Measuring the way a dependent variable changes with changes in a qualitative independent variable
It is sometimes necessary to determine the change in one variable associated with changes in a qualitative independent factor; i.e., one which varies in ways that cannot be measured quantitatively. Thus if one is studying the effect of various factors affecting the values of individual farms, one might wish to include the kind of road on which the farm was located. Yet different kinds of roads, such as arterial highway, all-weather, gravel, or dirt, cannot be stated in the same way that the measurements for continuously variable factors can.
Measuring Simple Correlation with a Qualitative Variable. Where a single qualitative factor is to be considered as an independent variable, and a quantitative factor as the dependent, the regression relation may be determined by grouping the observations according to the category of the qualitative factor, and calculating the average value of the dependent factor for each group.
The intensity of correlation between the quantitative dependent factor and a qualitative independent factor is measured by the correlation ratio, which corresponds to the correlation coefficient or index in that it measures the proportion of the variation in the dependent factor explained by its association with the independent factor. Using n0 to represent the number of cases in each successive group according to the independent factor, and M0 to represent the mean of the dependent variable Y in each such group, the correlation ratio ηyx is defined