Equal Precision Measurement Structural Models

Abstract

Error variance in structural models is often specified as a conditional variance associated with a manifest variable or as a regression path from a standardized error variance. This paper describes scenarios in which specification of both terms is useful: (1) Corrections for attenuation in single indicator factor models; (2) Tests of the equality in the proportion of explained variance across multiple dependent variables in regression models; (3) Factor models with two indicators; and (4) Tests of invariance in the proportion of measurement error variance across occasions in growth curve models. Examples of such models using both simulated and real-world data are presented.

Keywords:

• Attenuation correction
• confirmatory factor analysis
• growth curve models
• psychometrics
• regression
• structural equation modeling

Notes

1 Note, that with only three measurement occasions, it is not possible to add a Quadratic factor with loadings fixed to the square of the linear loadings as this would involve fitting ten parameters to the nine unique elements of the average sums of squares and cross products matrix in the data.

2 Some structural modeling software platforms, such as Ωnyx (von Oertzen et al., 2021) and Amos (Arbuckle, 2014), do not permit specification of non-linear constraints. Specification of equivalent equal precision models on such platforms involves the use of phantom variables (Rindskopf, 1984) with the precision parameter specified a common proportional standard deviation as shown in Figure 4 for the linear growth model in which factor loadings are fixed to elapsed time since the first measurement occasion.