# 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

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

Independent-measures analysis of variance
(Chapter 12)
Individual differences (page 347)
13-1 An Overview of the Two-Factor, IndependentMeasures ANOVA

Analysis of variance

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)

Complex analysis of variance

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)

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)

Factorial designs

Consider more than one factor

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

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)

The three F-ratios have the same basic structure

Numerator measures treatment mean differences
Denominator measures treatment mean differences
when there is no treatment effect
Main Effects

Mean differences among levels of one factor

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

The mean differences between individual treatment
conditions, or cells, are different from what would
be predicted from the overall main effects of the
factors

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)

Interaction can be defined as the unique effect of
two factors working together
Alternative definitions (slightly different
perspectives)

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)

Dependence of factors

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

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

Three distinct tests

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

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

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

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

First stage

Identical to independent samples ANOVA
Compute SStotal, SSbetween treatments, and
SSwithin treatments
Second stage

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)

This stage determines the numerators for the three
F-ratios by partitioning SSbetween treatments
Stage 2 of the Two-Factor Analysis (2 of 2)

Degrees of freedom

dftotal = N – 1

dfwithin treatments = Σdfinside each treatment

dfbetween treatments = k – 1
dfA = number of rows – 1

dfB = number of columns – 1

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

The hypothesis test for the two-factor ANOVA
follows the same four-step procedure that we have
seen before

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

Compute a value for η2 (eta squared), the
percentage of variance explained by the treatment
effects
Reporting the Results of a Two-Factor ANOVA

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 • • • • 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 • 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 • 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 • Testing simple main effects – – – 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) • • • 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) • Solution to the problem of high variance: use the specific variable associated with consistent individual differences as a second factor – – 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 • The validity of the ANOVA depends on three assumptions common to other hypothesis tests (t test and single-factor ANOVA) – – – 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) • 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) • 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) • Decide if each of the following statements is True or False. • T/F – • Two separate single-factor ANOVAs provide exactly the same information that is obtained from a twofactor analysis of variance. T/F – 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) • False – • 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 – 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

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