Ch 14: 2-Factor Analysis of Variance (Independent Measures)
An Overview of the 2-Factor Independent-Measures ANOVA
in the context of ANOVA, an independent or quasi-independent variable is
called a factor
o research studies with 2 factors are called: factorial designs or 2-factor
designs
the 2 factors are identified as A and B and the structure of a 2-factor design
can be represented as a matrix with the levels of factor A determining the
rows and levels of factor B determining the columns
the 2 factor ANOVA allows us to examine 3 types of mean differences within 1
analysis
o traditionally, the 2 independent variables in a 2-factor experiment are
identified as factor A and factor B
o for the study presented in table 14.1, gender is factor A (rows) and the
level of violence in the game is factor B (columns); there are 2 levels
for each factor
o presented as a matrix:
o 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 2
factors acting together
Main Effects
the mean differences among the levels of 1 factor are referred to as the main
effect of that factor
o when the design of the research study is represented as a matrix (with
1 factor determining the rows and the second factor determining the
columns) 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
hypothetical data for an experiment, table 14.2:
, the evaluation of main effects counts for 2 of the 3 hypothesis tests in a 2-
factor ANOVA
o we state hypothesis concerning the main effect of factor A and the
main effect of factor B and then calculate 2 separate f-ratios to
evaluate the hypotheses
o in symbols:
H0 : μA1 = μA2, HI : μA1 ≠ μA2
null hypothesis means no difference between factor A
H0 : μB1 = μB2, HI : μB1 ≠ μB2
null hypothesis means no difference between factor B
Interactions
an interaction between 2 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
the null hypothesis is that there is no interaction:
o H0 : there is no interaction between factors A and B; the mean
differences between treatment conditions are explained by the main
effects of the 2 factors
the alternative hypothesis is that there is an interaction between the two
factors
o HI : there is an interaction between factors
to evaluate the interaction, the 2-factor ANOVA first identifies mean
differences that are not explained by the main effects
modified hypothetical data for an experiment, table 14.3:
o the data show the same main effects as the values in the other table,
but the individual treatment means have been modified to create an
interaction
if the 2 factors are independent, so that one factor does not influence the
effect of the other, then there is no interaction
o on the other hand, when the 2 factors are not independent, so that the
effect of 1 factor depends on the other, then there is an interaction
when the effect of 1 factor depends on different levels of a second factor,
then there is an interaction between the factors
the concept of an interaction can also be defined in terms of the pattern
displayed in the graph
An Overview of the 2-Factor Independent-Measures ANOVA
in the context of ANOVA, an independent or quasi-independent variable is
called a factor
o research studies with 2 factors are called: factorial designs or 2-factor
designs
the 2 factors are identified as A and B and the structure of a 2-factor design
can be represented as a matrix with the levels of factor A determining the
rows and levels of factor B determining the columns
the 2 factor ANOVA allows us to examine 3 types of mean differences within 1
analysis
o traditionally, the 2 independent variables in a 2-factor experiment are
identified as factor A and factor B
o for the study presented in table 14.1, gender is factor A (rows) and the
level of violence in the game is factor B (columns); there are 2 levels
for each factor
o presented as a matrix:
o 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 2
factors acting together
Main Effects
the mean differences among the levels of 1 factor are referred to as the main
effect of that factor
o when the design of the research study is represented as a matrix (with
1 factor determining the rows and the second factor determining the
columns) 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
hypothetical data for an experiment, table 14.2:
, the evaluation of main effects counts for 2 of the 3 hypothesis tests in a 2-
factor ANOVA
o we state hypothesis concerning the main effect of factor A and the
main effect of factor B and then calculate 2 separate f-ratios to
evaluate the hypotheses
o in symbols:
H0 : μA1 = μA2, HI : μA1 ≠ μA2
null hypothesis means no difference between factor A
H0 : μB1 = μB2, HI : μB1 ≠ μB2
null hypothesis means no difference between factor B
Interactions
an interaction between 2 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
the null hypothesis is that there is no interaction:
o H0 : there is no interaction between factors A and B; the mean
differences between treatment conditions are explained by the main
effects of the 2 factors
the alternative hypothesis is that there is an interaction between the two
factors
o HI : there is an interaction between factors
to evaluate the interaction, the 2-factor ANOVA first identifies mean
differences that are not explained by the main effects
modified hypothetical data for an experiment, table 14.3:
o the data show the same main effects as the values in the other table,
but the individual treatment means have been modified to create an
interaction
if the 2 factors are independent, so that one factor does not influence the
effect of the other, then there is no interaction
o on the other hand, when the 2 factors are not independent, so that the
effect of 1 factor depends on the other, then there is an interaction
when the effect of 1 factor depends on different levels of a second factor,
then there is an interaction between the factors
the concept of an interaction can also be defined in terms of the pattern
displayed in the graph
