistribution of the data. Of course simulations are much better at establishingthat something is wrong than they are at establishing that everythingis okay. Still, our intuition is that likelihood ratio tests based on the normalmodel are likely to work well for measurement error regression modelsin general. Though there are plenty of available methods (for example seeFuller, 1989), we still suggest that a normal likelihood ratio test should bethe practitioner’s first choice, regardless of the distribution of the data. Itis particularly convenient that normal likelihood methods are available in allthe commercial structural equation modelling software with which we arefamiliar, so it is convenient to actually perform the kind of analysis we arerecommending.It is worth noting that the robustness we observe for the normal-theorylikelihood ratio test under the marked kurtosis of the heavy-tailed t andPareto distributions goes beyond what one would expect based on the literature(for example Browne, 1984; Satorra and Bentler, 1990; Lee and Xia,2006).35

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