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First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

Table 3: Estimated Type

Table 3: Estimated Type I error in 10,000 simulated data sets as a functionof parameter configuration, base distribution, and sample size. Error termof the latent regression is normal.Mild Parameter ConfigurationNormal Base Pareto Base T Base Uniform Basen LR Wald LR Wald LR Wald LR Wald50 0.0587 † 0.0547 0.0586 † 0.0526 0.0656 † 0.0603 † 0.0619 † 0.0586 †100 0.0574 0.0546 0.0556 0.0508 0.0560 0.0522 0.0510 0.0474250 0.0522 0.0483 0.0491 0.0452 0.0490 0.0459 0.0496 0.0453500 0.0490 0.0452 0.0522 0.0471 0.0494 0.0452 0.0483 0.04391000 0.0518 0.0474 0.0544 0.0495 0.0503 0.0450 0.0470 0.0432Severe Parameter ConfigurationNormal Base Pareto Base T Base Uniform Basen LR Wald LR Wald LR Wald LR Wald50 0.0327 † 0.1141 † 0.0476 0.1187 † 0.0351 † 0.1185 † 0.0295 † 0.1099 †100 0.0383 † 0.1597 † 0.0443 0.1323 † 0.0402 † 0.1509 † 0.0370 † 0.1612 †250 0.0513 0.1234 † 0.0534 0.1194 † 0.0494 0.1220 † 0.0473 0.1292 †500 0.0560 0.0742 † 0.0494 0.0725 † 0.0498 0.0714 † 0.0503 0.0689 †1000 0.0532 0.0569 0.0506 0.0563 0.0536 0.0565 0.0553 0.0553†Significantly different from 0.05 at the 0.05 level, Bonferroni corrected for 80 tests.This application of the normal model is formally justified only when thebase distribution is normal. Our purpose is to get an idea of how serious theproblems might be when the convenient normal assumption — so close tothe default settings for most software — is violated.For the mild parameter configuration, where there is not much measurementerror and not a very strong correlation between the latent independentvariables, both the likelihood ratio test and the Wald test do a good job ofcontrolling Type I error for sample sizes greater than 50; even for n = 50,departures from the Type I error rate of 0.05, while statistically significant,are not much of a practical problem. It is encouraging that this patternholds regardless of the base distribution, indicating some robustness of themethods based on normal likelihood.30

For the severe parameter configuration (with considerable measurementerror and a strong correlation between the latent independent variables), thelikelihood ratio test is a bit conservative for smaller sample sizes with thenormal, t and uniform base distributions; but the Wald test was subject toa Type I error rate distinctly greater than the putative 0.05 level, and didnot adequately protect against Type I error for n less than 1,000. This phenomenonwas observed for all four base distributions, including the normal.Table 3 suggests that in terms of controlling Type I error when the errorterm in the latent regression is normal, the likelihood ratio test based ona normal model does a good job regardless of the base distribution, whilethe Wald test can be unreliable for small to moderate sample sizes. Ofcourse, using a Wald test is immensely superior to ignoring measurementerror altogether.In Table 3, we kept the error term in the latent regression normal, foreasy comparison to the simulations in Section 1.2. In Table 4, we repeat thesimulations with the error terms coming from the base distribution, in anattempt make the normal likelihood methods misbehave. We also add testsbased on Browne’s (1984) weighted least squares method, which makes nodistributional assumptions beyond the existence of fourth moments.31

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