- Text
- Variables,
- Measurement,
- Regression,
- Parameter,
- Latent,
- Variable,
- Wald,
- Models,
- Likelihood,
- Squares,
- Draft,
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First Draft of the paper - University of Toronto

3 Modelling measurement errorThe clear implication **of** Section 1 is that measurement error in **the** independentvariables should be modelled, not ignored. Here, we present **the**simplest approach we know, using classical structural equation models **of** **the**sort described by Jöreskog (1978) and Bollen (1989). These models are quitegeneral; special cases include confirmatory factor analysis and path analysisas well as regression with measurement error.The classical structural equation models have no intercepts, and assumethat all independent variables and error terms have expected value zero.In practice, one centers all variables by subtracting **of**f **the** sample means,which for large samples is approximately **the** same as subtracting **of**f **the**population means. Since all **the** inference is asymptotic anyway, **the**re is noserious problem with this. We will discuss models with intercepts, and arguethat intercepts are **of**ten more trouble than **the**y are worth. A multivariatenormal assumption is common, but easy to relax.We prefer discuss **the**se relatively primitive methods (ra**the**r than thosedescribed, for example by Fuller, 1987) because **the** calculations are easier topresent to students and clients, and also because **the**y are close to **the** defaultsettings in widely available commercial s**of**tware for structural equationmodelling.3.1 Model identificationIn our experience, **the** greatest obstacle to using structural equation modelsin practice is that is is quite easy to come up with scientifically plausiblemodels that are not identified. Thus, to apply even **the** simplest structuralequation models to measurement error in regression, we need to discuss modelidentification.Suppose we have a vector **of** observable data D = (D 1 , . . . , D n ), and astatistical model (a set **of** assertions implying a probability distribution) forD, and this model depends on a parameter θ, which is usually a vector. If**the** probability distribution **of** **the** data corresponds uniquely to θ, **the**n wesay that **the** model is identified.It is possible for certain functions **of** **the** parameter vector to be identified,even when **the** entire model is not identified. If full knowledge **of** **the**probability distribution **of** **the** data implies knowledge **of** some function **of** **the**parameter vector, **the**n that function is said to be identified, and consistent20

estimation **of** it is a possibility. One example is **the** so-called “estimable functions”**of** **the** parameters **of** **the** over-parameterized linear models describedby Scheffé (1959).To show that a model is not identified, one need only produce two distinctparameter values that give rise to **the** same probability distribution. Forexample, let D 1 , . . . , D n be i.i.d. Poisson random variables with mean λ 1 +λ 2 ,where λ 1 > 0 and λ 1 > 0. The parameter is **the** pair θ = (λ 1 , λ 2 ). The modelis not identified because any pair **of** λ values satisfying λ 1 + λ 2 = c willproduce exactly **the** same probability distribution. Notice also how maximumlikelihood estimation will fail in this case; **the** likelihood function will havea ridge, a non-unique maximum along **the** line λ 1 + λ 2 = y. The functionλ = λ 1 + λ 2 , **of** course, is identified.For any statistical model, **the** probability distribution **of** **the** data is afunction **of** **the** parameter. If **the** parameter is also a function **of** **the** probabilitydistribution, **the**n **the** function is one-to-one and **the** model is identified.Now, in **the** classical structural equation models, D 1 , . . . , D n are i.i.d. multivariatenormal with mean zero, so that **the**ir joint probability distribution iscompletely determined by **the**ir common variance-covariance matrix. Followingstandard practice, we will denote this matrix by Σ = Σ(θ), where θ is avector **of** **the** model parameters. As **the** notation indicates, Σ is a function **of**θ. If it is also possible to solve for **the** elements **of** θ uniquely in terms **of** **the**elements **of** Σ so that θ is also a function **of** Σ, **the**n **the** structural equationmodel is identified. O**the**rwise, it is not.In Section 1, we gave a model for multiple regression with two independentvariables measured with error, represented by Equations (2). For simplicity,suppose that all **the** intercepts and expected values equal zero, all error termsare uncorrelated with **the** latent variables and with each o**the**r, and everythingis multivariate normal.We have D i = (Y i , X i,1 , X i,2 ), so that Σ has six unique elements. Theparameter θ has eight elements: γ 1 , γ 2 , ψ, three more for **the** unique elements**of** Φ, and two more for **the** error variances θ 1,1 and θ 2,2 . Attempting torecover **the**se eight parameter values from **the** six elements **of** Σ amounts tosolving six equations in eight unknowns. No unique solution is possible, andhence **the** model is not identified. We see from this simple example that for**the** kind **of** data set usually encountered in regression analysis, even a verymodest model for measurement error in **the** independent variables will notbe identified in general. How should we proceed?When a structural equation model is determined not to be identified (and21

- Page 1 and 2: Inflation of Type I error in multip
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