Coefficient of determination

The coefficient of determination, usually written R², summarizes how closely a statistical model’s predictions account for variation in an outcome. Its interpretation depends on the model and how R² is calculated.

Total sum of squares: The sum of squared deviations of observed outcomes from their mean. It supplies the baseline variation that ordinary least-squares R² partitions.
Simple linear regression: A linear model relating one outcome to one predictor. Its R² equals the squared sample correlation between predictor and outcome when an intercept is included.
Pearson correlation coefficient: A measure of the strength and direction of linear association between two variables. In simple linear regression with an intercept, R² is its square, but this relation does not generalize unchanged.
Negative R-squared: An R² value below zero, possible when model predictions perform worse than a chosen baseline under a defined scoring rule. It reveals that some R² definitions do not guarantee a zero-to-one range.
Residual sum of squares: The sum of squared differences between observed outcomes and model predictions. In ordinary least squares, its size relative to total variation determines R².
Multiple linear regression: A linear model relating an outcome to two or more predictors. Its R² summarizes fit from the predictors jointly, not each predictor's separate contribution.
Mean squared error: The average squared difference between observed values and predictions. Unlike R², it retains the outcome's units squared and directly measures prediction error.
Goodness of fit: The degree to which a statistical model agrees with observed data or its intended target. R² is one fit summary, but no single value establishes that a model is adequate.
Least squares: A method that estimates model parameters by minimizing squared residuals. The usual R² formula follows naturally from least-squares variance decomposition.
Logistic regression: A regression model for outcomes whose conditional distribution is categorical, commonly binary. Its likelihood-based fit motivates pseudo-R² measures rather than the standard variance ratio.

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