Statistical Methods
for Economists
Lecture (7 & 8)b
Two-Way Analysis of Variance
(ANOVA)
•David Bartl
•Statistical Methods for Economists
•INM/BASTE

Outline of the lecture
•Introduction
•Two-way ANOVA without interactions
•Two-way ANOVA with interactions

Two-factor ANOVA:  Motivation:  Example



Two-factor ANOVA:  Motivation & Introduction



Two-factor ANOVA:  Motivation & Introduction
the levels of the factor  A  (groups)
the levels of the factor  B  (blocks)

Two-factor ANOVA:  Introduction:  Assumptions



Two-factor ANOVA:  Introduction:  Assumptions



Two-factor ANOVA:  Introduction:  Assumptions



Two-factor ANOVA:  Introduction:  Assumptions
Factor B


Two-factor ANOVA:  Introduction:  Assumptions



Two-factor ANOVA:  Introduction:  The Goal



Two-factor ANOVA:  Hypothesis  HA



Two-factor ANOVA:  Hypothesis  HB



Two-factor ANOVA:  Hypothesis  HAB
means


Two-factor ANOVA:  Simplification



Two-Way ANOVA without
interactions
Yijk=μ+αi+βj+εijk
•
•

Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions:  Notation



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


Two-way ANOVA with no interactions:  Notation



Two-way ANOVA with no interactions:  Notation



Two-way ANOVA with no interactions:  Notation



Two-way ANOVA with no interactions:  Dimensions



Two-way ANOVA with no interactions:  Dimensions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions
0
(the subspace corresponding to the model)
(the orthogonal
complement =
= the space of
the residuals)
the vector of the numerical
outcomes
of the random experiment

Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions



Two-way ANOVA with no interactions:  Test for  HA



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

Two-way ANOVA with no interactions:  Test for  HA



Two-way ANOVA with no interactions:  Test for  HA



Two-way ANOVA with no interactions:  Test for  HA



Two-way ANOVA with no interactions:  Test for  HB



Two-Way ANOVA with
interactions
Yijk=μ+αi+βj+γij+εijk
•
•

Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions:  Notation



Two-way ANOVA with interactions:  Notation
(cf. one-way ANOVA)


Two-way ANOVA with interactions:  Notation



Two-way ANOVA with interactions:  Notation



Two-way ANOVA with interactions:  Notation



Two-way ANOVA with interactions:  Notation



Two-way ANOVA with interactions:  Dimensions



Two-way ANOVA with interactions:  Dimensions



Two-way ANOVA with interactions:  Dimensions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions
0
(the subspace corresponding to the model)
(the orthogonal
complement =
= the space of
the residuals)
the vector of the numerical
outcomes
of the random experiment

Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions



Two-way ANOVA with interactions:  Test for  HA



Two-way ANOVA with interactions:  Test for  HA
0
subspace of dimension
(the orthogonal
complement =
= the space of
the residuals)
subspace
of dimension
subspace of dimension
It holds:

Two-way ANOVA with interactions:  Test for  HA



Two-way ANOVA with interactions:  Test for  HA



Two-way ANOVA with interactions:  Test for  HA



Two-way ANOVA with interactions:  Test for  HB



Two-way ANOVA with interactions:  Test for  HAB



Two-way ANOVA with interactions:  Test for  HAB
0
subspace of dimension
(the orthogonal
complement =
= the space of
the residuals)
subspace
of dimension
subspace of dimension
It holds:

Two-way ANOVA with interactions:  Test for  HAB



Two-way ANOVA with interactions:  Test for  HAB



Two-way ANOVA with interactions:  Test for  HAB