- Text
- Variables,
- Measurement,
- Regression,
- Parameter,
- Latent,
- Variable,
- Wald,
- Models,
- Likelihood,
- Squares,
- Draft,
- Toronto

First Draft of the paper - University of Toronto

Since Expression (5) for **the** asymptotic bias does not depend on any **of** **the**expected 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 **the**re 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 **the**1dependent 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 **the**ory (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** **the**error 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 revisiting**the** 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 **the**measurement error terms by multiplying standardized random variablesby constants to give **the**m **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. O**the**rwise,**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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