# The consequences for the correlations

The reason for employing these definitions and their criteria becomes evident when examining the effect of the characteristics of the measurement model on the correlations between observed variables.

It can be shown from Figure 4.1 that the correlation between the observed variables r(y_{1j},y_{2j}) is equal to the joint effect of the variables that we want to measure (f_{1} and f_{2}), plus the spurious correlation due to the method factor [Sar84] as demonstrated in equation 4.1.

r(y_{1j},y_{2j}) = r_{1j}v_{1j} r(f_{1},f_{2})v_{2j}r_{2j} + r_{1j}m_{1j}m_{2j}r_{2j} | equation 4.1 |

or

r(y_{1j},y_{2j}) = q_{1j} r(f_{1},f_{2})q_{2j} + cmv_{12} | equation 4.2 |

where q_{ij} = r_{ij}v_{ij} and cmv_{12} = r_{1j}m_{1j}m_{2j}r_{2j}

Note that the equations show that, in general, the observed correlation is only equal to the correlation between the variables of interest (correlation corrected for measurement error), i.e. r(y_{1j},y_{2j}) = r(f_{1},f_{2}), when reliability and validity are equal to 1 and, consequently the method effects are zero. However, a situation in which there are no random errors (r_{ij}=1) is very unlikely.

Note also that r_{ij} and v_{ij}, which are always smaller than 1, will decrease the correlation (see the first term in equation 4.1: r_{1j}v_{1j} r(f_{1},f_{2})v_{2j}r_{2j}) while the method effects, if they are not zero, can generate an increase in the correlation (see the second term in equation 4.1: r_{1j}m_{1j}m_{2j}r_{2j}). This result suggests that it is possible that the low correlations for Methods 1 and 3 in Table 1.3 are due to the lower reliability of Methods 1 and 3 compared to Method 2. However, it is also possible that the correlations of Method 2 are higher because of common method variance for this method.

Before we leave this subject, we would like to mention that from equation 4.2 it immediately follows that:

r(f_{1}f_{2}) = (r(y_{1}y_{2}) - cmv_{12})/q_{1}q_{2} | equation 4.3 |

This means that, if we know the quality of the measures and we know the common method variance (cmv), then we can also correct the observed correlation for measurement error and obtain the correlation corrected for measurement error (r(f_{1}f_{2})). This would be the solution to the correction for measurement errors because, if the correlations are corrected, they can be used for the estimation of the coefficients of the regression and the causal models, as will be illustrated in the next three chapters. So, the problem will be solved if we know the qualities and the cmv. In the last chapter, we have seen that this information is provided by SQP. Thus, the procedure for correcting for measurement errors has, in principle, been specified. In the next chapters, we will show how this can be done in an easy and efficient way.

#### References

- [Sar84] Saris, W. E. and Stronkhorst, L. H. (1984). Causal modelling in nonexperimental research: an introduction to the LISREL approach.
*Sociometric Research Foundation*.