9.4 Remedial Measures for Assumption Violations

When assumptions in linear regression are violated, it's crucial to take action. Non-normality, heteroscedasticity, and multicollinearity can mess up your model's accuracy and reliability. Luckily, there are ways to fix these issues.

For non-normal residuals, try transforming your data. Weighted least squares can tackle heteroscedasticity. And if you're dealing with multicollinearity, ridge regression or principal component regression might be your best bet. Choose wisely based on your specific situation.

Addressing Non-normality of Residuals

Detecting Non-normality

Top images from around the web for Detecting Non-normality

• 

  Test de Shapiro-Wilk para contrastar la normalidad en R Commander View original

  Is this image relevant?

• 

  interpretation - Interpreting QQplot - Is there any rule of thumb to decide for non-normality ... View original

  Is this image relevant?

• 

  regression - Linear mixed effects models: what to do when the residual QQ-plot looks non-normal ... View original

  Is this image relevant?

• 

  Test de Shapiro-Wilk para contrastar la normalidad en R Commander View original

  Is this image relevant?

• 

  interpretation - Interpreting QQplot - Is there any rule of thumb to decide for non-normality ... View original

  Is this image relevant?

1 of 3

Top images from around the web for Detecting Non-normality

• 

  Test de Shapiro-Wilk para contrastar la normalidad en R Commander View original

  Is this image relevant?

• 

  interpretation - Interpreting QQplot - Is there any rule of thumb to decide for non-normality ... View original

  Is this image relevant?

• 

  regression - Linear mixed effects models: what to do when the residual QQ-plot looks non-normal ... View original

  Is this image relevant?

• 

  Test de Shapiro-Wilk para contrastar la normalidad en R Commander View original

  Is this image relevant?

• 

  interpretation - Interpreting QQplot - Is there any rule of thumb to decide for non-normality ... View original

  Is this image relevant?

1 of 3

• Residuals in linear regression should follow a normal distribution for valid inference and hypothesis testing
• Violations of this assumption can be detected through visual inspection of residual plots (histogram, Q-Q plot) or statistical tests (Shapiro-Wilk test, Anderson-Darling test)
• Non-normality can manifest as skewness, heavy tails, or outliers in the residual distribution
• Ignoring non-normality can lead to biased standard errors, invalid confidence intervals, and incorrect p-values

Applying Transformations

• Common transformations to address non-normality include logarithmic (log), square root, and Box-Cox transformations
  1. Logarithmic transformations are suitable when the residuals exhibit right-skewness (e.g., income data)
  2. Square root transformations are appropriate for moderately right-skewed residuals (e.g., count data)
  3. Box-Cox transformations provide a more flexible approach by estimating an optimal power transformation parameter ($\lambda$)
• After applying a transformation to the response variable, the model should be refitted, and the residuals should be reassessed for normality
  • If the transformation successfully addresses the non-normality, the model assumptions are considered satisfied
  • If the non-normality persists, alternative transformations or non-parametric methods may need to be considered
• It is important to interpret the coefficients and predictions in the transformed scale and, if necessary, back-transform them to the original scale for meaningful interpretation
  • For example, in a log-transformed model, a coefficient of 0.5 indicates a $e^{0.5} \approx 1.65$ times increase in the original scale for a one-unit increase in the predictor variable

Handling Heteroscedasticity

Identifying Heteroscedasticity

• Heteroscedasticity occurs when the variance of the residuals is not constant across the range of predicted values, violating the assumption of homoscedasticity
• Visual inspection of residual plots (residuals vs. fitted values) can reveal patterns of increasing or decreasing variance
• Statistical tests like the Breusch-Pagan test or White's test can formally assess the presence of heteroscedasticity
• Ignoring heteroscedasticity can lead to inefficient estimates, biased standard errors, and invalid hypothesis tests

Implementing Weighted Least Squares (WLS)

