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An often cited advantage of Structural Equation Modeling (SEM) is that it is able to account for measurement error in the observed indicator variables, therefore allowing for consistent estimates in the presence of error-in-variables (in contrast to standard linear regression). It is not clear to me, however, what types of measurement error are accounted for (i.e., what the specific assumptions concerning this measurement error are).

  • What if we are not dealing with random measurement error (pure noise, unrelated to both Y and X), but some kind of systematic measurement error (e.g., self-report measures of dietary intake are systematically lower than the actual intake for those at the higher end of the spectrum)?
  • Do these two types of measurement error correspond to classical and non-classical measurement error?
  • What are the implications for parameter retrieval? I understood that measurement error usually leads to attenuation bias (coefficient biased towards zero), but in other books I read that the effect can also be overestimated. Under what circumstances would it possibly be overestimated instead of underestimated? Multiple predictors, relaxing assumptions,...?

Any explanations or referrals to clarifying textbook chapters would be greatly appreciated.

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It depends on your measurement design and model(s) what types of measurement "error" (systematic vs. unsystematic) you can account for and whether the different sources of "error" can be separated from one another. For example, to separate random error in self report measures of depression, you need at least two measures (indicators) of depression to separate the true score (reliable) variance from random measurement error variance. If you also wanted to isolate "error" (or "bias") due to self reports, you would need to additionally include multiple indicators of another method (e.g., spouse reports in addition to self reports). If you wanted to isolate/separate indicator- (or item-)specific effects, you would need to use a repeated measures (longitudinal) design in which the same multiple indicators (e.g., items or tests) are measured multiple times.

Statistical mediation analysis is an example where the effects of measurement error on parameter estimation can be complex. This is described, for example, in the following article:

Fritz, M. S., Kenny, D. A., & MacKinnon, D. P. (2016). The combined effects of measurement error and omitting confounders in the single-mediator model. Multivariate Behavioral Research, 51(5), 681-697.

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5166584/

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