Suppose you have fit a linear model to some data and now...

Suppose you have fit a linear model to some data and now take a look at the residuals. For each of
the following possible residuals plots, tell whether you would try a re-expression and, if so, why.

Key Concepts

Linear Regression Assumptions

Linear regression relies on several key assumptions: linearity of the relationship between predictors and the response variable, independence of errors, constant variance (homoscedasticity), and normality of the error terms. Checking these assumptions helps ensure that the model is valid and that inferences drawn from it are reliable.

Residual Analysis

Residual analysis involves examining the differences between the observed values and the values predicted by the model. Plotting these residuals against predicted values or individual predictors can reveal patterns that suggest violations of model assumptions, such as curvature, unequal spread, or outliers, which can compromise the validity of the model.

Variable Transformation/Re-expression

When residual plots indicate non-linearity or heteroscedasticity (non-constant variance), a transformation or re-expression of the dependent or independent variables can be useful. Transformations such as logarithmic, square root, or polynomial adjustments are applied to improve the linearity of the relationship and stabilize the variance, leading to a better-fitting model.

Diagnostic Plots in Regression

Diagnostic plots, including residuals versus fitted values and quantile-quantile plots, are critical tools in regression analysis. They help identify issues like non-linear trends, unequal variances, and non-normal error distributions. Interpreting these plots correctly is important for deciding whether a transformation or refinement of the model is needed.

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