By Chapter 5, something interesting happens.

Until now, we modeled:

• continuous outcomes,
• count outcomes,
• probabilities.

But what if there is no obvious “response variable”? What if we simply want to understand: Are variables related?

Examples:

• Does shape affect sales?
• Does color affect purchase?
• Does clarity influence conversion?
• Does region affect product preference?

Now we move into: Loglinear Models

One of the most underrated parts of GLMs.

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The Main Question of Chapter 5

Suppose we observe:

Round sold 120,
Ovals sold 60.

Question: Is shape related to sales? Or: Are they independent?

That’s what loglinear models solve.

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What Type of Data Are We Modeling?

Counts.

But not ordinary counts.

We model: counts inside categories.

Example:

Shape	Sold	Not Sold
Round	120	80
Oval	60	140

Each cell contains: a count.

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Why Ordinary Regression Does Not Work

Suppose we predict:

Problem:

These are:

• frequencies,
• discrete counts.

Also:

• expected values must remain positive.

So ordinary regression becomes awkward.

Instead:

Loglinear models assume:

for cell counts.

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The Core Loglinear Model

We model:

where:

• $\mu_{ij} = expected count,$
• $\lambda = overall mean,$
• $\lambda_i^A = effect of factor A,$
• $\lambda_j^B = effect of factor B.$

At first this looks scary.

But the idea is simple.

We model: expected cell counts.

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Example — Diamond Shape × Sales

Suppose:

Shape	Count
Round	100
Oval	60
Cushion	40

Then:

The model estimates: expected counts for each category.

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Independence — The Most Important Idea

Suppose we want to know:

Does shape affect whether something sells?

If independent: Expected counts become:

Shape	Sold	Not Sold	Total
Round	60	40	100
Oval	20	80	100
Total	80	120	200

Expected Round Sold:

Observed: 60

Expected 40.

Difference suggests: dependence.

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Independence Model

Model:

Interpretation:

• Shape influences counts.
• Sales influences counts.
• But no interaction.

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Interaction — The Heart of Chapter 5

Now suppose shape changes sales.

Add interaction:

This term means: the effect of one variable depends on another.

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Numerical Example

Observed:

Shape	Sold	Not Sold
Round	90	10
Oval	10	90

Expected under independence:

Shape	Sold	Not Sold
Round	50	50
Oval	50	50

Huge mismatch.

Interaction becomes necessary.

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Three-Way Tables

Chapter 5 becomes powerful when adding more variables.

Example:

| Shape | Color | Sold |

Now model:

Possible interactions:

• Shape × Color
• Shape × Sale
• Color × Sale
• Shape × Color × Sale

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Interpretation of Interaction

Suppose:

Round diamonds:

• sell well in DEF.

Oval:

• sell well in GHI.

Now:

Shape effect changes by color. That becomes interaction.

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Hierarchical Principle

One of the most important ideas.

If you include:

Shape × Color

you must include:

• Shape
• Color

Do not include interaction alone.

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Likelihood Ratio Tests

How do we know if interaction matters?

Compare:

Model 1: independence.

Model 2: interaction.

Statistic:

where:

• O = observed,
• E = expected.

Large values: interaction exists.

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Pearson Chi-Square

Alternative measure:

Measures:

difference between:

• observed,
• expected.

Large values:
→ poor fit.

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Connection to Logistic Regression

Interesting fact:

Logistic regression is actually connected to loglinear models.

Example:

If outcome is fixed:

| Shape | Sold |

Logistic and loglinear models often become equivalent.

That’s why these chapters sit next to each other.

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Shape × Weight × Sold

Question:

Which combinations perform best?

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Director × Customer Segment × Sale

Question:

Are some directors better in certain segments?

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Deployment × Region × Conversion

Question:

Does deployment work differently by region?

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Customer Type × Product Type × Churn

Question:

Are some combinations unstable?

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Inventory Example

Suppose:

Category	Sold
LGRD 0.23–0.27	20
LGRD 0.28–0.32	40

Expected: 30 for both.

Observed differs.

Interaction may exist.

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Why Chapter 5 Matters

Before Chapter 5: you modeled outcomes.

After Chapter 5: you start modeling: relationships between categories.

That is a major shift.

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The Deep Lesson

Chapter 5 teaches: counts are not enough.

What matters is: how categories combine.

And interactions often explain reality better than averages.

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Final Thought

Loglinear models are easy to overlook.

But they quietly power:

• market basket analysis,
• contingency analysis,
• segmentation,
• categorical analytics,
• association discovery,
• business intelligence.

And once you learn them,

you stop asking:

“How many?”

and start asking:

“What combinations matter?”