ISYE 6414 — Georgia Tech Module 2 Study Guide | 2026/2027 Updated A+ Questions and Answers | Probability, Statistical Modeling & Data Analysis

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Module 2: Study Guide
ISYE 6414 — Module 2 Study Guide
Multiple Linear Regression
A cleaner, clearer re-teaching of the lecture material

Table of Contents
1. What Is Multiple Linear Regression?
2. Model Types: First Order, Second Order, and Interactions
3. Qualitative Predictors in Multiple Regression
4. Estimation (OLS in Matrix Form)
5. Interpretation: Marginal vs. Conditional Models
6. Roles of Variables: Controlling, Explanatory, Predictive
7. Statistical Inference
8. Estimation vs. Prediction (Generalized)
9. Assumptions & Diagnostics
10. Model Evaluation & Multicollinearity
11. Case Studies Summary
12. Large Sample Size & Inflated P-values

🔗 Bridge from Module 1
Module 1 covered two special cases:
- Simple Linear Regression — one quantitative predictor
- ANOVA — one qualitative predictor (converted to k−1 dummy variables)
Module 2 is the unification of both. Multiple linear regression allows any number
of predictors, both quantitative and qualitative, in the same model. All the tools


Module 2: Study Guide 1

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from Module 1 — OLS estimation, inference, R², diagnostics — carry forward and
generalize.
The only things that change:
- The notation shifts to matrices (more variables = vectors of betas)
- The denominator in MSE changes from n − 2 to n − p − 1 (where p = number
of predictors)
- Coefficient interpretation must always be stated conditional on other variables
being held fixed
- New challenges appear: multicollinearity and interpretation pitfalls

1. What Is Multiple Linear Regression?
The Model
Yi = β0 + β1 xi1 + β2 xi2 + ⋯ + βp xip + εi
​ ​ ​ ​ ​ ​ ​ ​ ​




Symbol Meaning
Yi ​ Response variable for observation i
xij ​ Value of predictor j for observation i
β0 ​
Intercept — expected Y when all predictors = 0
Expected change in Y for 1-unit increase in Xj , holding all other
βj
​




predictors fixed
εi ​
Error term
p Number of predictors

The Four Assumptions (Same as Module 1)
1. Linearity (Mean-Zero): E[εi ] = 0 — no systematic pattern the model is
​




missing
2. Constant Variance: Var(εi ) = σ 2 for all i
​




3. Independence: error terms are independent of each other
4. Normality: εi ∼ N (0, σ 2 ) — needed for inference
​




Module 2: Study Guide 2

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Matrix Formulation
Stack everything into vectors and matrices:

Y = Xβ + ε

Y: n × 1 vector of responses
X: n × (p + 1) design matrix — first column is all 1’s (for intercept), then one
column per predictor
β : (p + 1) × 1 vector of regression coefficients
ε: n × 1 vector of error terms
Why matrix form? Because it makes the math compact and general — all formulas
for estimation, inference, and prediction can be written cleanly without writing out
every term.

The Three Goals (Same as Module 1)
1. Prediction — Predict Y for new values of the predictors
2. Modeling — Understand which predictors explain Y and how
3. Hypothesis testing — Is the relationship statistically significant?

2. Model Types: First Order, Second Order, and
Interactions
“Linear” in linear regression refers to linearity in the parameters (β ’s), not
necessarily in the predictors themselves. This means you can include squared
terms and interaction terms and it’s still a linear regression.

First Order Model (Baseline)
Y = β 0 + β 1 x1 + β 2 x2 + ε
​ ​ ​ ​ ​




Each predictor has an additive, constant effect
If you plot E[Y ] vs. x2 for different values of x1 : parallel lines (constant
​ ​




intercept shifts, same slope)

Module 2: Study Guide 3

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