Introduction to (log) Odds Ratio

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Introduction to (log) Odds Ratio

Statistics and Methodology Anna Lobanova: a.lobanova@ai.rug.nl

Who is more likely to drink beer on Queen's Day - students or teachers?

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Example: Who is more likely to drink beer on Queen's Day students or teachers?

Students Teachers Total

Drink 90 20 110

Don't drink 10 80 90

Total 100 100 200

Group 1 = students, group 2 = teachers

Event ? drinking beer at Queen's Day

Question: Is one group more likely to drink beer on Queen's Day than the other group? Or is this event independent of professional status?

Terminology & Notation:

A Two-way Contingency Table

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Drink

Students n11

Teachers n21

Total

n+1

Don't drink

Total

n12

n1+

n22

n2+

n+2

n

Terminology & Notation:

A Two-way Contingency Table

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response variable

explanatory variable

Drink Don't drink

Total

Students Teachers Total

n11 = 90 n12 = 10 n21 = 20 n22 = 80 n+1 = 110 n+2 = 90

n1+ = 100 n2+ = 100 n = 200

joint distribution

marginal distributions

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Example 1: Let's use odds ratio to find out!

Students Teachers Total

Drink n11 = 90 n21 = 20 n+1 = 110

Don't drink n12 = 10 n22 = 80 n+2 = 90

Total n1+ = 100 n2+ = 100 n = 200

Step 1: the odds of a student drinking beer is 90 to 10 or 9/1 and the odds of a teacher drinking beer is 20 to 80 or 1/4 = 0.25:1

Step 2: the probability of success for every cell is

11 = n11/n1+ = 90/100 = 0.9 21 = n21/n2+ = 20/100 = 0.2

12 = n12/n1+ = 10/100 = 0.1 22 = n22/n2+ = 80/100 = 0.8

Step 3: Odds Ratio () = 0.9 / 0.1 = 0.72 = 36

0.2 / 0.8 0.02

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11 22

Odds Ratio () =

21 12

Inference from odds ratio:

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If

Then

odds ratio = 1

the event is equally likely in both groups

odds ratio > 1

the event is more likely in Group 1

odds ratio < 1

the event is more likely in Group 2

the greater the number the stronger the association

is never a negative number

In example 1:

odds ratio = 36

students are much more likely to drink beer than teachers!

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