⛽️Business Analytics Unit 6 – Regression Analysis

What's Regression Analysis?

• Statistical technique used to model and analyze the relationship between a dependent variable and one or more independent variables
• Helps understand how changes in independent variables are associated with changes in the dependent variable
• Enables prediction of the dependent variable based on known values of independent variables
• Assumes a linear relationship between the dependent and independent variables
• Useful for identifying trends, making forecasts, and supporting decision-making processes
• Can be used to test hypotheses about the relationships between variables
• Provides a measure of the strength and direction of the relationship between variables through correlation coefficients

Types of Regression Models

• Simple linear regression
  • Models the relationship between one independent variable and one dependent variable
  • Equation: $y = \beta_0 + \beta_1x + \epsilon$, where $y$ is the dependent variable, $x$ is the independent variable, $\beta_0$ is the y-intercept, $\beta_1$ is the slope, and $\epsilon$ is the error term
• Multiple linear regression
  • Models the relationship between multiple independent variables and one dependent variable
  • Equation: $y = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n + \epsilon$, where $y$ is the dependent variable, $x_1, x_2, ..., x_n$ are independent variables, $\beta_0$ is the y-intercept, $\beta_1, \beta_2, ..., \beta_n$ are slopes for each independent variable, and $\epsilon$ is the error term
• Logistic regression
  • Used when the dependent variable is categorical (binary or multinomial)
  • Models the probability of an event occurring based on independent variables
  • Equation: $\ln(\frac{p}{1-p}) = \beta_0 + \beta_1x_1 + \beta_2x_2 + ... + \beta_nx_n$, where $p$ is the probability of the event occurring
• Polynomial regression
  • Models non-linear relationships between the dependent and independent variables by including higher-order terms (squared, cubed, etc.) of the independent variables
• Stepwise regression
  • Iterative process of adding or removing independent variables to find the best-fitting model
  • Forward selection starts with no variables and adds them one by one
  • Backward elimination starts with all variables and removes them one by one

Key Concepts and Assumptions

• Linearity assumes a linear relationship between the dependent and independent variables
• Independence assumes that observations are independent of each other (no autocorrelation)
• Homoscedasticity assumes constant variance of the residuals across all levels of the independent variables
• Normality assumes that the residuals are normally distributed
• No multicollinearity assumes that independent variables are not highly correlated with each other
• Outliers and influential points can significantly impact the regression results and should be identified and addressed
• Residuals are the differences between the observed and predicted values of the dependent variable
• Coefficient of determination ($R^2$) measures the proportion of variance in the dependent variable explained by the independent variables

Building a Regression Model

• Define the problem and identify the dependent and independent variables
• Collect and preprocess data, handling missing values, outliers, and transforming variables if necessary
• Split the data into training and testing sets for model validation
• Select the appropriate regression model based on the nature of the problem and the relationships between variables
• Estimate the model parameters using the training data
  • Ordinary Least Squares (OLS) is a common method for estimating parameters in linear regression
  • Maximum Likelihood Estimation (MLE) is often used for logistic regression
• Assess the model's performance using evaluation metrics such as Mean Squared Error (MSE), Root Mean Squared Error (RMSE), and $R^2$
• Refine the model by adding or removing variables, transforming variables, or trying different regression techniques
• Validate the model using the testing data to ensure its generalizability

Interpreting Regression Results

• Coefficient estimates indicate the change in the dependent variable associated with a one-unit change in the corresponding independent variable, holding other variables constant
• P-values determine the statistical significance of each independent variable
  • A low p-value (typically < 0.05) suggests that the variable has a significant impact on the dependent variable
• Confidence intervals provide a range of plausible values for the coefficient estimates
• Standardized coefficients (beta coefficients) allow for comparison of the relative importance of independent variables
• Residual plots help assess the model's assumptions and identify patterns or issues in the residuals
• Interaction terms can be included to model the combined effect of two or more independent variables on the dependent variable

Checking Model Fit and Diagnostics

• Residual analysis
  • Plot residuals against predicted values to check for patterns or heteroscedasticity
  • Plot residuals against each independent variable to identify non-linear relationships
• Normality tests (Shapiro-Wilk, Kolmogorov-Smirnov) assess the normality of residuals
• Homoscedasticity tests (Breusch-Pagan, White's test) check for constant variance of residuals
• Multicollinearity diagnostics (Variance Inflation Factor, correlation matrix) identify highly correlated independent variables
• Influential point analysis (Cook's distance, leverage values) identifies observations that have a disproportionate impact on the regression results
• Cross-validation techniques (k-fold, leave-one-out) assess the model's performance on unseen data
• Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) compare the relative quality of different models

Real-World Applications in Business

• Demand forecasting predicts future product demand based on historical sales data, price, and other relevant factors
• Price optimization determines the optimal price for a product or service based on demand, competition, and other market factors
• Customer churn prediction identifies customers likely to stop using a product or service based on their characteristics and behavior
• Credit risk assessment evaluates the likelihood of a borrower defaulting on a loan based on their credit history, income, and other factors
• Marketing campaign effectiveness measures the impact of marketing activities on sales, customer acquisition, or other key performance indicators
• Inventory management optimizes stock levels based on demand forecasts, lead times, and other supply chain factors
• Sales performance analysis identifies the factors that contribute to sales success, such as salesperson characteristics, product features, or market conditions

Common Pitfalls and How to Avoid Them

• Overfitting occurs when a model is too complex and fits the noise in the data rather than the underlying pattern
  • Regularization techniques (Ridge, Lasso) can help prevent overfitting by shrinking coefficient estimates
  • Use cross-validation to assess model performance on unseen data
• Underfitting occurs when a model is too simple and fails to capture the true relationship between variables
  • Consider adding more relevant variables or using a more flexible model (polynomial, interaction terms)
• Ignoring multicollinearity can lead to unstable coefficient estimates and difficulty interpreting the model
  • Check correlation matrix and VIF to identify highly correlated variables
  • Consider removing one of the correlated variables or using dimensionality reduction techniques (PCA)
• Extrapolating beyond the range of the data can lead to unreliable predictions
  • Be cautious when making predictions for values outside the range of the training data
• Ignoring outliers and influential points can distort the regression results
  • Identify and investigate outliers and influential points using diagnostic measures (Cook's distance, leverage)
  • Consider removing or adjusting these observations if they are due to data entry errors or other issues
• Misinterpreting p-values and statistical significance
  • A statistically significant result does not necessarily imply practical significance or a strong effect size
  • Consider the magnitude of the coefficients and the context of the problem when interpreting results