COEFFICIENT OF DETERMINATION

One very convenient and useful way of interpreting the value of coefficient of correlation between two variables is to use the square of coefficient of correlation, which is called coefficient of determination. The coefficient of determination thus equal r² . If the value of r = 0.9, r² will be 0.81 and this would mean that 81 percent of the variation in the dependent variables has been explained by the independent variable. The maximum value of r² is unity because it is possible to explain all of the variation in Y, but it is not possible to explain more than all of it.

The coefficient of determination ( r_² ) is defined as the ration of the explained variance to the total variance.


The ratio of unexplained variance to total variance is frequently called the coefficient of non- determination. The coefficient of non-determination is denoted by K² and its square root is called the coefficient of alienation, or K. The K² and K values may also be used as the measure of the degree of relationship between two variables. For example, the higher the unexplained variance with respect to total variance, the higher will be the value of K² and the value of K. However, r² and r are more convenient in interpreting the result of correlation analysis.

It is much easier to understand the meaning of r² the r and, therefore, the coefficient of determination should be preferred in presenting the results of correlation analysis. Tuttle has beautifully pointed out that “the coefficient of correlation has been grossly overrated and is used entirely too much. Its square, the coefficient of determination, is a much more useful measure of the linear covariation of two variables. The reader should develop the habit of squaring every correlation coefficient he finds cited or stated before coming to any conclusion about the extent of the linear relationship between the two correlated variables.”

The relationship between r and r² may be noted --- as the value of r decreases from its maximum value of 1, the value of r² decreases much more rapidly. r will of course always be larger than r², unless r² = 0 or 1, when r = r² .

r              r2          r         r2

0.90       0.81    0.60    0.36

0.80       0.64    0.50    0.25

0.70       0.49    0.40    0.16

The coefficient of determination is a highly useful measure. However, it is often misinterpreted. The term itself may be misleading in that it implies the variable X stands in a determining or causal relationship to the variables Y. The statistical evidence itself never establishes the existence of such causality. All that the statistical evidence can do is to define covariation, that term being used in perfectly neutral sense. Whether causality is present or not and which way it runs if it is present, must be determined on the basis of evidence other than the quantitative observations.

However, r² is always a positive number. It cannot tell whether the relationship between the two variables is positive or negative. Thus the square root of r², i.e., √ r² = + r is frequently computed to indicate the direction of the relationship, in addition to indicating the degree of relationship. Since the range of r² is from 0 to 1, the coefficient of correlation r will vary within the range of √0 to √1 , or from + 1. The + (plus) sign of r will indicate positive correlation, whereas the – (minus) sign will mean a negative correlation.
 
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