Limitations of R-squared

While the Coefficient of Determination, or R-squared ($R^2$), gives us a handy percentage representing the proportion of variance explained by our model, it's important to understand its limitations. Relying solely on $R^2$ can sometimes paint an incomplete or even misleading picture of your regression model's performance.

R-squared Can Be Artificially Inflated

One significant issue is that $R^2$ almost always increases (or stays the same, but never decreases) when you add more independent variables (predictors or features) to your model. This happens even if the variables you add have no real relationship with the target variable you're trying to predict.

Think about it: adding more variables gives the model more flexibility to fit the training data, potentially capturing noise or random fluctuations rather than genuine patterns. A model with many irrelevant variables might show a high $R^2$ on the data it was trained on, but it likely won't perform well when making predictions on new, unseen data. This encourages building overly complex models that don't generalize well.

Note: More advanced metrics like Adjusted R-squared exist, which try to penalize the score for adding variables that don't improve the model significantly. However, for this introductory course, the main takeaway is to be wary of chasing a high $R^2$ simply by adding more inputs.

High R-squared Doesn't Guarantee a Good Model

A high $R^2$ value doesn't automatically mean your model is "good" or appropriate for your task. Here's why:

1. Bias: R-squared tells you nothing about whether your model's predictions are systematically too high or too low (biased). A model could have a high $R^2$ but consistently overestimate or underestimate the actual values. Other metrics like MAE or RMSE, along with visualizing prediction errors (residuals), are needed to check for bias.
2. Model Assumptions: Many regression techniques rely on certain assumptions about the data (like linearity). R-squared doesn't tell you if these assumptions are met. You could achieve a high $R^2$ with a model that fundamentally misunderstands the underlying relationship in the data. Again, visual checks, like plotting predictions against actual values, are often necessary.
3. Overfitting: As mentioned earlier, a model might fit the training data extremely well, leading to a high $R^2$, but fail miserably on new data. This is called overfitting. R-squared calculated on the training data alone cannot detect this. Evaluating on a separate test set is essential.

R-squared Doesn't Imply Causation or Correctness

R-squared measures the strength of correlation captured by the model, not whether the relationship makes sense or if one variable causes another. You might find a high $R^2$ between two variables that are coincidentally related or both influenced by a third, unobserved factor. It quantifies fit, not the theoretical soundness or causal validity of the model.

Context Matters for Interpretation

What counts as a "good" $R^2$ score is highly dependent on the context of the problem:

• In fields like physics or engineering, where measurements are precise, you might expect very high $R^2$ values (e.g., above 0.95).
• In social sciences, economics, or marketing, where human behavior is involved and data is inherently noisy, an $R^2$ of 0.5 or even 0.3 might be considered quite good or informative.

Furthermore, $R^2$ doesn't tell you if the prediction errors (measured by MAE or RMSE) are acceptably small for your specific application. A model could explain 90% of the variance ($R^2 = 0.9$) but still have an average error (MAE) that is too large for practical use.

Use R-squared Wisely

R-squared is a valuable metric for understanding the proportion of variance explained by your regression model. However, it shouldn't be the only metric you consider. Always evaluate it alongside error metrics like MAE, MSE, and RMSE, and use visualizations (like scatter plots of predicted vs. actual values, or residual plots) to get a more complete understanding of your model's strengths and weaknesses. Think of $R^2$ as one important indicator among several needed to judge how well your model performs.