Model Assumptions and Diagnostics in R

Regression analysis is a powerful tool for understanding relationships between variables. However, the validity of its results hinges on several key assumptions. This module explores these assumptions and how to diagnose potential violations using R.

Core Assumptions of Linear Regression

For a standard Ordinary Least Squares (OLS) linear regression model to provide unbiased and efficient estimates, several assumptions must hold true. Violations of these assumptions can lead to incorrect inferences and unreliable predictions.

Linearity: The relationship between the independent variables and the mean of the dependent variable is linear.

The model assumes a straight-line relationship. If the true relationship is curved, the linear model will not capture it accurately.

The expected value of the dependent variable (Y) is a linear combination of the independent variables (X1, X2, ..., Xk). Mathematically, E(Y|X) = β₀ + β₁X₁ + ... + βkXk. This means that a one-unit change in an independent variable is associated with a constant change in the dependent variable, holding other variables constant.

Independence of Errors: The errors (residuals) are not correlated with each other.

Each observation's error term should be independent of all other error terms. This is often violated in time-series data or clustered data.

The error term (εᵢ) for observation 'i' is independent of the error term (εⱼ) for observation 'j' where i ≠ j. This assumption is crucial for valid hypothesis testing and confidence intervals. Autocorrelation (serial correlation) is a common violation, where errors are correlated with previous errors.

Homoscedasticity: The variance of the errors is constant across all levels of the independent variables.

The spread of the residuals should be roughly the same for all values of the predictors. Unequal variance is called heteroscedasticity.

Var(εᵢ | X) = σ² for all i. This means the variability of the dependent variable around the regression line is consistent. If the variance increases or decreases with the independent variables, the model is heteroscedastic, which can lead to inefficient estimates and incorrect standard errors.

Normality of Errors: The errors are normally distributed.

The distribution of the residuals should be approximately normal, especially for smaller sample sizes.

εᵢ ~ N(0, σ²). This assumption is important for hypothesis testing and constructing confidence intervals, particularly when the sample size is small. For large sample sizes, the Central Limit Theorem often helps mitigate violations of this assumption for coefficient estimates.

No Perfect Multicollinearity: Independent variables are not perfectly linearly related to each other.

Independent variables should not be exact linear combinations of each other. High multicollinearity can inflate standard errors.

There is no exact linear relationship among two or more independent variables. If perfect multicollinearity exists, the regression coefficients cannot be uniquely estimated. High (but not perfect) multicollinearity can still cause problems, making it difficult to interpret individual predictor effects and increasing the variance of coefficient estimates.

Diagnosing Violations in R

R provides excellent tools for diagnosing potential assumption violations. We'll focus on graphical methods and statistical tests.

Residual Plots

Residual plots are the primary tool for diagnosing linearity, homoscedasticity, and identifying outliers or influential points. The

code

plot()

function applied to a linear model object in R generates a series of diagnostic plots.

The first plot, 'Residuals vs Fitted', is crucial for checking linearity and homoscedasticity. A random scatter of points around the horizontal line at 0 indicates good linearity and homoscedasticity. A curved pattern suggests a linearity violation. A 'fan' or 'cone' shape (residuals widening or narrowing) indicates heteroscedasticity.

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The second plot, 'Normal Q-Q', helps assess the normality of residuals. Points should fall roughly along the diagonal line. Deviations suggest non-normality. The third plot, 'Scale-Location', is another check for homoscedasticity, plotting the square root of the standardized residuals against the fitted values. A horizontal line with random scatter is ideal. The fourth plot, 'Residuals vs Leverage', helps identify influential points that might disproportionately affect the model.

Checking Independence of Errors

For time-series data or data with a natural ordering, checking for autocorrelation is vital. The Durbin-Watson test is commonly used.

The Durbin-Watson statistic ranges from 0 to 4. A value near 2 suggests no autocorrelation. Values significantly less than 2 indicate positive autocorrelation, and values significantly greater than 2 indicate negative autocorrelation.

In R, you can use the

code

durbinWatsonTest

function from the

code

car

package.

Checking Multicollinearity

Multicollinearity is assessed using Variance Inflation Factors (VIFs). High VIF values indicate that a predictor variable is highly correlated with other predictor variables in the model.

In R, the

code

vif()

function (often from the

code

car

package) can be used. A common rule of thumb is that VIFs above 5 or 10 suggest problematic multicollinearity.

Which residual plot is primarily used to check for linearity and homoscedasticity?

The 'Residuals vs Fitted' plot.

What statistical test is commonly used to detect autocorrelation in residuals?

The Durbin-Watson test.

What metric is used to diagnose multicollinearity among predictor variables?

Variance Inflation Factors (VIFs).

Addressing Violations

If assumption violations are detected, several strategies can be employed:

Non-linearity: Transform predictor variables (e.g., log, square root, polynomial terms) or use non-linear regression models.
Heteroscedasticity: Use weighted least squares (WLS) or robust standard errors.
Non-normality of Errors: For large samples, OLS is often robust. For smaller samples, consider transformations or non-parametric methods.
Autocorrelation: Use time-series specific models (e.g., ARIMA) or generalized least squares (GLS) methods.
Multicollinearity: Remove one of the highly correlated variables, combine variables, or use regularization techniques like Ridge or Lasso regression.

Learning Resources

R Documentation: Linear Model Diagnostics(documentation)

Official R documentation for the `lm` function, which includes details on diagnostic plots and model fitting.

An Introduction to Statistical Learning with Applications in R(blog)

A comprehensive book with chapters dedicated to linear regression, model selection, and diagnostics, often with R examples.

DataCamp: Regression Analysis in R(tutorial)

An interactive course covering the fundamentals of regression analysis in R, including assumption checking.

UCLA Statistical Consulting: Regression Analysis(tutorial)

A series of tutorials from UCLA's statistical consulting group, offering practical guidance on regression in R with diagnostic examples.

Towards Data Science: Understanding Regression Assumptions(blog)

A blog post explaining the core assumptions of linear regression and how to check them, often with R code snippets.

Stack Overflow: Checking Regression Assumptions in R(blog)

A collection of Q&A from Stack Overflow, providing practical solutions and discussions on R regression diagnostics.

R-bloggers: Regression Diagnostics(blog)

Aggregates blog posts from various R users, often featuring practical tips and code for regression diagnostics.

Cross Validated: Regression Diagnostics(blog)

A Q&A site for statisticians, featuring in-depth discussions and explanations of regression diagnostics and their interpretation.

The `car` Package Documentation(documentation)

Comprehensive PDF documentation for the `car` (Companion to Applied Regression) package, which provides advanced regression diagnostics like VIF and Durbin-Watson tests.

YouTube: Regression Diagnostics in R(video)

A video tutorial demonstrating how to perform and interpret regression diagnostics in R, often covering residual plots and key tests.