Statistical Methods
for Economists
Lecture (7 & 8)c
Three-Way Analysis of Variance
(ANOVA)
— Latin Squares
•David Bartl
•Statistical Methods for Economists
•INM/BASTE

Outline of the lecture
•Three-way ANOVA:  Introduction
•Latin squares
•Three-way ANOVA simplified by using Latin squares

Three-factor ANOVA:  Motivation:  Example
•We have:
•a set of distinct cars
•a set of distinct drivers
•several types of car-fuel (e.g. fuel with various additives)
•
•We wish to test whether the mileage (the fuel consumption per 100 km) of the car depends also upon
the driver who drives the car and on the type of the fuel.

Three-factor ANOVA:  Motivation:  Example



Three-factor ANOVA:  Motivation & Introduction



Three-factor ANOVA:  Introduction:  Assumptions



Three-factor ANOVA:  Introduction:  Assumptions



Three-factor ANOVA:  Introduction:  Assumptions



Three-factor ANOVA:  Introduction:  Assumptions
and


Three-factor ANOVA:  Simplification



Three-factor ANOVA:  Simplification



Latin squares
in
Three-way ANOVA
•
•

Latin square
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Latin square:  Remark
•The name “Latin square”  is inspired by the work of Leonhard Euler (1707–1783), a Swiss
mathematician, physicist, astronomer, geographer, logician, and engineer  who used the upper-case
letters of the Latin alphabet as the symbols in the square:
A
B
C
B
C
A
C
A
B
A
B
C
C
A
B
B
C
A
A
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B
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C
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B
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E
A
C
D
E
A
B
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E
A
B
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E
A
B
C
D

Three-factor ANOVA:  Simplification by using Latin sq.



Three-factor ANOVA:  Simplification by using Latin sq.



Three-factor ANOVA:  Simplification by using Latin sq.



Three-factor ANOVA:  Further assumptions



Three-factor ANOVA:  Further assumptions



Three-factor ANOVA:  Further assumptions



Three-Way ANOVA without
interactions
and simplified
by using
Latin squares
Yijk1=μ+αi+βj+γk+
                         +εijk1
•
•

Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions:  Notation



Three-way ANOVA with no interactions:  Notation



Three-way ANOVA with no interactions:  Notation
(cf. one-way ANOVA)


Three-way ANOVA with no interactions:  Notation



Three-way ANOVA with no interactions:  Notation



Three-way ANOVA with no interactions:  Notation



Three-way ANOVA with no interactions:  Notation



Three-way ANOVA with no interactions:  Dimensions



Three-way ANOVA with no interactions:  Dimensions



Three-way ANOVA with no interactions:  Dimensions



Three-way ANOVA with no interactions
0


Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions



Three-way ANOVA with no interactions:  Test for  HC



Three-way ANOVA with no interactions:  Test for  HC
0
subspace of dimension
(the orthogonal
complement =
= the space of
the residuals)
subspace
of dimension
subspace of dimension
It holds:

Three-way ANOVA with no interactions:  Test for  HC



Three-way ANOVA with no interactions:  Test for  HC



Three-way ANOVA with no interactions:  Test for  HC