Which of these is guaranteed to stay the same if we add an additional independent variable to a model?

Question

BlackTom AI · Accepted Answer

Question recap: we’re asked which quantity is guaranteed to stay the same if an additional independent variable is added to a linear model.

Option 1: 'The R Squared' — R^2 is the proportion of variance explained by the model, calculated as 1 − (RSS/TSS). When a new predictor is added, RSS can only stay the same or decrease, and TSS is unaffected by the model. Since the ratio can increase or stay the same, R^2 is not guaranteed to stay the same; it often increases but can, in some cases, remain unchanged if the new variable provides no additional explanatory power.

Option 2: 'The Adjusted R Squared' — Adjusted R^2 adjusts R^2 for the number of predictors in the model to penalize for adding variables that don’t improve fit. Because of this penalty term, adding a useless variable can decrease Adjusted R^2, and a genuinely useful variable can increase it. Therefore, Adjusted R^2 is not guaranteed to stay the same.

Option 3: 'The TSS' — The Total Sum of Squares (TSS) measures the total variability in the response variable around its mean and is a function of the observed y-values only. It does not depend on the model specification or the predictors included. Hence, adding an independent variable cannot change TSS; it remains constant.

Option 4: 'The RSS' — The Residual Sum of Squares (RSS) compares observed values to the model’s fitted values. When a new predictor is added, the model can fit the data at least as well as before, so RSS either stays the same or decreases. It is not guaranteed to stay the same, since a useful added variable typically reduces RSS.

Summary of the distinctions: TSS is determined solely by the data (y values) and is unaffected by the model; all other quantities depend on the model specification and can change with the addition of predictors.

Which of these is guaranteed to stay the same if we add an additional independent variable to a model?

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