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The adjusted R-squared (or adjusted R-squared value) is a statistic used to evaluate the goodness-of-fit of a regression model. It's a modification of the standard R-squared (coefficient of determination) that takes into account the number of predictors (independent variables) in the model. Adjusted R-squared adjusts the R-squared value based on the complexity of the model by penalizing the inclusion of unnecessary predictors.

Here's the formula for adjusted R-squared:

Adjusted R-squared = 1 - [(1 - R^2) * (n - 1) / (n - k - 1)]

Where:

• R^2 is the standard R-squared value, which measures the proportion of the variance in the dependent variable that is explained by the independent variables.
• n is the number of observations or data points in the dataset.
• k is the number of predictors (independent variables) in the model.

Key points to understand about adjusted R-squared:

1. Interpretation: The adjusted R-squared value ranges from 0 to 1. A higher adjusted R-squared value indicates a better fit of the model to the data.

2. Penalizes Complexity: Adjusted R-squared penalizes the inclusion of additional predictors that do not significantly improve the model's fit. It accounts for overfitting by reducing the adjusted R-squared value if unnecessary predictors are added.

3. Comparing Models: When comparing multiple regression models, you can use the adjusted R-squared values to assess which model provides a better balance between goodness of fit and model complexity. Generally, a higher adjusted R-squared suggests a better model.

4. Trade-Off: It's important to strike a balance between model complexity and goodness of fit. While a higher adjusted R-squared is desirable, adding too many predictors can lead to a complex and overfit model.

In practice, when comparing different regression models, you would prefer the model with the highest adjusted R-squared value, as long as it doesn't become overly complex with too many predictors. The adjusted R-squared helps you choose a model that explains a significant portion of the variance in the dependent variable while avoiding unnecessary complexity.

Let's consider two hypothetical regression models and their adjusted R-squared values for comparison:

Model 1:

• Adjusted R-squared: 0.85
• Number of predictors (features): 5
• Sample size: 100

Model 2:

• Adjusted R-squared: 0.90
• Number of predictors (features): 4
• Sample size: 150

In this example, we have two regression models, Model 1 and Model 2, with different adjusted R-squared values.

• Model 2 has a higher adjusted R-squared value (0.90) compared to Model 1 (0.85). This indicates that Model 2 provides a better fit to the data in terms of explaining the variance in the dependent variable.

• Model 2 also has fewer predictors (features) than Model 1 (4 vs. 5), which suggests that it might be a more parsimonious model, potentially providing a simpler explanation of the data.

• The sample size for Model 2 is larger (150) compared to Model 1 (100), which can also influence the adjusted R-squared value. In general, larger sample sizes tend to result in higher adjusted R-squared values, all else being equal.

Therefore, in this hypothetical example, Model 2 with an adjusted R-squared of 0.90 is considered to have a better fit to the data compared to Model 1 with an adjusted R-squared of 0.85. However, it's essential to consider other factors like the number of predictors and the sample size when interpreting and comparing adjusted R-squared values.

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