Coefficient of Determination

Represents the proportion of variance in the dependent variable that is accounted for by the independent variable(s). It is estimated by r², where r is the correlation (or multiple correlation) between the variables. When r² is multiplied by 100, one speaks of the percentage (rather than proportion) of variance accounted for in the dependent variable by the independent variable(s).

As an example, consider a Pearson correlation of .50. The corresponding coefficient of determination would equal .25 as a proportion and 25.0% as a percentage. Thus, a correlation of .50 implies that 25% of the variance in the dependent variable is shared or accounted for by the independent variable(s).

The theoretical range of the coefficient of determination is .00 to 1.0, however, in practice, the maximum range may be considered substantially less than 1.0, particularly in cases where the independent and dependent variables are differentially skewed (i.e., one positively skewed and one negatively skewed; see Dunlap, Burke, & Greer, 1995).

It will be noted that some have argued that correlation coefficients may be better interpreted as the coefficient of determination (Ozer, 1985), however, this contention does not appear to have been adopted by the wider research community. An exception would include research involving twins, where heritability coefficients (e.g., Falconer's formula) are in fact based on Pearson correlations, but they are interpreted as coefficients of determination (Falconer, 1996).

In contrast to the Pearson correlation, it may be considered inappropriate to calculate a coefficient of determination based on a Spearman correlation coefficient, as the concept of variance is less meaningful in data that are rank ordered.

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References

● Dunlap, W. P., Burke, M. J., Greer, T. (1995). The effect of skew on the magnitude of product-moment correlations. Journal of General Psychology, 122(4), 365-377.

● Ozer, D. J. (1985). Correlation and the coefficient of determination. Psychological Bulletin, 97(2), 307-315.

● Falconer, DS, MacKay TFC. Introduction to Quantitative Genetics, 4th Ed. 1996. Longmans Green, Harlow, Essex, UK.

Further reading

● Abelson, R. P. (1985). A variance explanation paradox: When a little is a lot. Psychological Bulletin, 97(1), 129-133.