Correlation: Interpretations
While the coefficient of covariance has no upper and lower limits, the coefficient of correlation can vary from positive one (indicating a perfect positive relationship), through zero (indicating the absence of a relationship), to negative one (indicating a perfect negative relationship). As a rule of thumb, correlation coefficients between .00 and .30 are considered negligible, those between .30 and .70 are moderate and coefficients between .70 and 1.00 are considered high. However, this rule should be always qualified by the circumstances.
The Coefficient of Determination
The primary meaning of the coefficient of correlation lies in the amount of variation in one variable that is accounted for by the variable it is correlated with. To obtain this information, square the coefficient of correlation. The squared correlation coefficient is called the coefficient of determination. The magnitude of the coefficient of determination indicates the proportion of variance in one variable, explained from knowledge of the second variable. Multiplied by 100, this proportion of variance indicates the percentage of variance that is known, accounted for, determined. The coefficient of determination is the primary information measure within the general linear model. Correlation coefficients of .30 account for about 10 percent of the variance. Correlation of .70 explains about 50 percent of variance.
The Coefficient of Alienation
Complementing the coefficient of determination is the coefficient of alienation, also called the coefficient of non-determination. The term coefficient of alienation was coined by the late Fred Kerlinger who was fully aware that it conflicts with another, rather obscure statistical index. Since the term coefficient of alienation rather nicely complements the term coefficient of determination and is more concise, it is an acceptable alternative to the term ‘coefficient of non-determination.’ The coefficient of alienation equals one minus the coefficient of determination.
Perfect Positive Relationships
The range of coefficient of correlation is from -1 to 1. An example of a perfect positive relationship,
is shown below.
The angular separation of the linear approximation of the above scatterplot is 45 degrees from the abscissa of the Cartesian coordinate system.
Moderately Positive Relationships
Moderately positive relationships are staple findings within the social sciences. Often referred to as the positive manifold, they are the embodiment of the dictum that everything is related to everything else. An example of a moderate positive relationship is
where r equals .70. This value can be found at the bottom of the product of the z-score column. This moderately positive relationship is shown in the figure below.
The extent to which the variables depart from a perfect relationship is reflected by the slope of the linear approximation of their values departing from the 45 degrees of angular separation from the abscissa of the Cartesian coordinate system.
Negligible Relationships
An example of two unrelated variables is shown below as
The coefficient of correlation equals .00. The plot of data from the example illustrating a negligible relationship is shown below.
When the variables are unrelated, the linear approximation of their values is parallel to the abscissa of the Cartesian coordinate system. Observing this scatter-plot, it may appear as a diamond, but in reality approximates a circle. The ellipsoid scatter-plot, typical of moderate relationships begins to change into a circular scatter-plot when variables exhibit characteristic of very low or null relationships. This corresponds to the proportional increase of the scatter around the second axis of the ellipse, perpendicular to the main axis.
Moderately Negative Relationships
An example of a moderately negative relationship is presented below.
The correlation was computed equals -.70. As the relationship increases, as in our discussed progression from the zero relationship to a moderately negative one, it begins again resemble an ellipse, and can be observed in the figure shown below.
The scatter along the primary axis of an ellipse begins to diminish, as the scatter around the secondary axis increases. In the course of this change in the distribution of the scatter-plot, the former main axis of the ellipse will change into the secondary axis, and the former secondary axis will take on the character of the principal axis of the ellipse. Thus, the orientation of the ellipse will change.
Perfect Negative Relationships
An example of a data representing a perfect negative relationship is shown below.
Here the correlation equals to -1.00. As in the case of a perfect positive relationship, the scatter-plot of a prefect negative relationship forms a straight line, this time, however, with a different the orientation, as shown below.
These basic types of relationships, discussed in this chapter, are prototypes of a multitude of relationships that can be observed in the course of real analysis of data.
Summary
The angular separation of the linear approximations of the observed values from the abscissa of the Cartesian coordinate system is 0 degrees when there is no linear relationship between the variables. In the case of the perfect positive relationship, this angular separation is 45 degrees, and in the case of a perfect negative relationship, 135 degrees.