Statistics – Spring 2008
Lab #9 – Factorial ANOVA
Defined:
Variables:
Relationship:
Example:
Assumptions:
Testing differences between group means
2+ IVs, where IV is categorical, DV is continuous
Group differences
How does IV#1 (violent, not violent) interact with IV#2 (training, no training) to improve memory
Normality. Heterogeneity.
In the last two labs, we discussed how to analyze data in situations where there is 1 IV, and the levels of the IV are
either 2 (t-test) or 3+ (ANOVA).
In today’s lab, we will discuss how to analyze data in situations where there are more than 1 IV, which is called
Factorial ANOVA. It is sometimes also called Two-WAY ANOVA (if you have 2 IV),
or Three-Way ANOVA (if you have 3 IV),
or Four-Way ANOVA (if you have 4 IV),
and so forth.
The most commonly used type of factorial ANOVA is the 2 x 2 design where there are two independent variables and
each variable has two levels. Factorial ANOVA can also be multi-level such as 2x3, 3x3, 4x10, etc.
1. Graphing
 The first step of any statistical analysis is to first graphically plot the data.
 Graphs are produced within the output for Factorial ANOVA, so see next section for how to produce plots.
2. Assumptions
 The assumptions for Factorial ANOVA are the same as for regular ANOVA, so see Lab #8 for more
information about how to test for the assumptions of ANOVA
3. Univariate - between
 In Lab #7 about t-test, we analyzed the relationship between sex (male, female) and errors1a. We found that
females scored significantly higher on the errors1a scale.
 In today’s lab, lets add another independent variable -- whether the participant has been a victim of a crime.
 Thus, we have:
a. IV1 = sex (male, female)
b. IV2 = victimcrime (yes, no)
c. DV = errors1a
 How to conduct Factorial ANOVA
1. Select Analyze --> General Linear Model --> Univariate
2. Move errors1a into the “Dependent Variable” box.
3. Move sex and victimcrime to the “Fixed Factors” box.
4. Click “Plots” and move the variables into the “Horizontal Axis” box and “Separate Lines” box.
Click “Add”, then “Continue”
FYI – it doesn’t matter which variable goes in which box, and you can do both plots if you want to.
5. Click “Options” and click “Descriptive Statistics” and “Estimates of effect size”.
6. Click OK.
 There are three parts to the output
1

Part 1 is the descriptive statistics.

Part 2 is the main part of the output. Levene’s test should be interpreted just like in Lab #7 and Lab #8. The
“Tests of Between-Subjects Effects” give you three critical pieces of information:
a. Main Effect of Variable 1 (sex): p = .055
b. Main Effect of Variable 2 (victimcrime): p = .047
c. Main Effect of Interaction (sex * victimcrime): p = .803
2

Part 3 is the plot: Notice that the information from the graph corresponds to the information from the ANOVA
table above. For example, the blue line is above the green line, which indicates that there is a Main Effect of
victimcrime. The mean for females (average of two right points) is higher than the mean for males (average of
two left points). And there is no interaction (crossing of the lines).


The output provides information about effect size, called partial eta squared.
WRITE-UP. The information included in the write up is the means, F value, df, and significance value. Lately
the field as a whole has recommended that you report effect size as well.
a. There was two main effects and no interactions. For example, there was a main effect of gender such
that males (M = 4.84) were significantly different than females (M = 5.61), F(1, 317) = 3.705, p = .055.
The effect size is .012. There was also a main effect of being a victim of a crime such that participants
who were victims (M = 4.99) were significantly different than participants who were not victims (M =
5.79), F(1, 317) = 3.971, p = .047. The effect size is .012
EVALUATION: Just like with evaluation all types of analysis involving group means (t-test, ANOVA), you
only really care about three pieces of information: (1) is the effect significant (p-value), (2) what is the size of
the effect (effect size), and (3) what is the direction of the effect (which mean is larger than the other).

3
4. Mixed – between factors and within factors
 We don’t have a paired-samples factor in our dataset so we can’t conduct a mixed factor design.
 However, in past labs we have assumed the system1 and system2 were the same variable, with system1
completed by the participants before the study began, and system2 completed by the participants after the
study ended.
 In today’s lab, lets add another IV -- sex (male, or female) which is a between-subjects factor.
 Thus, we have:
a. IV1 = repeated factors variables (system1, system2)
b. IV2 = sex (male, female)
 How to conduct Mixed ANOVA
1. Select Analyze --> General Linear Model --> Repeated Measures
2. Enter the number of level into “Number of Levels”. In this case its “2”. Click “Add”. Click “Define”.
3. Move the two variables over to the “Within-Subjects” box.
4. Move “sex” to the Between-Subjects Factor
5. Click “Plots”, move sex into the “Horizontal Axis” box, and “factor1” into “Separate Lines”
Click “Add”, then “Continue”
6. Click “Options”, click “Descriptive”, “Effect Size”, click “Continue”
7. Click OK.
 There are four parts to the output
 Part 1 is the descriptive statistics.


Part 2 is the test of the WITHIN factor. First you check the middle box to identify Sphericity, which is
whether the variance across conditions and across pairs is equal.
a. If the test is non-significant (p > .05), then sphericity exists. If sphericity exists, then look at the third
box where it says “Sphericity Assumed” to find the Omnibus F, which is p = .000
b. If the test is significant (p < .05), then sphericity does not exist. If Sphericity does not exist, you have
two options: (1) look at the Greenhouse-Geisser or Huynh-Feldt correction), or (2) look at the
Multivariate Tests box. The corrections correct for sphericity, whereas the Multivariate tests are not
based upon the assumption of sphericity. In all cases, the same result occurs, p = .000.
Part 2 is also the test of the MIXED factor results of whether there is an interaction between the two variables.
In this case, the interaction (factor1*sex) is significant, p = .042
4

Part 3 is the test of the BETWEEN factor. In this case, sex is NOT significant, p = .784
5

Part 4 is the graphical plot of the data: Notice that the information from the graph corresponds to the
information from the ANOVA table above. For example, the blue line is above the green line, which indicates
that there is a Main Effect of factor1 (pretest to posttest). The data are females (average of two right points) is
EQUAL to the mean for males (average of two left points) so there is NO MAIN EFFECT OF GENDER. And
there is an interaction because the lines (will) cross.


The output provides information about effect size, called partial eta squared.
WRITE-UP. The information included in the write up is the means, F value, df, and significance value. Lately
the field as a whole has recommended that you report effect size as well. (FYI – I am reporting results for
Greenhouse-Geisser).
a. There was an interaction and a main effect for the repeated measures factor. For example, there was an
interaction between gender and the repeated measures factor, F(1, 321) = 4.187, p = .042. The effect
size is 013. In other words, males and females responded differently across the pre-test and post-test
repeated measures. Male responses were more polarized, such that males scored higher on the pre-test
than females (M = 7.68, M = 7.24), and then scored lower on the pre-test than females (M = 6.42, M =
6.74). There was also a main effect of the repeated measures factor, F(1, 321) = 21.846, p < .000. The
effect size is .064. In other words, collapsing across gender, the pre-test was higher than the post-test
(M = 7.34, M = 6.67)
EVALUATION: Just like with evaluation all types of analysis involving group means (t-test, ANOVA), you
only really care about three pieces of information: (1) is the effect significant (p-value), (2) what is the size of
the effect (effect size), and (3) what is the direction of the effect (which mean is larger than the other).

6