How to Calculate the Coefficient of Determination (R-Squared) Step-by-Step

The coefficient of determination also known as R-squared (R2), is a statistic that measures how well a regression model fits the data. R-squared shows the proportion of variation in the response variable that can be explained by the predictors in the model. Calculating R-squared is simple once you understand the basic formula and components.

In this comprehensive guide, I’ll walk you through the step-by-step process of calculating R-squared using its formula I’ll also provide examples using regression analysis in statistics software to demonstrate how to interpret R-squared Let’s get started!

What is the Coefficient of Determination?

The coefficient of determination (R2) measures the degree to which the variance of the response variable Y is explained by the predictors (X variables) in a regression model

R-squared is defined as the proportion of total variation in Y that is explained by the regression model. It ranges from 0 to 1, with higher values indicating more of the response variable variation is accounted for by the predictors.

• R2 near 0 means the model does not fit the data well (low explanatory power).
• R2 near 1 indicates the regression model fits the data very well.

R-squared is also called the explained variation because it represents the amount of variation in Y that is explained by the X variables in the model. The remaining unexplained variation is captured by the error term.

R-Squared Formula

R-squared is calculated using the formula:

$R^2=\frac{SSR}{SST}$R2=SSTSSR​

Where:

• SSR = Regression sum of squares (variation explained by model)
• SST = Total sum of squares (total variation in Y)

This formula is based on partitioning the total sum of squares (SST) into:

• Sum of squares regression (SSR): Variation explained by X variables
• Sum of squares error (SSE): Unexplained variation (residuals)

$SST=SSR+SSE$SST=SSR+SSE

Substituting SST in the formula gives:

$R^2=\frac{SSR}{SSR+SSE}$R2=SSR+SSESSR​

Which is the proportion of explained variation out of total variation.

How to Calculate R-Squared Step-by-Step

Follow these 5 steps to calculate the coefficient of determination R-squared:

Step 1: Perform Regression Analysis

First, perform a regression analysis between the response (Y) and predictor variables (X). This gives the regression equation relating X and Y.

For example, let’s say we perform a simple linear regression of Y on X. This gives:

$Y=b_0+b_{1}X$Y=b0​+b1​X

Step 2: Calculate SST

Find the total sum of squares (SST) using the formula:

$SST={\sum }_{i=1}^n(y_i-\bar{y}{)}^2$SST=∑i=1n​(yi​−yˉ​)2

Where:

• y = response value
• $bar{y}$ = mean of response

This measures the total deviation of each y value from the mean.

Step 3: Calculate SSR

Next, calculate the regression sum of squares (SSR) using:

$SSR={\sum }_{i=1}^n({\hat{y}}_i-\bar{y}{)}^2$SSR=∑i=1n​(y^​i​−yˉ​)2

Where $hat{y}_i$ are the predicted y values from the regression equation.

SSR measures the variation explained by the model.

Step 4: Calculate SSE

The sum of squares error (SSE) is found by:

$SSE={\sum }_{i=1}^n(y_i-{\hat{y}}_i{)}^2$SSE=∑i=1n​(yi​−y^​i​)2

SSE represents the residual variation not explained by the model.

Step 5: Compute R-Squared

Finally, compute R-squared using its formula:

$R^2=\frac{SSR}{SST}=\frac{SSR}{SSR+SSE}$R2=SSTSSR​=SSR+SSESSR​

Substitute the values calculated for SSR and SST. This gives the coefficient of determination.

And we’re done! Those are the key steps involved in calculating R-squared manually from a regression analysis. Now let’s look at some examples.

R-Squared Example in Excel

Let’s see how to find R-squared for a simple linear regression example in Excel.

Given this X and Y data:

We perform a linear regression of Y on X, giving regression equation:

$Y=-0.2+2.6X$Y=−0.2+2.6X

Use Excel to find the R-squared value as follows:

1. Input the X and Y data

2. Click the Data Analysis button and select Regression

3. Select the Y and X input ranges

4. Check the Residuals box

5. Click OK

This gives the following regression output with R-squared and the sums of squares:

• SST (Total SS) = 108
• SSR (Regression SS) = 104.8
• SSE (Residual SS) = 3.2

Finally, calculate R-squared using the formula:

$R^2=\frac{SSR}{SST}=\frac{104.8}{108}=0.969$R2=SSTSSR​=108104.8​=0.969

We get the same R-squared value of 0.969 as in the Excel output. This confirms our manual R-squared calculation.

R-Squared in R Programming

Calculating R-squared in R is straightforward using the lm() function for linear regression.

Let’s use the same X and Y data from the Excel example:

# Store data in vectors X <- c(1, 2, 3, 4, 5)  Y <- c(3, 5, 7, 8, 12)# Linear modelmodel <- lm(Y ~ X)# Summary gives R-squaredsummary(model)$r.squared

This prints the R-squared value of 0.969. The summary() function applied on the linear model returns a detailed table including R-squared.

We don’t have to manually calculate the sums of squares – R computes them automatically behind the scenes!

R-Squared in Python

Here is how to find R-squared in Python using sklearn:

from sklearn.linear_model import LinearRegressionfrom sklearn.metrics import r2_score# X and Y data X = [[1], [2], [3], [4], [5]]  y = [3, 5, 7, 8, 12]# Create and fit modelmodel = LinearRegression()model.fit(X, y)  # R-squared scorer_squared = model.score(X, y)print(r_squared)

The LinearRegression.score() method returns the R-squared value. We get 0.969 again for this data.

Interpreting R-Squared

When interpreting R-squared, consider:

• Scale: R-squared is measured on a 0 to 1 scale. Values close to 1 indicate a better model fit.

• Context: The acceptability of an R-squared value depends on the problem context. Higher is better, but may not be possible or expected.

• Use cautiously: While R-squared indicates model fit, a higher R-squared does not necessarily mean the model has better predictive performance. It also does not indicate causation between X and Y variables.

• Multiple regression: With multiple X variables, adjusted R-squared is used to account for the number of predictors in the model. But interpretation remains similar.

Overall, R-squared gives the percentage of variation explained by the model – a valuable statistic for evaluating and comparing regression analyses. Used properly, it can aid model selection, improvement, and predictive accuracy.

Common Questions about R-Squared

Here are answers to some frequently asked questions about the coefficient of determination:

How is R-squared calculated for multiple regression?

The R-squared formula remains the same for multiple regression models with several X variables. The process for calculating SST, SSR and SSE is identical.

What is the difference between R-squared and Adjusted R-squared?

Adjusted R-squared penalizes model complexity, so will always be lower than R-squared. It is used more for comparing models rather than measuring fit.

Can R-squared be negative?