Residual analysis is a crucial step in multiple regression, helping us verify if our model meets key assumptions. By examining patterns in residual plots, we can spot issues like non-linearity, heteroscedasticity, and outliers that might mess up our results.

We'll look at how to create and interpret these plots, check for normality in residuals, and deal with heteroscedasticity. Understanding these concepts will help us build more reliable regression models and make better predictions.

Residual Plots for Model Assumptions

Graphical Representation and Purpose

• Residual plots are graphical representations of the residuals, the differences between the observed values and the predicted values from a regression model
• They are used to assess the validity of model assumptions in multiple linear regression
• Residual plots help evaluate key assumptions such as linearity, homoscedasticity (constant variance), independence of errors, and normality of residuals

Creating and Interpreting Residual Plots

• Residual plots are typically created by plotting the residuals against the predicted values, the independent variables, or the order of data collection
• A residual plot that shows a random scatter of points around the horizontal axis, with no discernible pattern, suggests that the linearity assumption is met
• Violations of the linearity assumption may be evident in residual plots as a curved or nonlinear pattern (quadratic or exponential), indicating that a linear model may not be appropriate for the data

Patterns in Residual Plots

Non-Random Patterns and Their Implications

• Non-random patterns in residual plots can indicate violations of model assumptions or other issues that may affect the validity of the regression results
• A funnel-shaped pattern in the residual plot, where the spread of residuals increases or decreases with the predicted values, suggests a violation of the homoscedasticity assumption (non-constant variance)
• A curved pattern in the residual plot may indicate that the relationship between the dependent and independent variables is nonlinear, suggesting that a higher-order term (quadratic or cubic) or a different model (exponential or logarithmic) may be needed

Outliers and Variable-Specific Patterns

• Outliers in the residual plot, represented by points that are far from the majority of the data, can have a significant impact on the regression results and should be investigated further
• Patterns in the residual plot that correspond to specific independent variables may suggest the need for interaction terms (product of two or more variables) or transformations of the variables (logarithmic or square root) to improve the model fit
• For example, if the residuals show a distinct pattern when plotted against a categorical variable (gender or treatment group), it may indicate that the effect of the variable on the response is not adequately captured by the current model

Normality of Residuals

Assessing Normality Visually

• The normality assumption in multiple regression requires that the residuals follow a normal distribution with a mean of zero
• Visual inspection of the histogram or density plot of the residuals can provide an initial assessment of the normality assumption, with a symmetric, bell-shaped distribution indicating normality
• A normal probability plot (Q-Q plot) compares the quantiles of the residuals to the quantiles of a normal distribution, with adherence to a straight line suggesting normality

Formal Tests and Implications of Violations

• Shapiro-Wilk and Kolmogorov-Smirnov tests are formal statistical tests that can be used to assess the normality of residuals, with p-values greater than the chosen significance level (0.05) indicating that the normality assumption is met
• Violations of the normality assumption may not have a substantial impact on the validity of the regression results if the sample size is large, due to the Central Limit Theorem
• If the normality assumption is violated, transformations of the dependent or independent variables (logarithmic or square root) may help to improve the normality of residuals

Homoscedasticity in Regression Models

Definition and Consequences of Heteroscedasticity

• Homoscedasticity assumes that the variance of the residuals is constant across all levels of the predicted values and independent variables
• Violations of homoscedasticity, known as heteroscedasticity, can affect the standard errors of the regression coefficients and lead to incorrect inferences
• Heteroscedasticity can lead to inefficient estimates of the regression coefficients and biased standard errors, which can impact hypothesis testing and confidence intervals

Detecting and Addressing Heteroscedasticity

• Visual inspection of the residual plot, with residuals plotted against predicted values or individual independent variables, can help identify patterns of heteroscedasticity
• A cone-shaped or fan-shaped pattern in the residual plot, where the spread of residuals increases or decreases with the predicted values, indicates the presence of heteroscedasticity
• Formal statistical tests for heteroscedasticity include the Breusch-Pagan test and the White test, which test the null hypothesis that the variance of the residuals is constant
• If heteroscedasticity is detected, remedial measures such as weighted least squares regression, robust standard errors, or transformations of the dependent or independent variables (logarithmic or square root) can be employed to mitigate its effects