Calculating R-squared (R²) in R: From Basic Formulas to Statistical Principles

Basic Concepts and Calculation Methods of R-squared

R-squared (R²), also known as the coefficient of determination, is a crucial statistical metric for assessing the goodness of fit of a model. In the R programming environment, while the standard library does not provide a dedicated function for computing R-squared between two vectors, it can be efficiently implemented using fundamental mathematical relationships.

The most concise calculation method is based on the square of the correlation coefficient. Given two vectors x and y of equal length, their R-squared equals the square of the Pearson correlation coefficient:

rsq <- function(x, y) cor(x, y) ^ 2

Usage example:

obs <- 1:5
mod <- c(0.8, 2.4, 2, 3, 4.8)
rsq_value <- rsq(obs, mod)
print(rsq_value)  # Output: 0.856

R-squared Calculation in the Linear Regression Framework

From the perspective of linear regression, R-squared can be computed as the ratio of regression sum of squares to total sum of squares. R's lm function offers a direct way to obtain the R-squared value:

rsq_lm <- function(x, y) {
  model <- lm(y ~ x)
  return(summary(model)$r.squared)
}

Although this approach involves slightly more code, it aligns with traditional statistical computation workflows and yields identical results to the correlation-based method.

Mathematical Principles and Statistical Derivation

To deeply understand the essence of R-squared, one must start from the fundamental principles of linear regression. Consider the simple linear regression model y ~ x:

Lemma 1: Centering Equivalence
The regression model y ~ x is statistically equivalent to y - mean(y) ~ x - mean(x). This centering transformation does not affect the computed R-squared value.

Lemma 2: Slope Coefficient Formula
The regression coefficient β can be obtained as the ratio of covariance to variance: β = cov(x, y) / var(x)

Lemma 3: Relationship between R-squared and Correlation
Through mathematical derivation, it can be proven that R-squared indeed equals the square of the correlation coefficient: R² = cor(x, y)²

The total sum of squares (TSS) can be decomposed into regression sum of squares (RSS) and error sum of squares (ESS): TSS = RSS + ESS
R-squared is defined as: R² = RSS / TSS = 1 - ESS / TSS

Limitations and Application Warnings for R-squared

Despite its computational simplicity, R-squared must be used cautiously in practical applications, particularly in predictive tasks:

Invariance to Constant Shifts
R-squared remains unchanged under constant shifts of vectors. For any constants a and b, R²(x + a, y + b) = R²(x, y). This property may render R-squared ineffective in certain scenarios for reflecting prediction quality.

Analysis of Extreme Example
Consider the following vectors:

preds <- 1:4 / 4
actual <- 1:4
rsq_result <- cor(preds, actual) ^ 2  # Result: 1

Although R-squared reaches a perfect 1, indicating a perfect linear relationship, preds is merely a linear scaling of actual and may not be practical in prediction tasks.

Distinction Between Training and Test Sets
R-squared is primarily suitable for assessing goodness of fit on training data. Applying it directly to test sets lacks statistical justification. In predictive modeling, metrics such as Mean Squared Error (MSE) or Root Mean Squared Error (RMSE) are recommended.

Common Mistakes and Correct Practices

Common errors in computing R-squared include using the residual sum of squares formula directly while omitting the regression step:

# Incorrect calculation method
preds <- c(1, 2, 3)
actual <- c(2, 2, 4)
rss <- sum((preds - actual) ^ 2)
tss <- sum((actual - mean(actual)) ^ 2)
rsq_wrong <- 1 - rss / tss  # May yield anomalous values

This approach can produce negative R-squared values, which are statistically unreasonable since the theoretical range of R-squared is [0,1].

Correct Alternative Formulas
Besides the correlation-based method, R-squared can also be computed via regression sum of squares:

regss <- sum((preds - mean(preds)) ^ 2)
rsq_correct <- regss / tss  # Correct result

Practical Application Recommendations

When selecting model evaluation metrics, consider the specific task requirements:

• Model Fit Assessment: Use R-squared on training data to evaluate the strength of linear relationships
• Prediction Performance Assessment: Employ metrics like MSE, RMSE, or MAE on test data
• Model Comparison: Conduct comprehensive evaluation using multiple metrics to avoid limitations of single indicators

In R programming practice, the simplified method based on correlation coefficients is recommended for quick computations, but formal analysis reports should clearly specify the calculation methods and statistical assumptions used.