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.

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

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