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First Draft of the paper - University of Toronto

Table 1: Estimated Type I error rates when independent variables and measurementerrors are all normal, and reliability **of** X 1 and X 2 are both equalto 0.9025% **of** Variance in Y is Explained by ξ 1Correlation Between ξ 1 and ξ 2N 0.0 0.2 0.4 0.6 0.850 0.0476 † 0.0505 † 0.0636 0.0715 0.0913100 0.0504 † 0.0521 † 0.0834 0.0940 0.1294250 0.0467 † 0.0533 † 0.1402 0.1624 0.2544500 0.0468 † 0.0595 † 0.2300 0.2892 0.46491000 0.0505 † 0.0734 0.4094 0.5057 0.743150% **of** Variance in Y is Explained by ξ 1Correlation Between ξ 1 and ξ 2N 0.0 0.2 0.4 0.6 0.850 0.0460 † 0.0520 † 0.0963 0.1106 0.1633100 0.0535 † 0.0569 † 0.1461 0.1857 0.2837250 0.0483 † 0.0625 0.3068 0.3731 0.5864500 0.0515 † 0.0780 0.5323 0.6488 0.88371000 0.0481 † 0.1185 0.8273 0.9088 0.990775% **of** Variance in Y is Explained by ξ 1Correlation Between ξ 1 and ξ 2N 0.0 0.2 0.4 0.6 0.850 0.0485 † 0.0579 † 0.1727 0.2089 0.3442100 0.0541 † 0.0679 0.3101 0.3785 0.6031250 0.0479 † 0.0856 0.6450 0.7523 0.9434500 0.0445 † 0.1323 0.9109 0.9635 0.99921000 0.0522 † 0.2179 0.9959 0.9998 1.00000†Not Significantly different from 0.05, Bonferroni corrected for 7,500 tests.16

marized in Table 1.2.2, which shows marginal means from **the** six-dimensionaltable **of** estimated Type I error rates.The trends just described can be readily deduced from Expression 5 (**the**asymptotic bias). What **the** simulation shows is that **the** inflation **of** TypeI error rate is very serious for sample sizes that could easily be encounteredin practice. Note that **the** main source **of** difficulty is **the** combination **of**correlation among **the** latent (true) independent variables, and measurementerror in **the** variable for which one is attempting to control.2 Type III ErrorConsider Model (2) again. Let **the** covariance between ξ 1 and ξ 2 be positive,**the** partial relationship between ξ 1 and Y be positive, and **the** partial relationshipbetween ξ 2 and Y be negative. That is, φ 1,2 > 0, γ 1 > 0, and γ 2 < 0.Again, suppose we ignore measurement error and fit **the** ordinary normallinear model (1), taking X 1 and X 2 as surrogates for ξ 1 and ξ 2 respectively,and testing H 0 : β 2 = 0 instead **of** H 0 : γ 2 = 0. We now describe a simulationshowing how small negative values **of** γ 2 can be overwhelmed by **the** positiverelationships between ξ 1 and ξ 2 , and between ξ 1 and Y , leading to rejection**of** **the** null hypo**the**sis at a high rate, accompanied by a positive estimatedregression coefficient for X 2 .This is particularly unpleasant from a scientist’s perspective, because **the**reality is that for each value **of** **the** first independent variable, **the** secondindependent variable is negatively related to **the** dependent variable. Butapplication **of** **the** standard statistical tool leads to **the** conclusion that **the**relationship is positive – **the** direct opposite **of** **the** truth. Almost certainly,this will muddy **the** literature and interfere with **the** development **of** anyworthwhile scientific **the**ory.As in **the** first set **of** simulations, we set all expected values in Model (2)to zero except for **the** intercept α = 1, and also let θ 1,2 = 0, γ 1 = 1, andφ 1,1 = φ 2,2 = 1. We **the**n employed a standard normal base distribution,toge**the**r with a sample size and set **of** parameter values guaranteed to causeproblems with Type I error: n = 500, φ 1,2 = 0.90, ψ = 1 (so that ξ 3 1 explains0.75 **of** **the** variance in Y ), θ 1,1 = 1 (so that **the** reliability **of** X 1 is 0.50), andθ 2,2 = 1 (so that **the** reliability **of** X 19 2 is 0.95). Later in **the** **paper**, this iscalled **the** “severe” parameter configuration.We **the**n varied γ 2 from minus one to zero, generating 10,000 data sets for17

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