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First Draft of the paper - University of Toronto

First Draft of the paper - University of Toronto

a valid measurement

a valid measurement error regression, using commercially available software.Many statisticians may prefer to use SAS proc calis (SAS Institute Inc.,1999) because they probably have access to it already and the lineqs syntaxis straightforward. Others may prefer AMOS because of the graphical interface,while still others (particularly psychometricians) may prefer LISRELbecause of its long history and the wealth of instructional material that isavailable. In any case, the hard part is not really with understanding themethods or using the software; it’s with having the right kind of data.In Section 4, we present a more limited set of simulations in which structuralequation models are applied to various simulated data sets that employthe test-retest design. The purpose of these simulations is not to becomprehensive, but just to provide some practical guidance. We find thatnormal-theory likelihood ratio tests work well even when the data are notnormal, and that for smaller samples, they generally protect better againstType I error than either Wald tests or tests based on Browne’s (1984) robustweighted least-squares method (though these are far, far better than ignoringthe measurement error). For substantial amounts of measurement error andstrong correlations between the independent variables, we found that testsbased on weighted least squares were biased even for n = 1, 000, while thenormal-theory likelihood ratio test was unbiased; this held for non-normal aswell as normal data.Finally, we ask a rhetorical question. If if ignoring measurement error inregression has such awful consequences, and there is a perfectly satisfactoryalternative, why are we still teaching our students to do it? In consultingsituations, why are we still helping our clients do it? In our view, the onlyreason is inertia. It is time for a change.1 Inflation of Type I error rateTo see how badly things can go wrong, consider a multiple regression modelin which there are two independent variables, both measured with simple additiveerror. The LISREL-type notation (for example Jöreskog, 1978; Bollen,1989) is employed for compatibility with the discussion of structural equationmodels later in this paper.Independently for i = 1, . . . , n,Y i = α + γ 1 ξ i,1 + γ 2 ξ i,2 + ζ i (2)8

X i,1 = ν 1 + ξ i,1 + δ i,1X i,2 = ν 2 + ξ i,2 + δ i,2 ,where α, γ 1 and γ 2 are unknown constants (regression coefficients), andE[ξi,1ξ i,2]=[κ1κ 2][ ]ξi,1V arξ i,2= Φ =[ ]φ11 φ 12φ 12 φ 22E[δi,1δ i,2]=[ ] 00[ ]δi,1V arδ i,2= Θ =[ ]θ11 θ 12θ 12 θ 22(3)E[ζ i ] = 0 V ar[ζ i ] = ψ.The true independent variables are ξ i,1 and ξ i,2 , but they are latent variablesthat cannot be observed directly. They are independent of the error term ζ iand of the measurement errors δ i,1 and δ i,2 ; the error term is also independentof the measurement errors. The constants ν 1 and ν 2 represent measurementbias. For example, if ξ 1 is true average minutes of exercise per day and X 1 isreported average minutes of exercise, then ν 1 is the mean amount by whichpeople exaggerate their exercise times.Also, it is reasonable to assume that errors of measurement may be correlated.Again, suppose that ξ 1 is true amount of exercise and X 1 is reportedamount of exercise, while ξ 2 is true age and X 2 is reported age. It is naturalto imagine that adults who exaggerate how much they exercise mighttend to under-report their ages. Thus, the covariance parameter θ 12 is quitemeaningful.When a model such as (2) holds, all one can observe are the triples(X i,1 , X i,2 , Y i ) for i = 1, . . . , n. Suppose the interest is in testing whetherξ 2 is related to Y , conditionally on the value of ξ 1 . The natural mistake is totake X 1 as a surrogate for ξ 1 and X 2 as a surrogate for ξ 2 , fit the modelY i = β 0 + β 1 X i,1 + β 2 X i,2 + ɛ i (4)by ordinary least squares, and test the null hypothesis H 0 : β 2 = 0 as asubstitute for H 0 : γ 2 = 0, using the usual t or F -test.Suppose that in fact γ 2 = 0, so that conditionally upon the value of ξ 1 ,the dependent variable Y is independent of ξ 2 . It turns out that except underspecial circumstances, the least squares quantity ̂β 2 converges almost surely9

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