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# First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

## Since Expression (5) for

Since Expression (5) for the asymptotic bias does not depend on any of theexpected value terms, we set all expected values to zero for the simulations,except for an arbitrary intercept α = 1 in the latent regression equation.Also, we let θ 1,2 = 0, so there is no correlation between the measurementerrors.This is a complete factorial experiment with six factors.1. Sample size: There were 5 values; n = 50, 100, 250, 500 and 1000.2. Correlation between latent (true) independent variables: Letting R 1 . R 2and R 3 be independent random variables with mean zero and varianceone, we calculatedξ 1 = √ 1 − φ 1,2 R 1 + √ φ 1,2 R 3 and (6)ξ 2 = √ 1 − φ 1,2 R 2 + √ φ 1,2 R 3 ,yielding V ar(ξ 1 ) = V ar(ξ 2 ) = 1 and a correlation of φ 1,2 between ξ 1and ξ 2 . A quiet but important feature of this construction is that whenφ 1,2 = 0, ξ 1 and ξ 2 are independent even when the distributions arenot normal. There were five correlation values in the simulation study:φ 1,2 = 0.00, 0.25, 0.75, 0.80 and 0.90.3. Variance explained by ξ 1 : With γ 1 = 1, γ 2 = 0 and V ar(ξ 1 ) = φ 1,1 =1, we have V ar(Y ) = 1 + ψ. So, the proportion of variance in the1dependent variable that comes from ξ 1 is . We used this as an1+ψindex of the strength of relationship between ξ 1 and Y , and adjustedit by changing the value of ψ. There were three values of explainedvariance, corresponding to a weak, moderate and strong relationshipbetween Y and ξ 1 : 0.25, 0.50 and 0.75.4. Reliability of X 1 : In psychometric theory (for example Lord and Novick,1968) the reliability of a test is the squared correlation between the observedscore and the true score. It is also the proportion of variancein the observed score that comes from the true score. From Model (2),we have[Corr(ξ 1 , X 1 )] 2 =[φ 1,1√φ1,1√φ1,1 + θ 1,1] 2=11 + θ 1,1.12

Thus we may manipulate the reliability by changing the value of theerror variance θ 1,1 . Five reliability values were employed, ranging fromlackluster to stellar: 0.50, 0.75, 0.80, 0.90 and 0.95.5. Reliability of X 2 : The same five values were used: 0.50, 0.75, 0.80, 0.90and 0.95.6. Base distribution: In all the simulations, the distribution of the errorsin the latent regression (ζ i ) are normal; we have no interest in revisitingthe consequences of violating the assumption of normal error inmultiple regression. But the distributions of the latent independentvariables and measurement errors are of interest. We constructed themeasurement error terms by multiplying standardized random variablesby constants to give them the desired variances. These standardizedrandom variables, and also the standardized variables R 1 . R 2 and R 3used to construct ξ 1 and ξ 2 – see Equations (6) – come from a commondistribution, which we call the “base” distribution. Four basedistributions were examined.• Standard normal• Student’s t with degrees of freedom 4.1, scaled to have unit variance.• Uniform on the interval (− √ 3, √ 3), yielding mean zero and varianceone.for x > 1) with α = 4.1, but stan-• Pareto (density f(x) =dardized.αx α+1The parameter value 4.1 for the t and Pareto was chosen so that fourthmoments would just barely exist, justifying the application of robustmethods later in the paper.1.2.1 Distributions and base distributionsBecause the simulated data values are linear combinations of standardizedrandom variables from the base distribution, the base distribution is the sameas the distribution of the simulated data only for the normal case. Otherwise,the independent variables (both latent and observed) are nameless linearcombinations that inherit some of the properties of the base distribution.13

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