Adjusted r-squared is a statistical measure used to assess the fit of a regression model while considering the number of predictors, providing a more accurate indication of model performance compared to regular r-squared. It penalizes unnecessary predictors to avoid overfitting and is preferred by researchers for model comparisons. A higher adjusted r-squared suggests a better fitting model that effectively explains variance without redundant variables.

1. Adjusted R-squared (R²)

2. Introduction
• Adjusted R-squared (R²) is a statistical measure used to evaluate the goodness of
fit of a regression model.
• It is an extension of the regular R-squared that accounts for the number of
predictors (independent variables) in the model.
• The adjusted R-squared value takes into consideration the number of predictors
and provides a more accurate representation of how well the model fits the data.

3. Continue..
• R-squared measures the proportion of the variance in the dependent variable
(the outcome) that is explained by the independent variables in the model.
• It ranges from 0 to 1, where 0 indicates that the model explains none of the
variance, and 1 indicates that the model explains all the variance.

4. Continue..
• However, R-squared tends to increase as more predictors are added to the model,
even if the additional predictors do not contribute significantly to explaining the
variance in the dependent variable.
• This is known as overfitting, where the model becomes too complex and might
not generalize well to new data.
• The adjusted R-squared adjusts the regular R-squared by penalizing the inclusion
of unnecessary predictors.
• It accounts for the number of predictors in the model and is calculated using the
following formula:

5. formula:
• Adjusted 𝑅2
= 1 −
1−𝑅2 𝑛−1
𝑛−𝑘−1
OR
𝑅2 = 1 −
𝛴𝑒
2
𝑛 − 𝑘
𝑦2 (n −1)
• Where:
• R² is the regular coefficient of determination (R-squared),
• n is the number of data points (sample size), and
• k is the number of independent variables (predictors) in the model.
• The adjusted R-squared value will be lower than the regular R-squared if the model
contains irrelevant or redundant predictors, and it will increase only if the additional
predictors contribute significantly to explaining the variance.

7. Conclusion
• Researchers and data analysts often prefer using adjusted R-squared
when comparing models with different numbers of predictors, as it
provides a more reliable assessment of model fit and helps prevent
the overfitting problem.
• A higher adjusted R-squared value generally indicates a better-fitted
model that explains more of the variation in the dependent variable
without unnecessarily including irrelevant predictors.