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# First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

## Table 1: Estimated Type

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 (theasymptotic 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 ofcorrelation 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 rejectionof the null hypothesis at a high rate, accompanied by a positive estimatedregression coefficient for X 2 .This is particularly unpleasant from a scientist’s perspective, because thereality 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 therelationship is positive – the direct opposite of the truth. Almost certainly,this will muddy the literature and interfere with the development of anyworthwhile scientific theory.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 then employed a standard normal base distribution,together 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 then varied γ 2 from minus one to zero, generating 10,000 data sets for17

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