By Chapter 6, regression starts becoming realistic.

Until now, all our models quietly assumed: each variable has one fixed effect.

Example:

If deployment increases sales by 10%, then deployment increases sales by 10% for everyone.

But real life rarely behaves that way.

Maybe:

• deployment helps large customers more,
• discounts work only in December,
• some shapes respond differently to pricing.

This is where Chapter 6 begins.

This chapter introduces one of the most important ideas in statistics: Interaction Effects.

---

What Is an Interaction?

An interaction means: the effect of one variable depends on another variable.

Without interaction:

With interaction:

That sounds abstract.

So let’s make it practical.

---

Example — Deployment and Customer Type

Suppose:

You deploy more diamonds. Sales increase.

Great.

But maybe:

Customer type	Extra Sales per $100k Deployment
Small	2
Large	20

Now deployment effect changes. That is interaction.

---

The Basic Interaction Model

Ordinary regression:

Interaction model:

That last term is interaction.

---

Interpretation of Each Coefficient

---

Intercept:

Baseline.

---

Main effect:

Effect of $X_1$ when $X_2=0.$

---

Second main effect:

Effect of $X_2$ when $X_1=0$.

---

Interaction:

How much one slope changes.

---

Numerical Example

Suppose:

Interpretation:

Base:

100

---

Without promotion:

Deployment adds: 20

---

With promotion:

Deployment adds:

20+30=50

Promotion amplifies deployment.

---

Why Interaction Exists in Real Life

Because systems are heterogeneous.

Examples:

---

Discount × Season

Discounts work differently in December.

---

Customer Count × Month

More customers matter differently by season.

---

Price × Shape

Price sensitivity differs by shape.

---

Aging × Inventory Type

Old inventory behaves differently.

---

Interaction vs Correlation

These are different.

Correlation:

Variables move together.

Interaction:

Variables change each other’s effects.

---

Example

Height and weight may correlate.

But: medicine effectiveness depending on age is interaction.

---

Interaction in GLMs

Interaction works exactly the same.

Example:

Poisson regression:

Interpretation changes:

Effects become multiplicative.

---

Example — Sales Counts

Suppose:

Monthly sales:

Exponentiate.

Discount effect outside December:

22% increase.

Discount effect inside December:

=2.01

Now discount doubles sales.

Huge difference.

---

Interaction in Logistic Regression

Model:

Interpretation:

Odds ratio changes.

---

Example:

Customer count changes purchase probability differently by month.

---

Continuous × Continuous Interaction

Example:

Price × Deployment.

Model:

Meaning:

Price sensitivity changes as deployment changes.

---

Continuous × Categorical

Example:

CustomerCount × Region.

Interpretation:

Customer growth behaves differently across regions.

---

Categorical × Categorical

Example:

Shape × Color.

Interpretation:

Preference combinations.

---

Why Interaction Terms Are Estimated

You asked this earlier:

Why do we add interaction terms?

Because:

we do not know interaction strength.

Model estimates:

from data.

If:

No interaction.

---

If: large: Strong interaction.

---

Hierarchical Principle

Very important.

If you include:

always include:

and:

Never include interaction alone.

---

Real Business Example

Suppose:

You want inventory forecasting.

Model:

Interpretation:

Customer growth behaves differently by season.

Exactly the type of problem you were discussing.

---

Splines and Interaction

Now suppose customer count effect is nonlinear.

Then:

instead of:

Now:

interaction becomes:

This enters: GAM territory.

Very powerful.

---

Visualization of Interaction

Without interaction:

Parallel lines.

With interaction:

Lines diverge.

That visual intuition is important.

---

Statistical Testing

How do we know interaction matters?

Compare:

Model 1:
No interaction

Model 2:
Interaction

Use:

• likelihood ratio test,
• deviance,
• AIC,
• BIC.

---

Inventory Example

Suppose:

December demand:

high.

Customer growth:

high.

Interaction captures:

December amplifies customer effect.

Without interaction:

inventory underestimated.

---

Chapter 6’s Big Lesson

Chapter 6 teaches:

variables do not act alone.

Effects depend on context.

And averages often hide reality.

---

Final Thought

Before Chapter 6:

you ask:

“What is the effect?”

After Chapter 6:

you ask:

“Does the effect change?”

That shift changes everything.

Interaction effects turn regression from:

simple prediction

into

understanding how systems actually behave.