Correcting Correlation and Covariance Matrices for Measurement Errors Before Further Analysis

ABSTRACT

Although the approach to correct for measurement errors in research has been known since the early 1970s, most researchers in the social sciences appear to ignore these recommendations. One possible reason is that the models with latent variables that were originally suggested may be too difficult to apply in practice. We suggest an alternative but simpler procedure to correct for measurement errors. If one knows the sizes of the random errors and method effects for the different measures, the correlation and covariance matrices for the observed variables can be corrected. This approach has already been shown for simple concepts measured by a single measure. In this paper, we show that this approach can be generalized to studies that use a mixture of simple and complex concepts measured by several questions. Using these corrected matrices as the basis for the estimation, the parameters of the models are automatically corrected for measurement errors. The conditions for the quality of these procedures will also be discussed.

KEYWORDS:

• SEM
• MTMM
• SQP
• correction for measurement error
• composite scores

Notes

¹ The random error variable in the unstandardized variable Y is normally represented by e while e´ represents the error variable after the variables are standardized while the random error variable is not standardized and does not have a variance equal to 1.

² Equation (11)(11) ${\sigma }_{f_i{f}_j}={\sigma }_{f_i}{\rho }_{F_i{F}_j}{\sigma }_{f_j}$(11) seems to suggest that the covariance is only corrected for random errors, but because the computation is based on ${\rho }_{F_i{F}_j}$, we can see in Equation (10)(10) ${\rho }_{F_i{F}_j}=\frac{{\rho }_{Y_i{Y}_j}-m_i{m}_j}{q_i{q}_j}$(10) that it is also corrected for common method variance.

³ This coefficient (λ) is similar to the validity coefficient v, which represents the strength of the relationship between the latent indicator $F_i$ and its true score $T_i$. The latter coefficients we will now call the indicator validity coefficients to indicate the similarity of the two coefficients and also the difference. The indicator validity coefficient indicates the strength of the relationship for an indicator between the latent variable and its true score. The concept validity coefficient indicates the strength of the relationship between the latent indicator and the concept of interest.

⁴ An alternative approach is suggested by Hayduk et al. (2019) but in that case they use fixed values for the coefficients and are not correcting for measurement errors.