What is a Coefficient of Determination?

The coefficient of determination (R squared), is defined as the proportion of the variance in the dependent variable that is predictable from the independent variable(s). It is a statistic that indicates the percentage of the change taking place in the dependent variable that can be explained by the change in the independent variable(s). The coefficient of determination is generally used for analyzing how changes in one variable can be explained by a change in a second variable. The coefficient of determination is the square of the correlation between forecasted y scores and actual y scores. Therefore, it falls between the range of 0 to 1.

How is the Coefficient of Determination Used?

The correlation of determination is either used for estimating future outcomes or testing of hypotheses, based on other given related information. It provides a measure of how outcomes that have been carefully monitored can be copied by the model, based on the proportion of total variation of outcomes elaborated by the model. There are cases where R-squared may produce negative values. This can occur when the forecasts that are being compared to the corresponding outcomes have not been deduced from a model-fitting method using those data. In the case where a model-fitting method has been used, R-squared may still be negative. For instance, when linear regression is conducted without including an intercept.

Advantages of Analyzing the Coefficient of Determination

In all scenarios where the coefficient of determination is used, the predictors are computed by the ordinary least-squares regression. In this situation, R-squared rises as the number of variables in the model rises. This indicates a disadvantage to one possible use of R-squared, where one might keep raising the number of variables to raise the R-squared value. R-squared lacks the ability to tell whether there is bias in the method used in predicting the data points. It also doesn't indicate whether the R-squared value is good or bad. For example, a low R-squared cannot be generally termed bad. It is up to the analyst to decide if the R-squared number is good or bad. Interpretation of the coefficient of determination. It is important to always be conscious about how the coefficient of determination is interpreted, as it should not be interpreted naively.

• An R-squared of 0 indicates that the dependent variable cannot be forecasted from the independent variable(s).
• An R-squared of 1 indicates the dependent variable can be forecasted without error from the independent variable(s).
• An R-squared that falls between 0 - 1 indicates the extent to which the dependent variable is predictable.

For example, an R-squared of 0.10 indicates that 10 percent of the variance in Y is forecasted from X.