Why may we opt for the adjusted r2 instead of the multiple r2 when assessing model fit of multiple linear regression?

Question

BlackTom AI · Accepted Answer

When evaluating why one would prefer adjusted r-squared over unadjusted (multiple) r-squared in the context of multiple linear regression, it's helpful to compare what each statistic measures and how they react to model complexity.
Option A: 'Adjusted r2 takes the number of predictors into account, whereas multiple r‑2 increases with more predictors and gives us a false understanding of our model fit.' This statement highlights a key concept: unadjusted r-squared always increases (or at least does not decrease) when you add more predictors, even if those predictors do not improve the model substantially. Adjusted r-squared incorporates a penalty for including additional predictors, effectively adjusting for the number of predictors relative to the sample size, which can prevent overestimation of model fit due to overfitting. This is a valid and commonly cited reason for preferring adjusted r-squared in model comparison.
Option B: 'None of the answers' This option would be incorrect if Option A accurately captures the motivation for using adjusted r-squared. Since the core rationale is that adjusted r-squared accounts for model complexity and helps avoid deceptive improvements from adding predictors, claiming there are no reasons would be inaccurate.
Option C: 'There is no difference; we could use either' This is misleading. There is a meaningful difference between the two statistics: unadjusted r-squared can increase with additional predictors regardless of their relevance, while adjusted r-squared can decrease if the added predictors do not provide enough explanatory power relative to the number of observations. Hence, they are not interchangeable for model selection.
Option D: 'Adjusted r2 takes the number of response variables into account, whereas multiple r2 increases with more response variables and gives us a false understanding of our model fit.' This statement is incorrect because standard multiple linear regression in a typical single-response context involves one response variable. Adjusted r-squared does not adjust for the number of response variables; it adjusts for the number of predictors. Misstating the role of response variables leads to a factual inaccuracy about what the metrics measure.
In summary, the most conceptually accurate explanation centers on how adjusted r-squared penalizes model complexity to counteract the tendency of unadjusted r-squared to improve simply by adding predictors, which helps provide a more truthful assessment of model fit in the presence of multiple predictors.

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