Statistical Methods
for Economists
Lecture (7 & 8)d
Four-Way Analysis of Variance
(ANOVA)
— Græco-Latin Squares
•David Bartl
•Statistical Methods for Economists
•INM/BASTE
[ Graeco / Greco ]

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

Four-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)
•several types of tyres
•
•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, the type of the fuel, and on the type of the tyres.

Four-factor ANOVA:  Motivation:  Example



Four-factor ANOVA:  Motivation:  Example



Four-factor ANOVA:  Motivation & Introduction



Four-factor ANOVA:  Introduction:  Assumptions



Four-factor ANOVA:  Introduction:  Assumptions



Four-factor ANOVA:  Introduction:  Assumptions



Four-factor ANOVA:  Simplification



Four-factor ANOVA:  Simplification



Græco-Latin squares
in
Four-way ANOVA
•
•

Latin square
1
2
3
2
3
1
3
1
2
1
2
3
3
1
2
2
3
1
1
2
3
4
2
3
4
1
3
4
1
2
4
1
2
3
1
2
3
4
3
4
1
2
4
3
2
1
2
1
4
3
1
2
3
4
5
2
3
4
5
1
3
4
5
1
2
4
5
1
2
3
5
1
2
3
4

Orthogonal Latin squares
1
2
3
2
3
1
3
1
2
1
2
3
3
1
2
2
3
1
&
1, 1
2, 2
3, 3
2, 3
3, 1
1, 2
3, 2
1, 3
2, 1
=

Orthogonal Latin squares
&
=
1
2
3
4
2
1
4
3
3
4
1
2
4
3
2
1
1
2
3
4
3
4
1
2
4
3
2
1
2
1
4
3
1, 1
2, 2
3, 3
4, 4
2, 3
1, 4
4, 1
3, 2
3, 4
4, 3
1, 2
2, 1
4, 2
3, 1
2, 4
1, 3

Orthogonal Latin squares



Orthogonal Latin squares



Orthogonal Latin squares



Orthogonal Latin squares:  The following is known:



Græco-Latin squares
•A pair of orthogonal Latin squares is also called a Græco-Latin square or
Graeco-Latin square or Greco-Latin square.
The name “Græco-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
and the lower-case letters of the Greek alphabet as the symbols in the respective square:
&
=
A
B
C
D
B
A
D
C
C
D
A
B
D
C
B
A
α
β
γ
δ
γ
δ
α
β
δ
γ
β
α
β
α
δ
γ
Aα
Bβ
Cγ
Dδ
Bγ
Aδ
Dα
Cβ
Cδ
Dγ
Aβ
Bα
Dβ
Cα
Bδ
Aγ

Four-factor ANOVA  &  Græco-Latin squares



Four-factor ANOVA  &  Græco-Latin squares



Four-factor ANOVA  &  Græco-Latin squares



Four-factor ANOVA:  Further assumptions



Four-factor ANOVA:  Further assumptions



Four-factor ANOVA:  Further assumptions



Four-Way ANOVA without
interactions
and simplified
by using
Græco-Latin squares
Yijkl1=μ+
  +αi+βj+γk+δl+εijk1
•
•

Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions:  Notation



Four-way ANOVA with no interactions:  Notation



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


Four-way ANOVA with no interactions:  Notation



Four-way ANOVA with no interactions:  Notation



Four-way ANOVA with no interactions:  Notation



Four-way ANOVA with no interactions:  Notation



Four-way ANOVA with no interactions:  Notation



Four-way ANOVA with no interactions:  Dimensions



Four-way ANOVA with no interactions:  Dimensions



Four-way ANOVA with no interactions:  Dimensions



Four-way ANOVA with no interactions:  Dimensions



Four-way ANOVA with no interactions:  Dimensions



Four-way ANOVA with no interactions:  Dimensions



Four-way ANOVA with no interactions
0


Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions



Four-way ANOVA with no interactions:  Test for  HD



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

Four-way ANOVA with no interactions:  Test for  HD



Four-way ANOVA with no interactions:  Test for  HD



Four-way ANOVA with no interactions:  Test for  HD