In our previous editions, we referred to the variable of interest that we aim to explain or predict as the response, outcome, or dependent variable.

In classical linear regression analysis, this dependent variable is typically numeric. To perform regression analysis using the Ordinary Least Squares (OLS) method, certain assumptions must be met. For a comprehensive overview, refer to our 7th Edition: Assumptions and Coefficient Estimates in Multiple Linear Regression.

However, in practical scenarios, these assumptions are not always fully satisfied. When the response variable is continuous but deviates from normality, variable transformations can often help in fitting the regression model successfully.

Common transformations include taking the square root, logarithm, or exponential of the outcome/dependent or predictor variables. These transformations are essential components of the feature engineering process, performed after thorough exploratory data analysis.

Despite these adjustments, the resulting regression model may still encounter issues like overfitting. In such cases, alternative regression techniques like Ridge Regression or Lasso Regression introduce regularization to address the limitations of OLS by penalizing large coefficients. These methods differ fundamentally in their fitting approach, substituting OLS with regularized estimations. We plan to delve deeper into these transformations and alternatives in future editions.

Moving Beyond Continuous Numeric Outcomes

In some cases, the response variable may not be numeric or may represent meaningful non-continuous values, such as binary categories or counts. For instance, you may want to explain a count variable or a binary outcome while still leveraging the interpretability of a linear combination of predictors. This is where the Generalized Linear Model (GLM) becomes invaluable.

A GLM extends traditional linear regression by accommodating a broader range of data distributions and relationships between variables. Instead of assuming a normal distribution for the response variable, GLMs allow it to follow any distribution from the exponential family (e.g., normal, binomial, Poisson).

A Closer Look at the Generalized Linear Model

In classical linear regression …

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