# Central Texas College Statistics of The Behavioral Sciences Essay

Locate the Key Concept identified in the chapter(s) PPT slides. Briefly explain, in 8-12 sentences, the Key Concept using complete sentences and your own words. (Plagiarized or quoted explanations will not receive credit.)

Chapter 13

Two-Factor Analysis of Variance

PowerPoint Lecture Slides

Essentials of Statistics for the Behavioral Sciences

Tenth Edition

by Frederick J Gravetter, Larry B. Wallnau, Lori-Ann B. Forzano, and James E. Witnauer

Chapter 13 Learning Outcomes

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Understand the logic of a two-factor study and a

matrix of group means

Describe main effects and interactions from a

pattern of group means in a two-factor ANOVA

Compute a two-factor ANOVA to evaluate means

for a two-factor independent-measures study

Measure effect size, interpret results, and articulate

assumptions for a two-factor ANOVA

Tools You Will Need

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Independent-measures analysis of variance

(Chapter 12)

Individual differences (page 347)

13-1 An Overview of the Two-Factor, IndependentMeasures ANOVA

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Analysis of variance

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Evaluates differences among two or more sample

means

In Chapter 12, the ANOVA was a single factor:

limited to one independent variable (IV) or one

quasi-independent variable

Independent-measures: the study uses a separate

sample for each of the different treatment conditions

being compared

An Overview of the Two-Factor, IndependentMeasures ANOVA (1 of 4)

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Complex analysis of variance

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Two independent variables are manipulated

(factorial ANOVA; only two-factor in this text)

Both independent variables and quasi-independent

variables may be employed as factors in a twofactor ANOVA

An independent variable (factor) is manipulated in

an experiment

An Overview of the Two-Factor, IndependentMeasures ANOVA (2 of 4)

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A quasi-independent variable (factor) is not

manipulated but defines the groups of scores in a

nonexperimental study

The two-factor ANOVA allows us to examine three

types of mean differences within one analysis

An Overview of the Two-Factor, IndependentMeasures ANOVA (3 of 4)

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Factorial designs

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Consider more than one factor

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•

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We will study two-factor designs only

Also limited to situations with equal n’s in each group

Joint impact of factors is considered

Three hypotheses tested by three F-ratios

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Large F-ratio ! greater treatment differences than

would be expected with no treatment effects

An Overview of the Two-Factor, IndependentMeasures ANOVA (4 of 4)

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The three F-ratios have the same basic structure

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Numerator measures treatment mean differences

Denominator measures treatment mean differences

when there is no treatment effect

Main Effects

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Mean differences among levels of one factor

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Differences are tested for statistical significance

Each factor (factor A and factor B) is evaluated

independently of the other factor(s) in the study

Interactions

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The mean differences between individual treatment

conditions, or cells, are different from what would

be predicted from the overall main effects of the

factors

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H0: There is no interaction between

factors A and B

H1: There is an interaction between

factors A and B

More about Interactions (1 of 2)

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Interaction can be defined as the unique effect of

two factors working together

Alternative definitions (slightly different

perspectives)

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If the two factors are independent, so one factor

doesn’t influence the effect of the other, then there

is no interaction

When the two factors are not independent, so the

effect of one factor depends on the other, then there

is interaction, and this can be seen in a graph

More about Interactions (2 of 2)

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Dependence of factors

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The effect of one factor depends on the level or

value of the other factor

Unique combinations of the factors produce unique

effects

Interaction can be observed in a graph

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Nonparallel lines (lines that cross, converge or

diverge) indicate that interaction is occurring

Is called the A × B interaction, or “ the A by B”

interaction

Figure 13.2 (a) Graph Showing the Treatment Means

for Table 13.3, for Which There Is No Interaction

FIGURE 13.2 (b) Graph for Table 13.4, for Which

There Is an Interaction

Independence of Main Effects and Interactions

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Three distinct tests

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Main effect of factor A

Main effect of factor B

Interaction of A and B

A separate F test is conducted for each

Results of one are independent of the others

Learning Check 1

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Which of the following accurately describes an

interaction between two variables?

A. Both variables produce a change in the

subjects’ scores

B. Both variables are equally influenced by a third

variable

C. The two variables are differentially affected by a

third variable

D. The effect of one variable depends on the levels

of the second variable

Learning Check 1 – Answer

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Which of the following accurately describes an

interaction between two variables?

A. Both variables produce a change in the

subjects’ scores

B. Both variables are equally influenced by a third

variable

C. The two variables are differentially affected by a

third variable

D. The effect of one variable depends on the

levels of the second variable

13-2 An Example of the Two-Factor ANOVA

and Effect Size

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The A-effect: the main effect of factor A typically

evaluates the mean differences between rows

The B-effect: the main effect of factor B typically

evaluates the mean differences between columns

The A × B interaction: evaluates mean differences

between treatment conditions that are not predicted

from the overall main effects

An Example of the Two-Factor ANOVA and Effect

Size

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First stage

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Identical to independent samples ANOVA

Compute SStotal, SSbetween treatments, and

SSwithin treatments

Second stage

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Partition the SSbetween treatments into three separate

components: differences attributed to factor A, to

factor B, and to any remaining mean differences

that define the interaction

Figure 13.3 Structure of the Analysis for a TwoFactor ANOVA

Stage 1 of the Two-Factor Analysis

Stage 2 of the Two-Factor Analysis (1 of 2)

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This stage determines the numerators for the three

F-ratios by partitioning SSbetween treatments

Stage 2 of the Two-Factor Analysis (2 of 2)

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Degrees of freedom

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dftotal = N – 1

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dfwithin treatments = Σdfinside each treatment

