Measurement Error Models With Uncertainty About the Error Variance

Abstract

It is well known that measurement error in observable variables induces bias in estimates in standard regression analysis and that structural equation models are a typical solution to this problem. Often, multiple indicator equations are subsumed as part of the structural equation model, allowing for consistent estimation of the relevant regression parameters. In many instances, however, embedding the measurement model into structural equation models is not possible because the model would not be identified. To correct for measurement error one has no other recourse than to provide the exact values of the variances of the measurement error terms of the model, although in practice such variances cannot be ascertained exactly, but only estimated from an independent study. The usual approach so far has been to treat the estimated values of error variances as if they were known exact population values in the subsequent structural equation modeling (SEM) analysis. In this article we show that fixing measurement error variance estimates as if they were true values can make the reported standard errors of the structural parameters of the model smaller than they should be. Inferences about the parameters of interest will be incorrect if the estimated nature of the variances is not taken into account. For general SEM, we derive an explicit expression that provides the terms to be added to the standard errors provided by the standard SEM software that treats the estimated variances as exact population values. Interestingly, we find there is a differential impact of the corrections to be added to the standard errors depending on which parameter of the model is estimated. The theoretical results are illustrated with simulations and also with empirical data on a typical SEM model.

Keywords:

• correction for measurement error
• error variance
• fixed parameters
• measurement error
• reliability
• standard errors
• standard error of reliability
• structural equation modeling

Notes

¹For the full questionnaire we refer to http://ess.nsd.uib.no/ess/round4/fieldwork.html

²The error variances and reliabilities were obtained by first fitting a confirmatory factor model to the indicators of these constructs. The error variance and reliability of the simple sum score were then obtained by adding “ghost variables” to the model. The standard errors of these quantities were obtained by bootstrapping (Raykov, 2009).

³We have used the OpenMx package in R (Bijer et al., 2010; R Development Core Team, 2010), and LISREL (Jöreskog & Sörbom, 1996) to double-check the results.

⁴All analyses were carried out using R versions 2.10 and 2.11 for Linux and the OpenMx package (Boker et al., 2010; R Development Core Team, 2010). The R code is available from the first author on request.

⁵In calculating these standard errors the estimation of a covariance matrix from a sample of size 1,500 was also taken into account.

⁶Full details of the Monte Carlo results are available on request from the first author.

⁷An implementation for the SEM package OpenMx in R is available on request from the first author.