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

First Draft of the paper - University of Toronto

The asymmetry in **the** right panel **of** Figure 3 is interesting. Suspectingthat **the** direction **of** **the** asymmetry arises from **the** positive correlationbetween latent variables, we did one additional set **of** simulations with **the**severe parameter configuration and a normal base distribution, but this timewith a negative correlation between **the** latent independent variables. Figure4 shows **the** results, with **the** Wald and weighted least squares tests biasedin **the** opposite direction from Figure 3, and **the** likelihood ratio test havinglower power on **the** positive ra**the**r than **the** negative side **of** **the** point γ 2 = 0.5 And **the**re’s moreThe main point **of** this **paper** is that when an independent variable is measuredwith error and we try to control for that independent variable withouttaking **the** measurement error into account, **the** “control” will be incomplete,and **the** result can be a drastic inflation **of** Type I error. We have illustratedthis for **the** normal linear model and simple additive measurementerror, but **the** problem is much more general. We would like to suggest thatregardless **of** **the** type **of** measurement error and regardless **of** **the** statisticalmethod used, ignoring measurement error in **the** independent variablescan seriously inflate Type I error. We will now support this assertion byreferences to **the** literature, supplemented by a collection **of** quick, smallscalesimulations. All **the** simulations in this section were carried out usingR Version 2.1.1 (R Development Core Team, 2006). Code is available atwww.utstat.toronto.edu/~brunner/MeasurementError.Logistic regression with additive measurement error In this smallsimulation, we constructed data sets with a pair **of** latent independent variablesξ 1 and ξ 2 , and also corresponding manifest variables X 1 and X 2 usinga normal base distribution and **the** “severe” parameter configuration **of** **the**preceding section. We **the**n constructed a binary dependent variable Y , with**the** log odds **of** Y = 1 equal to γ 0 +γ 1 ξ 1 +γ 2 ξ 2 , where γ 0 = γ 1 = 1 and γ 2 = 0.Ignoring **the** measurement error, we fit a standard logistic regression modelin which **the** log odds **of** Y = 1 equals β 0 +β 1 X 1 +β 2 X 2 , and used a likelihoodratio test **of** H 0 : β 2 = 0 as a surrogate for H 0 : γ 2 = 0. The parallel to whatwe did with ordinary least squares regression should be clear.In 1,000 simulations with n = 250, we incorrectly rejected **the** null hypo**the**sis957 times. This shows that **the** problem described in this **paper**36

Figure 3: Power **of** **the** normal likelihood ratio test versus normal Wald andweighted least squares, for **the** null hypo**the**sis γ 2 = 0Parameter ConfigurationBase Distribution Mild, n = 250 Severe, n = 1, 000NormalPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLSPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLS−0.4 −0.2 0.0 0.2 0.4−0.4 −0.2 0.0 0.2 0.4γ 2γ 2ParetoPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLSPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLS−0.4 −0.2 0.0 0.2 0.4−0.4 −0.2 0.0 0.2 0.4γ 2γ 2Student’s tPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLSPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLS−0.4 −0.2 0.0 0.2 0.4−0.4 −0.2 0.0 0.2 0.4γ 2γ 2UniformPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLSPower0.0 0.2 0.4 0.6 0.8 1.0LRWaldWLS−0.4 −0.2 0.0 0.2 0.4−0.4 −0.2 0.0 0.2 0.4γ 2γ 237

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