REVIEW SHEET for Exam 3
Chapter 14: Two-Factor ANOVA
Intro:
• In the context of ANOVA, an independent variable (or a quasi-independent variable) is
called a factor, and research studies with two factors are called factorial designs or
simply two-factor designs.
• The two factors are identified as A and B, and the structure of a two-factor design can be
represented as a matrix with the levels of factor A determining the rows and the levels
of factor B determining the columns.
• In a two-factor ANOVA, we conduct 3 separate hypotheses tests for the same data, with
a separate F-ratio for each test. The three F-ratios have the same basic structure:
F= variance (differences) between treatments_______
variance (differences) expected if there is no treatment effect
• The two-factor ANOVA allows us to examine three types of mean differences within one
analysis.
• The goal of the study is to evaluate the mean differences that may be produced
by either of these factors acting independently or by the two factors acting
together.
Main Effects:
• The mean differences among the levels of one factor are referred to as the main effect
of that factor.
• When the design of the research study is represented as a matrix, then the mean
differences among the rows describe the main effect of one factor, and the mean
differences among the columns describe the main effect for the second factor.
• In the case of a two-factor study, any main effects that are observed in the data
must be evaluated with a hypothesis test to determine whether they are
statistically significant effects.
• The evaluation of main effects accounts for two of the three hypothesis tests in a two-
factor ANOVA.
• Each of the main effects hypothesis tests in a two-factor ANOVA will have its own F-ratio
Interactions:
• An interaction between two factors occurs whenever the mean differences between
individual treatment conditions, or cells, are different from what would be predicted
from the overall main effects of the factors.
• E.g.: specific combinations of game violence and gender acting together may
have different effects than the effects of gender or game violence acting alone.
• To evaluate the interaction, the two-factor ANOVA first identifies mean differences that
are not explained by the main effects.
• The extra mean differences are then evaluated by an F-ratio with the following structure:
F= variance (mean differences) not explained by main effects
variance (differences) expected if there are no treatment effects
• The null hypothesis is that there is no interaction
, H0: There is no interaction between factors A and B. The mean differences between
treatment conditions are explained by the main effects of the two factors.
• The alternative hypothesis is that there is an interaction between the two factors:
H1: There is an interaction between factors.
• If the two factors are independent, so that one factor does not influence the effect of
the other, then there is no interaction.
• On the other hand, when the two factors are not independent, so that the effect
of one factor depends on the other, then there is an interaction.
• When the effect of one factor depends on the different levels of a second factor, then
there is an interaction between the factors.
• When the results of a two-factor study are presented in a graph, the existence of
nonparallel lines (lines that cross or converge) indicates an interaction between the two
factors. Two examples:
The Two-Factor ANOVA:
• The two-factor ANOVA consists of three hypothesis tests, each evaluating specific mean
differences: the main effect of A, the main effect of B, and the A x B interaction.
• These are three separate tests, but they are also independent of each other.
• The outcome for any one of the three tests is totally unrelated to the
outcome for either of the other two.
• Thus, it is possible for data from a two-factor study to display any possible
combination of significant and/or not significant main effects and
interactions.
• All three F-ratios have the same basic structure:
F= variance (differences) between treatments__________
variance (differences) expected if there are no treatment effects
• The general technique for measuring effect size with an ANOVA is to compute �2, the
percentage of variance that is explained by the treatment effects.
• For a two-factor ANOVA, we compute 3 separate values for �2
• The validity of this ANOVA depends on the same three assumptions we have
encountered with other independent-measures designs (the t test and the single factor
ANOVA):
1. The observations within each sample must be independent.
2. The populations from which the samples are selected must be normal.
Chapter 14: Two-Factor ANOVA
Intro:
• In the context of ANOVA, an independent variable (or a quasi-independent variable) is
called a factor, and research studies with two factors are called factorial designs or
simply two-factor designs.
• The two factors are identified as A and B, and the structure of a two-factor design can be
represented as a matrix with the levels of factor A determining the rows and the levels
of factor B determining the columns.
• In a two-factor ANOVA, we conduct 3 separate hypotheses tests for the same data, with
a separate F-ratio for each test. The three F-ratios have the same basic structure:
F= variance (differences) between treatments_______
variance (differences) expected if there is no treatment effect
• The two-factor ANOVA allows us to examine three types of mean differences within one
analysis.
• The goal of the study is to evaluate the mean differences that may be produced
by either of these factors acting independently or by the two factors acting
together.
Main Effects:
• The mean differences among the levels of one factor are referred to as the main effect
of that factor.
• When the design of the research study is represented as a matrix, then the mean
differences among the rows describe the main effect of one factor, and the mean
differences among the columns describe the main effect for the second factor.
• In the case of a two-factor study, any main effects that are observed in the data
must be evaluated with a hypothesis test to determine whether they are
statistically significant effects.
• The evaluation of main effects accounts for two of the three hypothesis tests in a two-
factor ANOVA.
• Each of the main effects hypothesis tests in a two-factor ANOVA will have its own F-ratio
Interactions:
• An interaction between two factors occurs whenever the mean differences between
individual treatment conditions, or cells, are different from what would be predicted
from the overall main effects of the factors.
• E.g.: specific combinations of game violence and gender acting together may
have different effects than the effects of gender or game violence acting alone.
• To evaluate the interaction, the two-factor ANOVA first identifies mean differences that
are not explained by the main effects.
• The extra mean differences are then evaluated by an F-ratio with the following structure:
F= variance (mean differences) not explained by main effects
variance (differences) expected if there are no treatment effects
• The null hypothesis is that there is no interaction
, H0: There is no interaction between factors A and B. The mean differences between
treatment conditions are explained by the main effects of the two factors.
• The alternative hypothesis is that there is an interaction between the two factors:
H1: There is an interaction between factors.
• If the two factors are independent, so that one factor does not influence the effect of
the other, then there is no interaction.
• On the other hand, when the two factors are not independent, so that the effect
of one factor depends on the other, then there is an interaction.
• When the effect of one factor depends on the different levels of a second factor, then
there is an interaction between the factors.
• When the results of a two-factor study are presented in a graph, the existence of
nonparallel lines (lines that cross or converge) indicates an interaction between the two
factors. Two examples:
The Two-Factor ANOVA:
• The two-factor ANOVA consists of three hypothesis tests, each evaluating specific mean
differences: the main effect of A, the main effect of B, and the A x B interaction.
• These are three separate tests, but they are also independent of each other.
• The outcome for any one of the three tests is totally unrelated to the
outcome for either of the other two.
• Thus, it is possible for data from a two-factor study to display any possible
combination of significant and/or not significant main effects and
interactions.
• All three F-ratios have the same basic structure:
F= variance (differences) between treatments__________
variance (differences) expected if there are no treatment effects
• The general technique for measuring effect size with an ANOVA is to compute �2, the
percentage of variance that is explained by the treatment effects.
• For a two-factor ANOVA, we compute 3 separate values for �2
• The validity of this ANOVA depends on the same three assumptions we have
encountered with other independent-measures designs (the t test and the single factor
ANOVA):
1. The observations within each sample must be independent.
2. The populations from which the samples are selected must be normal.
