Coefficient of determination: meaning, use in decisions, and whether it is always positive

(i) Coefficient of determination. The coefficient of determination is the square of the correlation coefficient, r². It tells us what share of the total variation in the dependent variable is accounted for by its linear relationship with the independent variable: r² = explained variation / total variation, while 1 − r², the coefficient of non-determination, is the share left unexplained. The DU guideline minutes include it in the teaching of Unit 3. For example, if the correlation between a firm's monthly digital advertising and its online orders is 0.8, then r² = 0.64: 64% of the month-to-month variation in orders goes with advertising, and 36% is due to other factors.

Importance in statistical decision making.

1. A clear measure of strength. r = 0.5 may sound fair, but r² = 0.25 shows that only a quarter of the variation is explained. Being a percentage, r² is easier to read and compare than r.
2. A first check on forecasts. A high r² means that the regression line accounts for most of the variation in the data it was fitted to, so its estimates are more promising than those from a line with a low r²; a low r² warns that other factors must be studied first. A high r² is not by itself a guarantee that forecasts will come close to actual values: a correlation can be spurious, both variables being moved by a third factor such as time, and that influence has to be removed before the relationship is trusted. So a manager uses r² alongside a check that the relationship makes sense, before basing budgets or targets on it.
3. Choosing between explanations. When several possible drivers of sales are examined one at a time, the one with the highest r² explains the most variation in the data, which makes it the first candidate to look at; it becomes a basis for planning only if the link is a genuine one rather than spurious.

(ii) Always positive: do we agree? We agree, with one qualification.

• r² is the square of r, and it is also a ratio of two variations (sums of squares), neither of which can be negative. So r² can never be negative: it lies between 0 and 1, whether r itself is positive or negative. r = +0.7 and r = −0.7 both give r² = 0.49.
• Strictly, it is non-negative rather than always positive: r² = 0 when r = 0, that is, when there is no linear relationship at all, and r² = 1 when the correlation is perfect (r = ±1).
• Because the sign is lost in squaring, r² shows how much variation is explained but not the direction of the relationship; the direction must be read from r or from the regression coefficient.