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# First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

## The asymmetry in

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 thesevere 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 rather than the negative side of the point γ 2 = 0.5 And there’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 thepreceding section. We then constructed a binary dependent variable Y , withthe 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 hypothesis957 times. This shows that the problem described in this paper36

Figure 3: Power of the normal likelihood ratio test versus normal Wald andweighted least squares, for the null hypothesis 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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