Statistics for the Behavioral Sciences (5th ed.)
Gravetter & Wallnau
Chapter 15
Two-Factor ANOVA:
Independent Measures, Equal ns
University of Guelph
Psychology 3320 — Dr. K. Hennig
Fall 2003 Term
Chapter 15 in outline
1)
2)
3)
4)
Overview
Main Effects and Interactions
An example
Assumptions
Overview




Goal of research: isolate two variables and
examine their relation, eliminating or reducing the
influence of extraneous/outside variables
Single factor analyses thus far
More typically behavior is more complex and is
influenced by a variety of variables acting and
interacting simultaneously
manipulate 2 (or more variables) => one DV
(univariate analysis)
Fig. 15.1
Effect of an audience on errors for low and high
self-esteem individuals


How many separate samples?
How many independent variables?
number of errors
no aud audience
no aud audience
High SE
Low SE
Table 15-1 (p. 479)
Structure of a two-factor experiment. Two levels for the
humidity factor (low and high), and three levels for the
temperature factor (70°, 80°, and 90°).
Table 15-2 (p. 481)
Hypothetical data from an experiment examining two
different levels of humidity (factor A) and three
different levels of temperature (factor B).
•Main effects for Factors A and B
•Consistent 10-pt difference - no interaction
Definition


When the effect of one factor depends on the
different levels of a second factor, then there is an
interaction between the factors
e.g., The effect of changing humidity depends on
the temperature, and there is an interaction.
Figure 15-4 (p. 482)
Two different levels of humidity (factor A) and three
different temperature conditions (factor B).
What is the effect of temperature on performance?
Depends…
(These data show the same main effects as the
data in Table 15.2, but the individual treatment
means have been modified to produce an
interaction.)
Figure 15-2 (p. 484)
(a) Graph showing the
data from Table 15.2,
where there is no
interaction. (b) Graph
showing the data from
Table 15.3, where there
is an interaction.
• Look for the existence
of non-parallel lines
Figure 15-4a (p. 486)
Different combinations of main effects and
interaction for a two-factor study.
Figure 15-4b (p. 486)
Different combinations of main and interaction effects
for a two-factor study.
Figure 15-4c (p. 486)
Three sets of data showing different combinations of main
effects and interaction for a two-factor study.
Figure 15-3 (p. 487)
(a) A line graph and (b) a
bar graph showing the
results from a two-factor
experiment.
Figure 15-4 (p. 489)
Structure of the analysis for a two-factor analysis of
variance.
the
extra
Fig.15-5 (p. 490): Arousal-Performance Study
Two levels of task difficulty (easy and hard) and
three levels of arousal (low, medium, and high),
i.e., six different treatment conditions (n = 5)
Steps

Factor A (humidity has no effect on performance):



Factor B (overall there are no differences in mean
performance among the three temperatures):




Ho: A1 = A2
H1: A1 <> A2
Ho: B1 = B2 = B3
H1: At least one mean is different from another
General formular: variance (differences between
rows(colmns)/ variance (differences) by chance
Interaction:
Ho: There is no interaction
H1: There is an interaction
Main effect
MS Abetw
FA 
MS with
MS Bbetw
FB 
MS with
SSbetw
 MS betw 
df betw
 MS with
SS with

df with
2
Trow
G2
SS Abw  

, df Abw # rows  1
nrow N
SS with   SScell , df with   df cell (n  1)
Interaction effect (the extra)
SS AxB  SSbetw  SS A  SS B
FAxB
MS AxB

MS with
T2
df AxB  df betw  df factor A  df factor B
G2
SSbetw  

, df betw # cells  1(6  1)
n
N
Effect size for 2-way ANOVA

As with repeated measures ANOVA, remove any
variability than arises from other sources
SS A
Factor A,  
SStotal  SS B  SS AxB
2


η2 = .004 (or 0.4%)
Reporting: F(1,76) = 4.51, p < .05, η2 = 0.056
Results from an experiment examining the
eating behavior of non-obese and obese
individuals who have either a full or an empty
stomach.
Table 1 (p. 500)
Figure 15-5 (p. 501)
A table and a graph showing the mean number of
crackers eaten for each of the four groups in Example
15.2.