- Text
- Variables,
- Measurement,
- Regression,
- Parameter,
- Latent,
- Variable,
- Wald,
- Models,
- Likelihood,
- Squares,
- Draft,
- Toronto

First Draft of the paper - University of Toronto

Table 3: Estimated Type I error in 10,000 simulated data sets as a function**of** parameter configuration, base distribution, and sample size. Error term**of** **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 **the**base distribution is normal. Our purpose is to get an idea **of** how serious **the**problems might be when **the** convenient normal assumption — so close to**the** default settings for most s**of**tware — is violated.For **the** mild parameter configuration, where **the**re 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 **of**controlling 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** **the**methods based on normal likelihood.30

For **the** severe parameter configuration (with considerable measurementerror and a strong correlation between **the** latent independent variables), **the**likelihood ratio test is a bit conservative for smaller sample sizes with **the**normal, 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, while**the** Wald test can be unreliable for small to moderate sample sizes. Ofcourse, using a Wald test is immensely superior to ignoring measurementerror altoge**the**r.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 **the**simulations 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

- Page 1 and 2: Inflation of Type I error in multip
- Page 3 and 4: But if the independent variables ar
- Page 5 and 6: sion coefficients are different fro
- Page 7 and 8: and the model is not formally ident
- Page 9 and 10: X i,1 = ν 1 + ξ i,1 + δ i,1X i,2
- Page 11 and 12: the same direction, but if they hav
- Page 13 and 14: Thus we may manipulate the reliabil
- Page 15 and 16: 1.2.2 ResultsAgain, this is a compl
- Page 17 and 18: marized in Table 1.2.2, which shows
- Page 19 and 20: each value of γ 2 . For each data
- Page 21 and 22: estimation of it is a possibility.
- Page 23 and 24: Γ is an m × p matrix of unknown c
- Page 25 and 26: giving further thought to model ide
- Page 27 and 28: It is instructive to see how this w
- Page 29: We emphasize that the simulations r
- Page 33 and 34: In Table 4, using the base distribu
- Page 35 and 36: weighted least squares test for the
- Page 37 and 38: Figure 3: Power of the normal likel
- Page 39 and 40: measurement error, this fits neatly
- Page 41 and 42: We started with two correlated bina
- Page 43 and 44: Well-established solutions are avai
- Page 45 and 46: is that the client has data, and li
- Page 47 and 48: University of Wisconsin, Madison.Be
- Page 49: Robustness in the Analysis of Linea