12.1 Multiple Regression Model and Assumptions

Multiple regression models analyze how several factors influence an outcome simultaneously. This powerful tool is used in business for forecasting demand, setting prices, evaluating employee performance, and predicting customer churn.

The model assumes linearity, independence, homoscedasticity, normality, and no multicollinearity. These assumptions are crucial for accurate predictions and reliable insights. Violating them can lead to biased estimates and invalid conclusions, affecting business decisions.

Multiple Regression Model

Concept of multiple regression

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• Models the relationship between a dependent variable and two or more independent variables (predictors)
• Analyzes how multiple factors simultaneously influence a single outcome
• Used in business for demand forecasting (predicting sales based on price, advertising, and competitor actions), pricing strategy (determining optimal price considering production costs, market trends, and consumer behavior), employee performance (analyzing impact of experience, education, and training), and customer churn (identifying key drivers of attrition based on demographic, behavioral, and transactional data)

Key assumptions of regression

• Linearity assumes the relationship between the dependent variable and each independent variable is linear
  • Violations can lead to biased estimates and inaccurate predictions
• Independence assumes the errors (residuals) are independent of each other
  • Autocorrelation or clustering of errors can invalidate the model
• Homoscedasticity assumes the variance of the errors is constant across all levels of the independent variables
  • Heteroscedasticity (non-constant variance) can affect the reliability of the model's estimates
• Normality assumes the errors are normally distributed with a mean of zero
  • Non-normality can impact the validity of statistical tests and confidence intervals
• No multicollinearity assumes the independent variables are not highly correlated with each other
  • High multicollinearity can lead to unstable estimates and difficulty interpreting individual effects of predictors

Multiple Regression Equation and Assumptions

Formulation of regression equations

• General form of a multiple regression equation: $Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + ... + \beta_pX_p + \epsilon$
  • $Y$: Dependent variable
  • $\beta_0$: Intercept (constant term)
  • $\beta_1, \beta_2, ..., \beta_p$: Regression coefficients for each independent variable
  • $X_1, X_2, ..., X_p$: Independent variables (predictors)
  • $\epsilon$: Error term (residual)
• Predicting sales ($Y$) based on advertising expenditure ($X_1$) and price ($X_2$): $Sales = \beta_0 + \beta_1Advertising + \beta_2Price + \epsilon$

Validity assessment of assumptions

• Graphical methods assess assumptions visually
  • Residual plot: Scatter plot of residuals against predicted values to check linearity and homoscedasticity
  • Normal probability plot: Compares distribution of residuals to normal distribution to assess normality
• Statistical tests quantify assumption violations
  • Durbin-Watson test checks for autocorrelation in residuals
  • Breusch-Pagan test assesses presence of heteroscedasticity
  • Shapiro-Wilk test tests for normality of residuals
• Multicollinearity diagnostics identify high correlations between independent variables
  • Correlation matrix identifies high correlations
  • Variance Inflation Factor (VIF) quantifies severity of multicollinearity for each predictor
    • VIF values above 5 or 10 indicate potential multicollinearity issues