In regression diagnostics, if the spread of residuals increases with the fitted values, which assumption is violated?

In regression diagnostics, if the spread of residuals increases with the fitted values, which assumption is violated?

• A. Normality
• B. Linearity
• C. Heteroscedasticity
• D. Independence

Answer: (C)

The situation described points to non-constant variance of the errors across levels of the fitted values. In regression, we assume the residuals have constant variance no matter where you are on the fitted value scale. When you see the residuals spread out more as the fitted value increases, that means the variance of the errors depends on the level of the fitted value. This is heteroscedasticity. Normality is about the shape of the residuals’ distribution, not how their spread changes with predictions. Linearity is about the relationship form between predictors and the outcome—nonlinearity would show a curved pattern in the residuals, not just increasing spread. Independence refers to residuals not being correlated with one another—autocorrelation would show a different pattern over observations, not a changing spread with fitted values. In practice this pattern can bias standard errors and affect inference, and you might address it with a transformation, weighted least squares, or robust standard errors.

The situation described points to non-constant variance of the errors across levels of the fitted values. In regression, we assume the residuals have constant variance no matter where you are on the fitted value scale. When you see the residuals spread out more as the fitted value increases, that means the variance of the errors depends on the level of the fitted value. This is heteroscedasticity.

Normality is about the shape of the residuals’ distribution, not how their spread changes with predictions. Linearity is about the relationship form between predictors and the outcome—nonlinearity would show a curved pattern in the residuals, not just increasing spread. Independence refers to residuals not being correlated with one another—autocorrelation would show a different pattern over observations, not a changing spread with fitted values.

In practice this pattern can bias standard errors and affect inference, and you might address it with a transformation, weighted least squares, or robust standard errors.