• Weighted least squares (WLS) is a method to address heteroscedasticity by assigning different weights to each observation based on the variance of the residuals
  1. Observations with smaller variances receive higher weights
  2. Observations with larger variances receive lower weights
• The weights in WLS are typically determined by estimating the variance function, which models the relationship between the variance of the residuals and the predictor variables
  • Common variance functions include the inverse variance ($1/\sigma_i^2$), the squared residuals ($\varepsilon_i^2$), or a parametric function of the predictors ($\sigma_i^2 = f(X_i)$)
• To implement WLS, the regression model is modified by multiplying both sides of the equation by the square root of the weights ($\sqrt{w_i}$)
  • The resulting weighted regression model is then estimated using ordinary least squares
• WLS provides more efficient and unbiased estimates compared to ordinary least squares when heteroscedasticity is present
  • However, it requires correctly specifying the variance function to obtain valid results
  • Misspecification of the variance function can lead to biased estimates and incorrect inferences

Mitigating Multicollinearity

Understanding Multicollinearity

• Multicollinearity refers to high correlations among the predictor variables in a multiple regression model
• It can lead to unstable and unreliable coefficient estimates, inflated standard errors, and difficulty in interpreting the individual effects of predictors
• Multicollinearity can be detected through correlation matrices, variance inflation factors (VIF), or condition indices
• Perfect multicollinearity (exact linear dependence among predictors) can prevent the estimation of the regression coefficients altogether

Ridge Regression

• Ridge regression is a regularization technique that addresses multicollinearity by adding a penalty term to the least squares objective function
  • The penalty term is proportional to the square of the magnitude of the coefficients, controlled by a tuning parameter ($\lambda$)
• As $\lambda$ increases, ridge regression shrinks the coefficient estimates towards zero, reducing their variance and mitigating the impact of multicollinearity
  • The optimal value of $\lambda$ is typically determined through cross-validation
• Ridge regression provides a trade-off between bias and variance
  • It introduces some bias in the coefficient estimates but reduces their variance, leading to improved prediction accuracy and stability
• Ridge regression retains all the original predictors in the model, making it useful when all predictors are considered relevant

Principal Component Regression (PCR)

• Principal component regression (PCR) is another approach to handle multicollinearity
• It involves transforming the original predictor variables into a set of uncorrelated principal components and then using a subset of these components as predictors in the regression model
• PCR reduces the dimensionality of the predictor space by selecting a smaller number of principal components that capture most of the variation in the original variables
  • This helps to alleviate multicollinearity and improve the stability of the coefficient estimates
• The number of principal components to retain can be determined based on the proportion of variance explained or through cross-validation
• PCR can be effective in reducing multicollinearity and improving model stability
  • However, it may sacrifice some interpretability as the principal components are linear combinations of the original predictors

Choosing Remedial Measures

Assessing Assumption Violations

• The choice of remedial measure depends on the specific assumption violation encountered in the linear regression model
  • Different violations require different approaches to address them effectively
• For non-normality of residuals, transformations such as logarithmic, square root, or Box-Cox transformations can be applied to the response variable
  • The choice of transformation depends on the pattern of non-normality observed in the residuals (e.g., right-skewness, heavy tails)
• When dealing with heteroscedasticity, weighted least squares (WLS) is a suitable remedial measure
  • WLS assigns different weights to observations based on the variance of the residuals, giving more weight to observations with smaller variances
• In the presence of multicollinearity, ridge regression and principal component regression (PCR) are commonly employed
  • Ridge regression adds a penalty term to the least squares objective function, while PCR transforms the predictors into uncorrelated principal components

Selecting the Most Suitable Measure

• The decision between ridge regression and PCR depends on the severity of multicollinearity and the desired interpretability of the model
  • Ridge regression retains all the original predictors, making it suitable when all predictors are considered relevant
  • PCR reduces the dimensionality by using a subset of principal components, which can improve stability but may sacrifice some interpretability
• It is important to assess the effectiveness of the chosen remedial measure by re-evaluating the model assumptions after applying the remedy
  • If the assumption violation persists, alternative measures or a combination of measures may need to be considered
• The selection of the most suitable remedial measure should also take into account the specific context, the goals of the analysis, and the interpretability of the resulting model
  • For example, if the primary goal is prediction accuracy, ridge regression or PCR may be preferred over transformations that alter the scale of the variables
• Consulting with subject matter experts and considering the practical implications of each remedial measure can help guide the decision-making process