AR2 - Adjusted Coefficient of Determination

The Adjusted Coefficient of Determination (denoted as AR2 or ACOD) is a modified version of the standard Coefficient of Determination (R2) that adjusts for the number of independent variables (predictors) in a model.

While the standard R2 always increases (or stays the same) when you add new variables—even if they are completely useless—AR2 penalizes you for adding variables that do not genuinely improve the model’s predictive power.

\[\text{AR2}(y, \hat{y}) = 1 - \left[ \frac{(1 - \text{R2})(N - 1)}{N - k - 1} \right]\]
Where:

  • \(\text{R2}\) is the standard Coefficient of Determination.

  • \(N\) is the total number of samples (data points).

  • \(k\) is the number of independent features/predictors used in the model.

Note

Nomenclature Consistency As with R2, many external sources and libraries (like Scikit-Learn) refer to this metric as “Adjusted R-squared”. Because permetrics strictly defines R-squared as the literal square of Pearson’s correlation, we explicitly denote this metric as Adjusted R2 (AR2) or Adjusted COD to prevent mathematical misunderstandings.

Description

Advantages:

  • Overfitting Prevention: AR2 is the ultimate safeguard against the “illusion of improvement.” It provides a much more accurate and honest measure of goodness-of-fit for multiple regression models by decreasing if a newly added predictor improves the model less than would be expected by chance.

  • Model Selection: Highly effective for comparing the explanatory power of regression models that contain different numbers of predictors.

Disadvantages:

  • Sample Size Constraint (Fatal Flaw): Because the denominator is \(N - k - 1\), the metric will mathematically crash (division by zero) if your number of samples equals your number of predictors plus one (\(N = k + 1\)). Furthermore, if \(N < k + 1\), the metric breaks completely. You must always have significantly more data points than features.

  • Negative Values: Like standard R2, AR2 can yield negative values if the model is exceptionally poor, which can sometimes confuse non-technical stakeholders who expect a percentage-like score.

Properties

Example Usage

Note: Depending on your training dataset, ensure that the parameter X_shape is correctly passed in your function called.

from numpy import array
from permetrics.regression import RegressionMetric

## 1. For 1-D array (Single-output)
y_true = array([3, -0.5, 2, 7])
y_pred = array([2.5, 0.0, 2, 8])

evaluator = RegressionMetric(y_true, y_pred)
# Calculate Adjusted Coefficient of Determination
print("AR2: ", evaluator.adjusted_coefficient_of_determination(X_shape=(100, 5))) # 100 samples, 5 features in training dataset

## 2. For > 1-D array (Multi-output)
y_true = array([[0.5, 1], [-1, 1], [7, -6]])
y_pred = array([[0, 2], [-1, 2], [8, -5]])

evaluator = RegressionMetric(y_true, y_pred)
# Return an array of scores for each column
print("AR2 (Multi-output): ", evaluator.AR2(X_shape=(100, 5), multi_output="raw_values"))