Coefficient 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 termed as coefficient of determination. The coefficient of determination thus equals to r². If the value of r = 0.9,  r² will be 0.81 and this would mean that the 81 per cent of the variation of the maximum value of r2 is unity as 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 (r²) is termed as the ratio of the explained variance to the total variance.

Coefficient of determination = explained variation/ total variance

The ratio of the explained variance to total variance is frequently known as  coefficient of non- determination. The coefficient of non- determination is represented by K² and its square root is known as the coefficient of alienation, or K. The K and K2 values may also be used as the measure of the degree or relationship between 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 the correlation analysis.

It is much easier to understand the meaning or r² than r and therefore, the coefficient of determination must  be preferred in presenting the result of correlation has been grossly overrated and is used entirely too much. Its square, the coefficient of determination, is much more useful measure of the linear covariation of two variables, the reader must develop the habit of squaring every correlation coefficient he finds or stated before coming to any conclusion about the extent of the linear relationship between the two correlated variables.

The relationship b/w r and r² may be noted as the value of r decreases from its maximum value of 1. The value of r² reduces much more rapidly r will of course always be larger than r² unless r² = 0 or 1.0 when r = r².

And, hence the coefficient of correlation is 0.707 when just half the variance in y is due to X.

It must be clearly noted that the fact that a correlation between two variables has a value of r =0.60 and the correlation between two other variables has a value of r = 0.30 does not demonstrate that the first correlation is twice as strong as the second/ the relationship between the two given values of r can better be understood by computing the value of r². When r =0.6, r² = 0.36 and when r =0.30, r² = 0.09. This implies that, in the first case 36% of the total variation is described whereas in the second case 9% of the total variation is explained.

The coefficient of determination is a highly useful measure. However it is frequently misinterpreted. The term itself may be misleading in that it implies that the variable x stands in determining or causal relationship to the variable y. The statistical evidence itself never establishes the existence of such causality. All that the statistical evidence can do is to define the conversation that the term being used in a perfectly neutral sense. Whether causality is present or not & 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 and it cannot tell whether the relationship between the two variables is positive or negative. Thus the square root of r², R² = ± r is frequently computed to point the direction of the relationship in addition to indicate the degree of relationship. As the range of r² is form 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 indict positive correlation whereas the (minus) sign will mean a negative correlation.