Hi. I'm currently trying to do sem on a model with a single indicator variable using mplus. based on what I read, I need to calculate the measurement error on my own.
θε = Var(y)(1 − ρy)
Var(y) is the variance of the observed indicator variable y, and 𝜌y is the reliability
of y.
Can anyone tell me how to calculate the Var(y) and 𝜌y?
Hello,
with a single indicator, you cannot estimate the measurment error. Instead you have to come up with a theoretically reasonable value. If you could manage to re-do the study (at least for a reasonable fraction the sample) you could use the re-test reliability at least as a basis. This however would equate the measurement error with the concept of "random error" inherent in estimates of reliability. An error in a model however, is more than that and signifies all other causal sources of the indcator (like the error in a structural model). Hence, the error has more to do with validity rather than reliability.
There are some discussions in the literature to come up with an error variance by using either expert knowledge on the indicator and its measurement quality, or intentionally fixing the error variance such that this reflects the *intented* conceptual gap between the indicator and the *supposed* latent variable. From this perspective you can actively control the conceptual distance and, thus, the meaning of the latent variable you capture (with small values implying a stronger conceptual similarity between indicator and latent vs. higher value decreasing the similarity).
Here are some paper
Alwin, D. F., Jackson, D. J., & Jackson, E. F. (1980). Measurement models for response errors in surveys: Issues and applications. Sociological Methodology, 11(1980), 68-119. doi:10.2307/270860
Duncan, O. D., Haller, A. O., & Portes, A. P. (1968). Peer influences on aspirations: A reinterpretation. American Journal of Sociology, 74(2), 119-137. doi:10.1086/224615
Hayduk, L. A., & Littvay, L. (2012). Should researchers use single indicators, best indicators, or multiple indicators. BMC Medical Research Methodology, 12(159), 1-17. doi:10.1186/1471-2288-12-159
Hayduk, L. a., Stratkotter, R. F., & Rovers, M. W. (1997). Sexual orientation and the willingness of catholic seminary students to conform to church teachings. Journal for the Scientific Study of Religion, 36(3), 455-467.
Oberski, D. L., & Satorra, A. (2013). Measurement error models with uncertainty about the error variance. Structural Equation Modeling: A Multidisciplinary Journal, 20(3), 409-428.
Onyskiw, J. E., & Hayduk, L. a. (2001). Processes underlying children's adjustment in families characterized by physical aggression. Family Relations, 50(4), 376-385. doi:10.1111/j.1741-3729.2001.00376.x
Savalei, V. (2019). A comparison of several approaches for controlling measurement error in small samples. Psychological Methods, 24(3), 352–370. doi:10.1037/met0000181
Best,
Holger
Agrandaiz Ramana Harahap What kind of variable are you dealing with? Is it a test or questionnaire sum score (composite) variable that represents an aggregate of multiple items? If so, you could distribute the items across two or more subscores ("item parcels") rather than computing a single sum score. That way, you would end up with multiple indicators. The error variances can then be identified and estimated as free model parameters.
With a single indicator, you would have to come up with a reasonable "guess" about reliability as mentioned by Holger Steinmetz . Sometimes, reliability estimates from previous studies using the same variable in the same population may be reasonable, but it is not always easy to ensure that such estimates apply to your study.
Holger Steinmetz The concept of "test-retest reliability" is based on very strong assumptions (1. tau equivalence or essential tau equivalence, 2. equal error variances, 3. uncorrelated error variables) which, in my experience are rarely met in practice. Especially (1) and (2) are often violated -- at least for social science variables. If a variable has at least some state-like properties (i.e., is reflecting situation variance in addition to trait/stable variance), the test-retest correlation will underestimate reliability and reflect mere stability, violating (1). Also, error variances are often unequal in longitudinal (test-retest) studies due to a Socratic effect (measurement error is often higher at Time 1 compared to subsequent time points), violating (2).
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