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dfbetween treatments = k – 1

dfA = number of rows – 1

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dfB = number of columns – 1

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dferror = dfwithin treatments – dfbetween subjects

Mean Squares and F-Ratios for the Two-Factor

ANOVA

Table 13.7 A Summary Table for the Two-Factor

ANOVA for the Data from Example 13.2

Source

SS

df

MS

F

440

3

Factor A (browsing type)

80

1

80

4.00

Factor B (relationship strength)

180

1

180

9.00

A×B

180

1

180

9.00

Within treatments

320

16

20

Total

760

19

Between treatments

The Hypothesis Test

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The hypothesis test for the two-factor ANOVA

follows the same four-step procedure that we have

seen before

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Step 1: State the hypotheses and select an alpha

level

Step 2: Locate the critical region

Step 3: Compute the F-ratios

Step 4: Make a decision

Measuring Effect Size for the Two-Factor ANOVA

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Compute a value for η2 (eta squared), the

percentage of variance explained by the treatment

effects

Reporting the Results of a Two-Factor ANOVA

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Report means and standard deviations (usually in a

table or graph due to the complexity of the design)

Report the results of all three hypothesis tests (A

and B main effects; A × B interaction)

For each test include F, df, p value, and η2

E.g., F (1, 16) = 9.00, p < .05, η2 = 0.36
Interpreting the Results from a Two-Factor
ANOVA
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Focus on the overall pattern of results
Significant interactions require particular attention
because even if you understand the main effects,
interactions go beyond what main effects alone can
explain
Extensive practice is typically required to be able to
clearly articulate results that include a significant
interaction
The key concept is explaining how interactions
differ from main effects.
Figure 13.4 Sample Means for the Data in
Example 13.2
Learning Check 2
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Which of the following accurately describes the two stages of a
two-factor ANOVA?
A. The first stage partitions the total variability, and the second
stage partitions the within-treatment variability
B. The first stage evaluates the hypothesis tests for the main
effects, and the second stage evaluates the hypothesis test
for an interaction effect
C. The first stage partitions the between-treatment variability,
and the second stage partitions the within-treatment
variability
D. The first stage partitions the total variability, and the second
stage partitions the between-treatment variability
Learning Check 2 – Answer
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Which of the following accurately describes the two stages of a
two-factor ANOVA?
A. The first stage partitions the total variability, and the second
stage partitions the within-treatment variability
B. The first stage evaluates the hypothesis tests for the main
effects, and the second stage evaluates the hypothesis test
for an interaction effect
C. The first stage partitions the between-treatment variability,
and the second stage partitions the within-treatment
variability
D. The first stage partitions the total variability, and the
second stage partitions the between-treatment
variability
13-3 More about the Two-Factor ANOVA
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Testing simple main effects
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A separate analysis for each of the individual
columns or rows; testing the significance of mean
differences within one column or row of a two-factor
design
The two-factor experiment is separated into a series
of separate single-factor experiments
Is done when there is a significant interaction: the
mean differences within one column (or row) show a
different pattern than the mean differences within
another column (or row)
Using a Second Factor to Reduce Variance
Caused by Individual Differences (1 of 2)
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Participant characteristics (such as age and
gender) may vary considerably from one person to
another
Participant characteristics can influence the scores
obtained in a study
Large variance tends to reduce the size of the t
statistic or F-ratio and, therefore, reduces the
likelihood of finding significant mean differences
Using a Second Factor to Reduce Variance
Caused by Individual Differences (2 of 2)
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Solution to the problem of high variance: use the
specific variable associated with consistent
individual differences as a second factor
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Instead of one group in each treatment, the
researcher divides the participants into two separate
groups within each treatment
One factor consists of the two treatments and the
second factor is, for example, based on gender or
on age (e.g., adolescents/adults)
Assumptions for the Two-Factor ANOVA
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The validity of the ANOVA depends on three
assumptions common to other hypothesis tests (t
test and single-factor ANOVA)
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The observations within each sample must be
independent of each other
The populations from which the samples are
selected must be normally distributed
The populations from which the samples are
selected must have equal variances (homogeneity
of variance)
Learning Check 3 (1 of 2)
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If a two-factor analysis of variance produces a
statistically significant interaction, then you can
conclude that _____.
A.
B.
C.
D.
either the main effect for factor A or the main effect
for factor B is also significant
neither the main effect for factor A nor the main
effect for factor B is significant
both the main effect for factor A and the main effect
for factor B are significant
the significance of the main effects is not related to
the significance of the interaction
Learning Check 3 – Answer (1 of 2)
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If a two-factor analysis of variance produces a
statistically significant interaction, then you can
conclude that _____.
A.
B.
C.
D.
either the main effect for factor A or the main effect
for factor B is also significant
neither the main effect for factor A nor the main
effect for factor B is significant
both the main effect for factor A and the main effect
for factor B are significant
the significance of the main effects is not related
to the significance of the interaction
Learning Check 3 (2 of 2)
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Decide if each of the following statements
is True or False.
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T/F
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Two separate single-factor ANOVAs provide exactly
the same information that is obtained from a twofactor analysis of variance.
T/F
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A disadvantage of combining two factors in an
experiment is that you cannot determine how either
factor would affect participants’ scores if it were
examined in an experiment by itself.
Learning Check 3 – Answers (2 of 2)
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False
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Main effects in a two-factor ANOVA are identical to
results of two one-way ANOVAs, but a two-factor
ANOVA provides interaction results, too!
False
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The two-factor ANOVA allows you to determine the
effect of one variable, controlling for the effect of the
other
Figure 13.5 The ANOVA for an IndependentMeasures Two-Factor Design
Clear Your Doubts, Ask Questions