Mixed ANOVA Models
combining between and within
Mixed ANOVA models

We have examined One-way and
Factorial designs that use:
– a single between-subjects IV
– multiple between-subjects IVs
– a single within-subjects IV
– multiple within-subjects IVs
Mixed ANOVA models

Mixed ANOVA models
– contain at least one betweensubjects IV and one within-subjects
IV
– two-way, three-way, or higher order
factorial designs can be created
using any combination of between
and within subjects IVs
Two-Way Mixed ANOVA
One between-subjects IV
 One within-subjects IV
 Commonly used design
 Very useful for addressing
frequently occurring research
questions
 Often called split-plot design from
origins in agricultural applications

Split-plot designs
 Two crops are compared

Each crop is exposed to three fertilizer
conditions
 The combined effect of crop and
fertilizer is examined
Fertilizer I
Crop A
Crop B
Fertilizer II
Fertilizer III
Two-Way Mixed ANOVA

The within-subjects IV can take all
three forms:
– the same subjects are measured on 3
or more occasions
– the same subjects are exposed to 3
or more treatments
– the same subjects provide three or
more ratings that are measured on
the same scale
Two-Way Mixed ANOVA


The between-subjects IV can be:
– randomly assigned - treatment vs.
control
– attribute variable - gender, grade,
age group, etc.
The most common use involves:
– between-subjects IV – treatment or
control condition
– within-subjects IV - growth over time
Examples

Treatment and control groups are
are assessed on pre, mid, and post
treatment occasions.
 Males and females are given three
different types of medication.
 Tenured and non-tenured teachers
rate three different aspects of
school climate.
Examples
 Children are randomly assigned to get
the treatment (Head Start) or not (At
home & daycare), AND are assessed on
pre, mid, and post treatment occasions.
Pre
Head Start
At home
Daycare
Mid
Post
Examples

Males and females rate the same
three reasons for teaching in a
private religious school.
Values
Males
Females
Support
Rel.Bel.
Two-Way Mixed ANOVA

Both the between-subjects IV and
the within-subjects IV can have
any number of levels (2+).
 Three research questions
 Three sets of null and alternative
hypotheses
 Two main effects, one interaction
Two-Way Mixed ANOVA

The question and hypotheses for
the between-subjects IV will follow
the same patterns we have used
before.
 The question and hypotheses for
the within-subjects IV will also
follow the same patterns we have
used before.
Two-Way Mixed ANOVA

Interpret the interaction effect
first.
 Follow the same interpretation
strategies we have used for other
types of factorial designs.
 Graphing is particularly helpful.
Profile Analysis Approach




Uses Multivariate Approach
No sphericity assumption
Homogeneity of Variance - Covariance
Main Effect for Group
– Height

Main Effect for Time
– Slope

Group X Time Interaction
– Parallelism
Examples
 The Mixed ANOVA approach is the best way
to analyze the data we have been working
with all semester.
58
56
54
52
50
48
46
Pre
Mid
Post
Head Start
At Home
Daycare
Steps for Interpretation



Follow the same interpretation guidelines as
other Factorial designs
Use the Tukey Spreadsheet on the web
Calculate the appropriate effects sizes that
“tell the story”
Steps for Interpretation

Step 1 – Interpret the interaction term

Step 2 – Interpret the main effects

Step 3 – Graph the data “both ways”, meaning
exchange the row and column variables to
determine which picture is most useful
Steps for Interpretation

Typically it is most helpful to illustrate “change
over time”, or whatever the within-subjects
variable is, on the X axis

Typically it is most helpful to put the group
variable, or whatever the between-subjects term
is, as the separate lines variable.
Time on X, Groups as Lines
Social Development by Schedule
4.000
3.500
3.000
2.500
2.000
1.500
Split Day
Fall
Winter
Spring
Split Week
Steps for Interpretation

Step 4 – If the interaction term is statistically
significant, qualify the interpretation of the
main effects.

Step 5 – If there is a statistically significant
main effect with only two levels, no more
analyses are needed for that effect. Simply
examine the two marginal means (row or
column totals).
Steps for Interpretation

Step 6 – If there is a main effect with more than
two levels, perform post hoc comparisons
among the marginal means (row or column
totals).

This may require running additional analyses
as SPSS only gives you Post Hoc comparisons
for Between-Subjects terms.
Steps for Interpretation
Step 7 – Next, turn to the interaction effects.
There is not one rule that fits all situations. The
exact comparisons needed to make
interpretations will vary from analysis to
analysis.
 Look for the portion of your graphs where the
lines are non-parallel.


Next use the Tukey spreadsheet.
Steps for Interpretation

Step 8 – Consider Simple Effects first. This
means look at the pattern of differences with
rows or columns in your design first. If they are
different, then you have your answer about
where the interaction is coming from.

If this does not completely explain the
interaction, then consider looking at cell mean
comparisons across rows and columns.
Steps for Interpretation


Step 9 – Effect size calculations. Again, there is
no one rule that will fit every situation. Your
job is to illustrate the findings from your study
with the effect sizes that fit the pattern in the
results.
Within-subjects terms = Dependent Case
 Between-subjects terms = Independent Case
Steps for Interpretation
Center, p<.005
Time, p<.001
Interaction, p<.001
Now what?