A bad residual plot may mean:

Nonlinear trend → try polynomial or transformation

Unequal spread → violates constant variance

Outliers → may influence fit heavily

Omitted variables → need multiple regression
Concept
Residuals
Diagnostic plots
Intrinsically linear
Polynomial regression
Adjusted 𝑅 2
Multiple regression
Signs of model failure
What to Know
Show error between observed and
predicted values
Help assess if linear model is appropriate
Can become linear with log/power
transformations
Handles curve; still fits via least squares
Penalizes extra variables; used in
polynomials & multiple regression
Models’ multiple predictors: each has its
own 𝛽𝑗
Patterns in residuals, non-normal errors,
missing predictors
WHAT IS MULTIPLE REGRESSION?

A statistical model that explains a response variable 𝑌 using multiple predictors
𝑥1 , 𝑥2 , … , 𝑥𝑘

The goal is to understand how each predictor relates to 𝑌, while holding others
constant
🔹 WHY USE MULTIPLE REGRESSION?

To model complex relationships where one variable isn't enough

To control for confounding variables

To improve prediction accuracy

To determine the individual effect of each predictor on the outcome

Each coefficient shows the effect of a predictor on 𝑌 while holding all other
variables constant

Positive coefficient: as the variable increases, 𝑌 tends to increase

Negative coefficient: as the variable increases, 𝑌 tends to decrease
🔹 MODEL FIT (CONCEPTUAL ONLY)

𝑅 2 : proportion of variation in 𝑌 explained by the model

Adjusted 𝑅 2 : accounts for the number of predictors; discourages overfitting
🔹 ASSUMPTIONS (QUALITATIVE KNOWLEDGE ONLY)
1. Linearity – Each predictor has a linear relationship with 𝑌
2. Independence – Observations are not related to each other
3. Normality – Errors (residuals) should follow a normal distribution
4. Constant Variance – Residuals should have roughly equal spread (no fanning)
5. No Perfect Multicollinearity – Predictors should not be perfectly correlated
🔹 COMMON ISSUES

Multicollinearity: predictors are highly correlated, making coefficients unreliable

Overfitting: too many variables fit the training data well but fail on new data

Omitted Variable Bias: leaving out important variables skews results

Non-random residuals: suggest the model may be missing important structure
(e.g., nonlinearity)
🔹 QUALITATIVE QUESTIONS TO EXPECT

What does coefficient represent in multiple regression?

Why use multiple regression instead of simple regression?

What does a high or low 𝑅 2 or adjusted 𝑅 2 tell you?

What is multicollinearity and why is it a problem?

How can you tell if a model is missing something important?