# 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’ (f_{1}) and ‘satisfaction with the economy’ (f_{2}). The measurement model for two variables of interest is presented in Figure 4.1.

In this model it is assumed that:

- f
_{i}is the trait/factor i of interest measured by a direct question. - y
_{ij}is the observed variable (for trait i measured by method j). - t
_{ij}is the ‘true score’ of the response variable y_{ij}. - M
_{j}is the method factor that represents a specific reaction of respondents to a method and therefore generates a systematic error. - e
_{ij}is the random measurement error term for y_{ij}.

Furthermore, from Chapter 2 we already know that:

The r_{ij} 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 (y_{ij}) and the true score (t_{ij}), which is r_{ij}^{2}.

The v_{ij} 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 (f_{i}) and the true score (t_{ij}), which is v_{ij}^{2}.

The m_{ij} 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, m_{ij}^{2} = 1 – v_{ij}^{2}, 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 (M_{j}) and the true score (t_{ij}) resulting in m_{ij}^{2}. The contribution of the method to the correlations, called **common method variance** or **cmv**, is equal to r_{1j}m_{1j}m_{2j}r_{2j}.

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 (r_{ij}v_{ij})^{2}.