Quantitative Results

The **coefficient of determination** is a statistical measurement that examines how differences in one variable can be explained by the difference in a second variable, when predicting the outcome of a given event.

- Values can range from 0.00 to 1.00, or 0 to 100%.
- In terms of regression analysis, the coefficient of determination is an overall measure of the accuracy of the regression model.
- In simple linear regression analysis, the calculation of this coefficient is to square the
*r*value between the two values, where*r*is the correlation coefficient. - In a multiple linear regression analysis,
*R*^{2}is known as the multiple correlation coefficient of determination. - It helps to describe how well a regression line fits (a.k.a., goodness of fit). An
*R*^{2}value of 0 indicates that the regression line does not fit the set of data points and a value of 1 indicates that the regression line perfectly fits the set of data points. - By definition,
*R*^{2}is calculated by one minus the Sum of Squares of Residuals (*SS*) divided by the Total Sum of Squares (_{error}*SS*):_{total}*R*^{2}= 1 – (*SS*)._{error }/ SS_{total} - In the case of a multiple linear regression, if the predictor variables are too correlated with one another (referred to as multicollinearity), this can cause the coefficient of determination to be higher in value.
- If, for whatever reason, there is multicollinearity in the regression model, the Adjusted
*R*Squared (Adjusted Coefficient of Determination) should be interpreted. The Adjusted*R*^{2}can take on negative values, but should always be less than or equal to the Coefficient of Determination.*Note*: The Adjusted*R*^{2}will only increase if more predictors variables are added to the regression model. - Inversely, the Coefficient of Non-Determination explains the amount of unexplained, or unaccounted for, variance between two variables, or between a set of variables (predictors) in an outcome variable. Where the Coefficient of Non-Determination is simply 1 –
*R*^{2}.

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