In simple linear regression, if R-squared is zero, what does that indicate about the model?

• A. The model perfectly explains all variability
• B. The predictor perfectly predicts the outcome
• C. The intercept is zero
• D. The predictor explains none of the variability in the outcome

Answer: (D)

The main idea here is what R-squared tells us about a simple linear regression model. R-squared is the proportion of the outcome’s variability that is explained by the linear relationship with the predictor. If this value is zero, the linear component you add does not reduce the unexplained variation at all. In other words, the best-fitting line is flat and does not depend on the predictor—the slope is zero and the predicted values equal the mean of the outcome. This means there is no linear association between the predictor and the outcome.

This does not mean there is no relationship at all, only that there is no linear relationship detected by this model. Nonlinear relationships could still exist, but the straight-line model wouldn’t capture them. It also doesn’t imply the intercept must be zero; with a zero slope, the intercept equals the mean of the outcome, which could be nonzero. And it certainly does not indicate a perfect explanation—that would correspond to R-squared equal to 1.