lecture6_s10 measurement - Consequences of measurement...

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Psychology 588: Covariance structure and factor models Feb 10, 2010 Consequences of measurement error
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Scaling indeterminacy of latent variables 2 • Scale of a latent variable is arbitrary and “determined” by a convention for convenience • Typically set to variance of one (factor analysis convention) or to be identical to an arbitrarily chosen indicator By centering indicator variables, we set latent variables’ means to zero • Consider the following transformation: * * , 1,. .., , , 0 jjj j j jj j xj J a b b a bb ν λξ δ ξ λ νλ δ =+ + = = + ⎛⎞ =− + + ⎜⎟ ⎝⎠
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• If all J indicators are considered simultaneously, vector notation is more convenient: * * , 1 ab a bb ξ ξξ =+ + ⎛⎞ =− + + ⎜⎟ ⎝⎠ x νλ δ λ δ meaning that the linear transformation of ξ can be exactly compensated in the accordingly transformed ν * = ν λ a / b and λ * = λ / b , leaving the errors δ unchanged (i.e., same fit)
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What’s great about measurement errors in equation 4 • Regression weights and correlations are interpreted, implicitly assuming that the “operationally defined” variables involve no measurement error --- hardly realized for theoretical constructs (e.g., self esteem, IQ, etc.) • Ignoring the measurement error will lead to inconsistent estimates • We will see consequences of ignoring measurement errors
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Univariate consequences 5 • Consider a mean-included equation for X (hours worked per week) to indicate ξ (achievement motivation): Given only one indicator per latent variable, the intercept and loading (i.e., weight) are simply scaling constants for ξ However, if the ξ scale is set comparable to the X scale (i.e., λ = 1 ), we see that var( X ) is an over-estimation of ϕ
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This note was uploaded on 06/11/2010 for the course PSYC 588 taught by Professor Sunjinghong during the Spring '10 term at University of Illinois at Urbana–Champaign.

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lecture6_s10 measurement - Consequences of measurement...

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