Types of R2	Defination	Interpretation	Use
R2 (Coefficient of Determination)	Represents the proportion of the variance in the dependent variable explained by the independent variables in a regression model.	Higher R2 indicates a better fit.	Assess overall goodness of fit.
Adjusted R2	Modification R2 that penalizes unnecessary predictors. Adjusts for the number of predictors in the model.	Reflects goodness of fit while considering model simplicity.	Particularly useful for comparing models with different numbers of predictors.
Weighted R2	Considers different weights for each data point when calculating goodness of fit.	Useful when some data points contribute more or less to the model.	Accounts for varying impact of data points.
Bayesian R2	In Bayesian statistics, R2 has a Bayesian interpretation, considering uncertainty in parameter estimates.	Incorporates Bayesian approach to model uncertainty.	Suitable for Bayesian statistical analyses.

What is a ‘good’ R-squared value?

What makes a (R²) value “good” depends on the situation. In social sciences, even a 0.5 (R²) can be seen as strong. In some fields, a high (R²) like 0.9 is considered good. In finance, an (R²) above 0.7 means a strong correlation, while below 0.4 is seen as a weak one. Remember, these aren’t strict rules; it varies based on the specific study or analysis.

R

# Generate some sample data
set.seed(123)
x <- rnorm(100)
y <- 2 * x + rnorm(100)

# Fit a linear regression model
model <- lm(y ~ x)

# Extract R-squared and Adjusted R-squared values
rsquared <- summary(model)$r.squared
adj_rsquared <- summary(model)$adj.r.squared

# Print R-squared and Adjusted R-squared
cat("R-squared:", round(rsquared, 4), "\n")
cat("Adjusted R-squared:", round(adj_rsquared, 4), "\n")

Output:

R-squared: 0.7721 

Adjusted R-squared: 0.7698 

The R-squared value is approximately 0.7721.

• The Adjusted R-squared value is approximately 0.7698.
• These values indicate that around 77.21% of the variability in the dependent variable (y) is explained by the independent variable (x) in the model. The adjusted R-squared accounts for the number of predictors and penalizes for model complexity.

Visualize the data and fitted model

R

# Visualize the data and fitted model
plot(x, y, main = "Linear Regression", col = "blue", pch = 16)
abline(model, col = "red")

Output:

The scatter plot displays the data points (x and y) in blue.

• The red line represents the fitted linear regression model.

Limitations

• A high R² value doesn’t mean one variable causes the other.
• R² is influenced by the choice of predictors , adding more predictors tends to inflate the value.
• A good R² doesn’t guarantee a well-fitting model it may fit training data well but perform poorly on new data.
• It’s sensitive to outliers so extreme values can disproportionately impact the result.

Conclusion

R-squared shows how well a model fits data, with higher values indicating better fit. It’s versatile, featuring various types like adjusted and weighted. A “good” R-squared varies by field; 0.5 may be strong in social sciences, while 0.9 is expected in some fields. However, it has limitations, such as sensitivity to outliers.

Good R Squared Value in R – FAQs

Is a higher (R²) always better?

Yes, in general, a higher (R²) is better. It indicates that a larger proportion of the variability in the dependent variable is explained by the model, suggesting a more effective fit.

What does a low (R²) value mean?

A low (R²) (close to 0%) suggests that the model doesn’t explain much of the variation in the dependent variable. It may indicate that the chosen predictors do not effectively capture the patterns in the data.

Can (R²) be too high?

Yes, (R²) can be too high, especially if the model is overfitting the data. Extremely high (R²) values may capture noise as if it were a true pattern, leading to poor generalization to new data.

Are there industry-specific standards for a good (R²) value?

Yes, standards for a good (R²) value can vary by industry. For instance, in social sciences, a relatively low (R²) might be considered good, while in finance, a higher (R²) is often expected for a model to be effective.

What is a good range for (R²) in regression analysis?

A good range for (R²) depends on the context, but values between 0.3 and 0.7 are often considered moderate, while values above 0.7 are generally seen as strong. However, the interpretation varies based on the field and complexity of the problem.

What does an R-squared value of 0.9 mean?

An R² value of 0.9 means that 90% of the variability in the dependent variable is explained by the independent variables in the model.