Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects ...

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Chapter 13 Factorial Analysis of Variance

Transcript of Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects ...

Page 1: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Chapter 13

Factorial Analysis of Variance

Page 2: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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

Page 3: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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

Page 4: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Recognizing and Interpreting Interaction Effects

Words– Interaction effect occurs when the effect of

one variable depends on the level of another variable

Numbers

Page 5: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Recognizing and Interpreting Interaction Effects

Graphically

Page 6: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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

Page 7: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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

Page 8: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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

Page 9: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Figuring a Two-Way ANOVA

Sums of squares

2Within )( MXSS -=Â

2Within )( MXSS -=Â

2Total )( GMXSS -=Â

2Total )( GMXSS -=Â

WithinnInteractioColumnsRowsTotal SSSSSSSSSS +++= WithinnInteractioColumnsRowsTotal SSSSSSSSSS +++=

Page 10: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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 =

Page 11: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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 =

Page 12: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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 =

Page 13: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Figuring a Two-Way ANOVA

Degrees of freedom

1RowsRows -=Ndf 1RowsRows -=Ndf

1ColumnsColumns -=Ndf 1ColumnsColumns -=Ndf

1ColumnsRowsCellsnInteractio ---= dfdfNdf 1ColumnsRowsCellsnInteractio ---= dfdfNdf

Page 14: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Figuring a Two-Way ANOVA

Degrees of freedom

Last21Within ... dfdfdfdf +++= Last21Within ... dfdfdfdf +++=

1Total -=Ndf 1Total -=Ndf

Page 15: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Figuring a Two-Way ANOVA

ANOVA table for two-way ANOVA

Page 16: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Assumptions in Two-Way ANOVA

Populations follow a normal curve Populations have equal variances Assumptions apply to the populations

that go with each cell

Page 17: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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

--=

Page 18: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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

+=

Page 19: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Power for Studies Using 2 x 2 or 2 x 3 ANOVA (.05 significance level)

Page 20: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Approximate Sample Size Needed in Each Cell for

80% Power (.05 significance level)

Page 21: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Extensions and Special Cases of the Factorial ANOVA

Three-way and higher ANOVA designs Repeated measures ANOVA

Page 22: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

Controversies and Limitations

Unequal numbers of participants in the cells

Dichotomizing numeric variables– Median split

Page 23: Chapter 13 Factorial Analysis of Variance. Basic Logic of Factorial Designs and Interaction Effects Factorial research design –Effect of two or more.

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.


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