Correction of the correlations for measurement errors
In order to illustrate how to correct the correlations for measurement errors, we extend the model in Figure 2.2 to two variables of interest (f), for example ‘satisfaction with the government’ (f1) and ‘satisfaction with the economy’ (f2). The measurement model for two variables of interest is presented in Figure 4.1.
In this model it is assumed that:
- fi is the trait/factor i of interest measured by a direct question.
- yij is the observed variable (for trait i measured by method j).
- tij is the ‘true score’ of the response variable yij.
- Mj is the method factor that represents a specific reaction of respondents to a method and therefore generates a systematic error.
- eij is the random measurement error term for yij.
Furthermore, from Chapter 2 we already know that:
The rij coefficients represent the standardized effects of the true scores on the observed scores. This effect is smaller if the random errors are larger. This coefficient is called the reliability coefficient. Reliability is defined as the strength of the relationship between the observed response (yij) and the true score (tij), which is rij2.
The vij coefficients represent the standardized effects of the variables of interest on the true scores for the observed variables that are really measured. Therefore, this coefficient is called the validity coefficient. Validity is defined as the strength of the relationship between the variable of interest (fi) and the true score (tij), which is vij2.
The mij coefficients represent the standardized effects of the method factor on the true scores, called the method effect. An increase in the method effect results in a decrease in validity and vice versa. It can be shown that, for this model, mij2 = 1 – vij2, and the method effect is therefore equal to the invalidity due to the method used. The systematic method effect is the strength of the relationship between the method factor (Mj) and the true score (tij) resulting in mij2. The contribution of the method to the correlations, called common method variance or cmv, is equal to r1jm1jm2jr2j.
The total quality of a measure is defined as the strength of the relationship between the observed variable and the variable of interest, that is (rijvij)2.