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# First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

## Table 2: Marginal Mean

Table 2: Marginal Mean Estimated Type I Error RatesSample Size50 100 250 500 10000.1908 0.2744 0.3946 0.4834 0.5651Correlation Between ξ 1 and ξ 20.0 0.2 0.4 0.6 0.80.0500 0.1660 0.5154 0.5505 0.6262Proportion of Variance Explained by ξ 10.25 0.50 0.750.2733 0.3847 0.4869Reliability of X 10.50 0.75 0.80 0.90 0.950.6064 0.4698 0.4207 0.2669 0.1445Reliability of X 20.50 0.75 0.80 0.90 0.950.3081 0.3751 0.3875 0.4125 0.4250Base DistributionNormal Pareto Student’s t Uniform0.3869 0.3690 0.3831 0.387518

each value of γ 2 . For each data set, we fit Model (1) and tested H 0 : β 2 = 0at α = 0.05 with the usual F -test. Each test was classified as significant witĥβ 2 > 0, significant with ̂β 2 < 0, or nonsignificant.Figure 2 shows the results. For substantial negative values of γ 2 , the nullhypothesis H 0 : β 2 = 0 is rejected at a high rate with ̂β 2 < 0, leading tothe correct conclusion even though the model is wrong. As the value of γ 2increases, the proportion of significant tests decreases to near zero aroundγ 2 = −0.76 Then for values of γ 2 closer to zero (but still negative), the nullhypothesis is increasingly rejected again, but this time with ̂β 2 > 0, leadingto the conclusion of a positive relationship, when in fact the relationshipwas negative. This example shows how ignoring measurement error in theindependent variables can lead to firm conclusions that are directly oppositeto reality.Figure 2: Probability of Rejecting H 0 : β 2 = 0Probability0.0 0.2 0.4 0.6 0.8 1.0With β^2 > 0With β^2 < 0−1.0 −0.8 −0.6 −0.4 −0.2 0.0γ 219

draft - Toronto and Region Conservation Authority
draft - Toronto and Region Conservation Authority
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