Post on 14-Jan-2016
Chapter 13
Factorial Analysis of Variance
Basic Logic of Factorial Designs and Interaction Effects
Factorial research design– Effect of two or more variables examined at
once– Efficient research design
Interaction effects– Combination of variables has a special
effect
Basic Logic of Factorial Designs and Interaction Effects
Two-way analysis of variance One-way analysis of variance Main effect Cell
– Cell mean
Marginal means
Recognizing and Interpreting Interaction Effects
Words– Interaction effect occurs when the effect of
one variable depends on the level of another variable
Numbers
Recognizing and Interpreting Interaction Effects
Graphically
Basic Logic of the Two-Way ANOVA
The three F ratios– Column main effect– Row main effect– Interaction effect
Logic of the F ratios for the row and column main effects
Logic of the F ratio for the interaction effect
Figuring a Two-Way ANOVA
Structural model for the two-way ANOVA– Each score’s deviation from the grand mean
• Score’s deviation from the mean of its cell• Score’s row’s mean from the grand mean• Score’s column’s mean from the grand mean• Remainder after other three deviations subtracted
from overall deviation from grand mean
Figuring a Two-Way ANOVA
Sums of squares
2
RowRows )( GMMSS -=Â2
RowRows )( GMMSS -=Â
2
ColumnsColumns )( GMMSS -=Â2
ColumnsColumns )( GMMSS -=Â
 -------= 2ColumnRownInteractio )]()()()[( GMMGMMMXGMXSS  -------= 2
ColumnRownInteractio )]()()()[( GMMGMMMXGMXSS
Figuring a Two-Way ANOVA
Sums of squares
2Within )( MXSS -=Â
2Within )( MXSS -=Â
2Total )( GMXSS -=Â
2Total )( GMXSS -=Â
WithinnInteractioColumnsRowsTotal SSSSSSSSSS +++= WithinnInteractioColumnsRowsTotal SSSSSSSSSS +++=
Figuring a Two-Way ANOVA
Population variance estimates
Rows
RowsRows
2Rows or
df
SSMSS =
Rows
RowsRows
2Rows or
df
SSMSS =
Columns
ColumnsColumns
2Columns or
df
SSMSS =
Columns
ColumnsColumns
2Columns or
df
SSMSS =
Figuring a Two-Way ANOVA
Population variance estimates
nInteractio
nInteractionInteractio
2nInteractio or
df
SSMSS =
nInteractio
nInteractionInteractio
2nInteractio or
df
SSMSS =
Within
WithinWithin
2Within or
df
SSMSS =
Within
WithinWithin
2Within or
df
SSMSS =
Figuring a Two-Way ANOVA
F ratios
Within
Rows2Within
2Rows
Rows or MS
MS
S
SF =
Within
Rows2Within
2Rows
Rows or MS
MS
S
SF =
Within
Columns2Within
2Columns
Columns or MS
MS
S
SF =
Within
Columns2Within
2Columns
Columns or MS
MS
S
SF =
Within
nInteractio2Within
2nInteractio
nInteractio or MS
MS
S
SF =
Within
nInteractio2Within
2nInteractio
nInteractio or MS
MS
S
SF =
Figuring a Two-Way ANOVA
Degrees of freedom
1RowsRows -=Ndf 1RowsRows -=Ndf
1ColumnsColumns -=Ndf 1ColumnsColumns -=Ndf
1ColumnsRowsCellsnInteractio ---= dfdfNdf 1ColumnsRowsCellsnInteractio ---= dfdfNdf
Figuring a Two-Way ANOVA
Degrees of freedom
Last21Within ... dfdfdfdf +++= Last21Within ... dfdfdfdf +++=
1Total -=Ndf 1Total -=Ndf
Figuring a Two-Way ANOVA
ANOVA table for two-way ANOVA
Assumptions in Two-Way ANOVA
Populations follow a normal curve Populations have equal variances Assumptions apply to the populations
that go with each cell
Effect Size in Factorial ANOVA
nInteractioColumnsTotal
Columns2Columns SSSSSS
SSR
--=
nInteractioColumnsTotal
Columns2Columns SSSSSS
SSR
--=
nInteractioRowsTotal
Rows2Rows SSSSSS
SSR
--=
nInteractioRowsTotal
Rows2Rows SSSSSS
SSR
--=
RowsColumnsTotal
nInteractio2nInteractio SSSSSS
SSR
--=
RowsColumnsTotal
nInteractio2nInteractio SSSSSS
SSR
--=
Effect Size in Factorial ANOVA
WithinColumnsColumns
ColumnsColumns2Columns ))((
))((
dfdfF
dfFR
+=
WithinColumnsColumns
ColumnsColumns2Columns ))((
))((
dfdfF
dfFR
+=
WithinRowsRows
RowsRows2Rows ))((
))((
dfdfF
dfFR
+=
WithinRowsRows
RowsRows2Rows ))((
))((
dfdfF
dfFR
+=
WithinnInteractionInteractio
nInteractionInteractio2nInteractio ))((
))((
dfdfF
dfFR
+=
WithinnInteractionInteractio
nInteractionInteractio2nInteractio ))((
))((
dfdfF
dfFR
+=
Power for Studies Using 2 x 2 or 2 x 3 ANOVA (.05 significance level)
Approximate Sample Size Needed in Each Cell for
80% Power (.05 significance level)
Extensions and Special Cases of the Factorial ANOVA
Three-way and higher ANOVA designs Repeated measures ANOVA
Controversies and Limitations
Unequal numbers of participants in the cells
Dichotomizing numeric variables– Median split
Factorial ANOVA in Research Articles
A two-factor ANOVA yielded a significant main effect of voice, F(2, 245) = 26.30, p < .001. As expected, participants responded less favorably in the low voice condition (M = 2.93) than in the high voice condition (M = 3.58). The mean rating in the control condition (M = 3.34) fell between these two extremes. Of greater importance, the interaction between culture and voice was also significant, F(2, 245) = 4.11, p < .02.