Coefficient of Determination (r²)

The coefficient of determination (r²) is a measure of how well a linear model fits a set of data. It is a squared value between 0 and 1, representing the proportion of variance in the dependent variable that is explained by the independent variables.

Formula:

r² = 1 - [(n - 1)SSres/(n - 1)SSTot]

where:

• r²: Coefficient of determination
• n: Number of observations
• SSres: Sum of squares of residuals
• SSTot: Sum of squares of total variation

Interpretation:

• r² = 1: The model perfectly fits the data, explaining all variability in the dependent variable.
• r² = 0: The model does not explain any variability in the dependent variable.

Significance:

• A high coefficient of determination indicates a good fit between the model and the data.
• A low coefficient of determination indicates a poor fit.
• The coefficient of determination is an important metric for model evaluation, but should not be the only factor considered.

Example:

A model with r² = 0.85 indicates that the model explains 85% of the variance in the dependent variable.

Additional Notes:

• The coefficient of determination is a squared value, so it can be negative.
• The coefficient of determination can be biased for models with a large number of independent variables.
• It is important to consider other model evaluation metrics, such as mean squared error (MSE) and root mean squared error (RMSE), in addition to r².