Comment expliquez-vous le concept de r carré ajusté à un public non technique?

Imagine R-squared as a report card grade that measures how well a student understands a subject. Now, think of adjusted R-squared as a modified grade that factors in the difficulty level of the coursework or any extra credit earned. In this context, while R-squared serves as a basic indicator of how much our statistical model comprehends the data, adjusted R-squared goes a step further by considering the challenges and complexities the model encounters. It's like giving the model a more thoughtful and fair evaluation, accounting for various factors to provide a more accurate measure of its performance in real-world scenarios.

There needs to be an explanation of what sum of squares and total sum of squares. Maybe an analogy can be used here? Or perhaps a real life example used throughout the article to bring readers with no background in statistics/econometrics along.

The R-squared or coefficient of determination is the percentage of variation explained by the model (explanatory variables included in the model). For instance, if the model r-squared is 82%, this means that 82% of the variation in the response (dependent variable) is explained (or captured) by the explanatory variable(s). It also means that the model fails to explain 18% of the variation in the response. For a simple linear regression (a single explanatory variable), the r-squared is the square of the correlation coefficient between Y (dependent) and X (explanatory variable.

Adjusted R-squared is like a score that tells us how good our model is at predicting things. It considers both how well our model fits the data and how many things we're trying to predict with. So, a higher adjusted R-squared means our model is doing a good job, but we don't want it to be too high, or we might be making things overly complicated.

Imagine you're trying to guess someone's age based on clues. Each clue helps improve your guess: R-squared tells you how much of the guessing game you’re getting right overall. If you use more clues, you’ll naturally guess better, so R-squared always increases as you add more clues—even if the new clues don’t add much value.

Adjusted R-squared is like a fairness check. It asks, "Did the new clues really help, or are you just adding more guesses to make it seem like you’re doing better?" If the new clue doesn’t genuinely improve your guess, adjusted R-squared actually penalizes you.

The more explanatory variables are added into a regression model, the larger the r-square becomes. The adjusted r-squared, as the name suggests, penalises explanatory variables that do not contribute to the explanatory power of the model. It is advisable to rely on the r-squared for multiple linear regression models (several explanatory variables). Discrepancy between the r-squared and the adjusted r-squared indicates that some explanatory variables part of the model do not represent variables that contribute to the explanatory performance of the model. A more parcimonious model is needed.

Adjusted R-squared is used to evaluate how well a regression model explains the data while accounting for model complexity. Unlike regular R squared, it penalizes the inclusion of irrelevant predictors, helping to avoid overfitting. This makes adjusted R squared particularly useful when comparing models with different numbers of predictors, ensuring a balance between simplicity and accuracy.

To compare models, look for the one with the highest adjusted R squared, as it indicates better explanatory power while penalizing complexity. However, if a simpler model has a slightly lower adjusted R squared, it may still be preferable due to easier interpretation and reduced overfitting risk. Always consider adjusted R squared alongside other metrics like Mean Squared Error or AIC for a comprehensive evaluation.

Adjusted R-squared indicates how well a regression model explains the variability in the dependent variable, considering the number of predictors used. A higher adjusted R squared means the model is better at explaining the data without being overly complex. If adjusted R squared decreases after adding a predictor, it suggests the new predictor adds little value to the model.