7.1.6 MLR - Residual Analysis¶

Objective: This session examines the application of both the F-test and T-test in multiple linear regression to assess the significance of model parameters and the overall model.
Context: Understanding these tests helps to validate the influence of independent variables on the dependent variable within the regression framework.

Scenario: Analyzing Hanumantha's dataset which explores monthly credit card expenditure as a function of annual income and household size.
Hypothesis Setup:
Null Hypothesis (\(H_0\)): \(\beta_1 = \beta_2 = 0\) (Neither annual income nor household size affects monthly spend).
Alternative Hypothesis (\(H_a\)): \(\beta_1 \neq 0\) or \(\beta_2 \neq 0\) (At least one of the coefficients significantly affects monthly spend).
F-test Results:
Derived from the regression output, the F-statistic calculated as the ratio of the mean square regression (MSR) to the mean square error (MSE) is 104.63, with a p-value close to 0, indicating a strong rejection of \(H_0\).

Purpose: To determine the significance of individual regression coefficients.
Procedure:
For each parameter (\(\beta_i\)), the null hypothesis (\(H_0\)) is \(\beta_i = 0\) (the variable does not affect the dependent variable).
The alternative hypothesis (\(H_a\)) is \(\beta_i \neq 0\) (the variable significantly affects the dependent variable).
Calculations:
The test statistic is computed as the ratio of the estimated coefficient (\(b_i\)) to its standard error (s of \(b_i\)).
For example, \(b_1\) (associated with annual income) was found to be significant with a test statistic of 10.28, and a p-value close to 0.
Similarly, \(b_2\) (associated with household size) showed a test statistic of 6.29, further supporting significant influence.

Insights from Analysis:
Both the F-test and individual T-tests validate significant relationships in the model, suggesting that both annual income and household size significantly impact monthly credit card expenditure.
Next Steps:
Building on these findings, future sessions will explore further diagnostic checks and additional regression methods to refine the predictive capabilities of the models developed.

Interactive Learning: The session included interactive discussions led by Tejas and Ashwini, helping to contextualize theoretical concepts within practical applications, enhancing learning outcomes for all participants.