Coefficient of determination
In a multiple linear model

Consider a linear model with more than a single explanatory variable, of the form

where, for the i th case, Y i {\displaystyle {Y_{i}}} is the response variable, X i, 1, …, X i, p {\displaystyle X_{i,1},\dots,X_{i,p}} are p regressors, and ε i {\displaystyle \varepsilon _{i}} is a mean zero error term. The quantities β 0, …, β p {\displaystyle \beta _{0},\dots,\beta _{p}} are unknown coefficients, whose values are estimated by least squares. The coefficient of determination R is a measure of the global fit of the model. Specifically, R is an element of and represents the proportion of variability in Y that may be attributed to some linear combination of the regressors (explanatory variables) in X.

R is often interpreted as the proportion of response variation "explained" by the regressors in the model. Thus, R = 1 indicates that the fitted model explains all variability in y {\displaystyle y}, while R = 0 indicates no 'linear' relationship (for straight line regression, this means that the straight line model is a constant line (slope = 0, intercept = y ¯ {\displaystyle {\bar {y}}}) between the response variable and regressors). An interior value such as R = 0.7 may be interpreted as follows: "Seventy percent of the variance in the response variable can be explained by the explanatory variables. The remaining thirty percent can be attributed to unknown, lurking variables or inherent variability."

A caution that applies to R, as to other statistical descriptions of correlation and association is that " correlation does not imply causation." In other words, while correlations may sometimes provide valuable clues in uncovering causal relationships among variables, a non-zero estimated correlation between two variables is not, on its own, evidence that changing the value of one variable would result in changes in the values of other variables. For example, the practice of carrying matches (or a lighter) is correlated with incidence of lung cancer, but carrying matches does not cause cancer (in the standard sense of "cause").

In case of a single regressor, fitted by least squares, R is the square of the Pearson product-moment correlation coefficient relating the regressor and the response variable. More generally, R is the square of the correlation between the constructed predictor and the response variable. With more than one regressor, the R can be referred to as the coefficient of multiple determination.
language:response_language[Italian], detectable_content:number_placeholders[3]