What are the assumptions of Linear Regression?

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BlackTom AI · Accepted Answer

Question restatement: The prompt asks to identify the assumptions of Linear Regression, and the provided answer set includes two items to evaluate.
Option 1: 'Normality in residuals' — This is commonly cited as an assumption when we want to perform statistical inferences (e.g., constructing t-tests and confidence intervals for the coefficients). In ordinary least squares (OLS), the estimator of the coefficients remains unbiased and efficient under several conditions (linearity, independence, homoscedasticity), even if the residuals are not normally distributed. However, normality of the residuals is not required for the model to provide unbiased estimates; it primarily enables valid hypothesis tests and confidence intervals in small samples. Thus, this assumption is context-dependent: it matters for inference, but not necessary for the estimation itself.
Option 2: 'Linearity between X and Y.' — This is a core assumption of linear regression. It states that the relationship between the predictors and the mean of the response is linear, meaning the expected value of Y is a linear function of X. If the true relationship is nonlinear, the linear model will be misspecified and predictions and inferences may be biased or misleading. This makes linearity a fundamental assumption for the model to be appropriate in the first place.
Nuanced distinction: Some frameworks emphasize different facets. For example, practitioners often assume linearity of the mean response, independence of errors, homoscedastic constant variance of residuals, and correct specification of the model. Normality of residuals is especially tied to the applicability of inferential procedures rather than to the validity of the estimated coefficients themselves. Misinterpretations commonly arise when people treat normality as a strict requirement for the model to fit; instead, it is primarily an assumption about the sampling distribution used for inference.
In summary, Option 2 reflects a primary, structural assumption of linear regression. Option 1 is conditionally important for inference (not strictly required for estimation) and depends on whether the goal is hypothesis testing and interval estimation versus point estimation alone. The distinction helps learners avoid conflating estimation validity with the need for residual normality in all scenarios.

Machine Learning: Fundamentals and Applications Quiz 1 - Linear Regression

What are the assumptions of Linear Regression?

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