1. Assessing Model Assumptions:
Definition: Assessing model assumptions is a crucial step in regression analysis to ensure that the statistical model used is appropriate for the data. Regression models, including linear regression, logistic regression, and Poisson regression, rely on several key assumptions:
Linearity: The relationship between the dependent and independent variables is linear. This assumption can be checked using diagnostic plots like scatterplots and residual plots.
Independence: The observations are independent of each other. In time series data, for example, this assumption may not hold, and special techniques like autoregressive models are needed.
Homoscedasticity: The variance of the residuals (the differences between observed and predicted values) is constant across all levels of the independent variable(s). Residual plots and statistical tests like the Breusch-Pagan test can help assess homoscedasticity.
Normality of Residuals: The residuals follow a normal distribution. This can be examined through quantile-quantile (Q-Q) plots or statistical tests like the Shapiro-Wilk test.
2. Identification of Outliers and Influential Observations:
Outliers: Outliers are data points that deviate significantly from the rest of the data. They can affect the model’s fit and assumptions. Techniques for identifying outliers include:
- Visual inspection of scatterplots or residual plots.
- Calculating standardized residuals (z-scores) and identifying values that fall outside a certain threshold (e.g., z-score greater than 3 or -3).
- Using robust regression techniques that are less sensitive to outliers, such as robust regression or the Theil-Sen estimator.
Influential Observations: Influential observations have a strong impact on the regression model. They can be outliers or data points that have a high leverage on the regression coefficients. Techniques for identifying influential observations include:
- Calculating Cook’s distance, which measures the influence of each observation on the regression coefficients. Large Cook’s distances indicate influential observations.
- Identifying observations with high leverage, often assessed using the hat matrix or leverage values.
- Checking for high leverage and high residual values, as these observations can be both outliers and influential.
3. Techniques for Model Improvement:
Model improvement techniques aim to enhance the model’s fit, accuracy, and predictive power. Some strategies include:
Variable Selection: Choose the most relevant independent variables by performing feature selection techniques like stepwise regression, Lasso regression, or forward/backward selection.
Transformation: Transform variables to achieve linearity or stabilize variance. Common transformations include logarithmic, square root, or Box-Cox transformations.
Residual Analysis: Examine the residuals to detect patterns or heteroscedasticity. If issues are identified, consider using weighted least squares or robust regression techniques.
Interaction Terms: Include interaction terms (cross-products) between independent variables to capture complex relationships that may not be apparent in the main effects model.
Multicollinearity Mitigation: Address multicollinearity (high correlation between independent variables) by removing or combining variables or using ridge regression.
Cross-Validation: Use cross-validation techniques like k-fold cross-validation to assess model performance and prevent overfitting.
Regularization: Apply regularization techniques like L1 (Lasso) or L2 (Ridge) regularization to prevent overfitting and improve model stability.
Model Comparison: Compare different models using information criteria (e.g., AIC or BIC) to choose the best-fitting model.
In summary, assessing model assumptions, identifying outliers and influential observations, and employing model improvement techniques are essential steps in regression analysis. These practices help ensure that regression models are valid, robust, and capable of providing accurate insights and predictions from the data.
