Questions: Assumptions in Linear Regression

Question 1

A researcher fits a linear regression and the residual-vs-fitted plot shows a clear U-shaped (curved) pattern. Which assumption is violated, and what is the primary consequence?

• A. Homoscedasticity — standard errors are inflated but coefficient estimates remain unbiased.
• B. Normality — p-values are unreliable, especially for small samples.
• C. Linearity — the coefficient estimates themselves are biased because the model misrepresents the true relationship.
• D. Independence — standard errors are incorrect but the curvature pattern is unrelated to independence.

Answer: (C)

A curved residual pattern is the signature of violated linearity: the model fits a straight line to a nonlinear relationship, so the slope coefficients are biased estimates of a nonlinear truth. This is the most fundamental violation — the estimates themselves are wrong, not just their standard errors. A fan shape indicates heteroscedasticity (option A); normality violations show up in a QQ-plot; independence violations leave no distinctive trace in a residual-vs-fitted plot.

Question 2

A residual-vs-fitted plot shows residuals tightly clustered near fitted values of 10 but widely scattered near fitted values of 100. This pattern most likely indicates:

• A. Autocorrelation — errors from nearby observations are correlated with each other.
• B. Non-linearity — the true relationship curves upward at higher predicted values.
• C. Heteroscedasticity — the variance of the errors grows with the fitted values.
• D. A normality violation — residuals at higher fitted values follow a non-normal distribution.

Answer: (C)

A fan or funnel shape in the residual plot is the classic diagnostic for heteroscedasticity — the error variance is not constant but increases (or decreases) with the fitted values. This violates the homoscedasticity assumption. The consequence is incorrect standard errors (making significance tests and confidence intervals unreliable), though coefficient estimates remain unbiased. A log transformation of the outcome often stabilizes this pattern.

Question 3

For large samples, the normality of errors assumption matters less because the Central Limit Theorem makes regression coefficient estimates approximately normal regardless of the error distribution.

• T. True
• F. False

Answer: True

True. The normality assumption is needed for exact inference in small samples, but the CLT ensures that as sample size grows, the sampling distribution of the regression coefficients becomes approximately normal even when the errors themselves are not. Normality is therefore considered the mildest of the four LINE assumptions — large-sample inference is robust to departures from it.

Question 4

If any regression assumption is violated, the regression model cannot be fitted and the software will refuse to compute coefficients.

• T. True
• F. False

Answer: False

False — regression always produces coefficients by minimizing squared residuals, regardless of whether assumptions are satisfied. The mechanical fitting process is blind to violations. Assumptions govern the validity of inferential statements (p-values, confidence intervals, standard errors), not the ability to compute a line. A regression fit on non-linear data, correlated observations, or heteroscedastic errors will produce numbers — but the inferential interpretations of those numbers will be wrong.

Question 5

Why is violation of the independence assumption often more damaging than violation of the normality assumption in linear regression?

Model answer: Independence violations corrupt standard errors in hard-to-predict ways and are often invisible to standard diagnostic plots. When observations are correlated — in time-series data, clustered data (students within schools), or repeated measures — the effective sample size is smaller than the nominal n, and standard errors based on n independent observations understate true uncertainty, sometimes severely. Normality violations mainly affect small-sample inference and become negligible as n grows due to the CLT. Independence violations do not self-correct with larger samples.

The practical danger is that independence violations leave no trace in a residual-vs-fitted plot, which examines magnitudes, not the order or grouping of observations. You need specialized diagnostics: Durbin-Watson for autocorrelation, or knowledge of study design to identify clustering. Normality you can inspect in a QQ-plot and often ignore at large n; independence problems require structural fixes such as clustered standard errors or mixed models.