Factorial ANOVAFactorial ANOVA

Chapter 12Chapter 12

Research DesignsResearch Designs

Between – Between (2 between subjects factors)Between – Between (2 between subjects factors) Mixed Design (1 between, 1 within subjects factor)Mixed Design (1 between, 1 within subjects factor) Within – Within (2 within subjects factors)Within – Within (2 within subjects factors) The purpose of this experiment was to determine The purpose of this experiment was to determine

the effects of testing mode (treadmill, bike) and the effects of testing mode (treadmill, bike) and gender (male, female) on maximum VOgender (male, female) on maximum VO22..

Testing mode is a within subjects factor with 2 levelsTesting mode is a within subjects factor with 2 levelsGender is a between subjects factor with 2 levelsGender is a between subjects factor with 2 levelsMaximum VOMaximum VO22 is the dependent variable. is the dependent variable.

A 3 x 2 DesignA 3 x 2 Design

The designs are sometimes identified by the The designs are sometimes identified by the number of factors and the levels of each factor.number of factors and the levels of each factor.

The purpose of this experiment was to The purpose of this experiment was to determine the effects of determine the effects of intensity (low, med, intensity (low, med, high)high) and and gender (male, female)gender (male, female) on strength on strength development. All subjects experience all three development. All subjects experience all three intensities.intensities.

A 3 x 2 factorial ANOVA was used to determine A 3 x 2 factorial ANOVA was used to determine the effects of intensity (low, med, high) and the effects of intensity (low, med, high) and gender (male, female) on strength gender (male, female) on strength development.development.

Gender is a between subjects factor, intensity is Gender is a between subjects factor, intensity is a within subjects factor.a within subjects factor.

Interaction?Interaction?

Interaction is the combined effects of the Interaction is the combined effects of the factors on the dependent variable.factors on the dependent variable.

Two factors interact when the Two factors interact when the differencesdifferences between the means on one factor depend between the means on one factor depend upon the level of the other factor.upon the level of the other factor.

If training programs affect men and If training programs affect men and women differently then training programs women differently then training programs interact with gender.interact with gender.

If training programs affect men and If training programs affect men and women the same they do not interact.women the same they do not interact.

No Interactions (Parallel No Interactions (Parallel Slopes)Slopes)

The red lines represent the average scores for BOTH A1 & A2 at each level of B.

The red lines are graphing B Main Effects.

No InteractionNo Interaction

Red line is the Average A1 mean (averaged across all levels of B).

Blue line is the average A2 mean.

Main effect for A compares the red and blue mean values.

Significant InteractionSignificant Interaction

Groups A1 and A2 are NOT EQUALLY affected by the levels of B.

Strong InteractionStrong Interaction

Groups A1 and A2 are NOT EQUALLY affected by the levels of B.

A1 goes DOWN

A2 goes UP

Draw in the means for A1 and A2?

Draw in means for B1, B2, B3.

Significant InteractionSignificant Interaction

Groups A1 and A2 are NOT EQUALLY affected by the levels of B.

Draw in the means for A1 and A2.

Draw in means for B1, B2, B3.

Factorial ANOVA AssumptionsFactorial ANOVA Assumptions Between-Between designs have the same Between-Between designs have the same

assumptions as One-way ANOVA.assumptions as One-way ANOVA. Dependent Variable is interval or ratio.Dependent Variable is interval or ratio. The variables are normally distributedThe variables are normally distributed The groups have equal variances (for between-The groups have equal variances (for between-

subjects factors)subjects factors) The groups are randomly assigned.The groups are randomly assigned.

Between-Within are similar to Repeated measures Between-Within are similar to Repeated measures ANOVA, but now sphericity must be applied to the ANOVA, but now sphericity must be applied to the pooled data (across groups) & the individual group, pooled data (across groups) & the individual group, this is referred to as this is referred to as multisample sphericitymultisample sphericity or or circularitycircularity. .

Sphericity :requires equal differences between Sphericity :requires equal differences between within subjects means. In other words within subjects means. In other words the changes the changes between each time point must be equal.between each time point must be equal.

A Between-Between Factorial ANOVAA Between-Between Factorial ANOVA

The purpose of this experiment was to The purpose of this experiment was to determine the effects of practice (1, 3, 5 determine the effects of practice (1, 3, 5 days/wk) and experience (athlete, non-days/wk) and experience (athlete, non-athlete) on throwing accuracy. athlete) on throwing accuracy.

9 athletes & 9 non-athletes were randomly 9 athletes & 9 non-athletes were randomly assigned to the practice groups (1, 3, 5 assigned to the practice groups (1, 3, 5 days/wk). days/wk).

A 3 x 2 Factorial ANOVA with two between A 3 x 2 Factorial ANOVA with two between subjects factors practice (1, 3, 5 days/wk) and subjects factors practice (1, 3, 5 days/wk) and experience (athlete, non-athlete) was used to experience (athlete, non-athlete) was used to test the effects of practice and experience on test the effects of practice and experience on throwing accuracy.throwing accuracy.

ANOVA TerminologyANOVA Terminology The purpose of this experiment was to compare The purpose of this experiment was to compare

the effects of Gender (M,F) and the dose of the effects of Gender (M,F) and the dose of Gatorade (none, 2 pints, 4 pints) on VO2. Gatorade (none, 2 pints, 4 pints) on VO2. Subjects were randomly assigned to Gatorade Subjects were randomly assigned to Gatorade groupsgroups..

The independent variables The independent variables Gatorade and Gatorade and Gender Gender are are FACTORSFACTORS. .

The Gatorade has The Gatorade has 3 LEVELS3 LEVELS (none, 2 pints, 4 (none, 2 pints, 4 pints)pints) , Gender has , Gender has 2 LEVELS2 LEVELS

The dependent variable in this experiment is The dependent variable in this experiment is VO2VO2

This a 2 x 3 ANOVA with two between subjects This a 2 x 3 ANOVA with two between subjects factors.factors.

The Effects of Gender & Gatorade on VO2The Effects of Gender & Gatorade on VO2

Create a categorical variable for all Between-Subjects Factors.

Gender (0 – Male, 1 – Female)Gatorade (1 – None, 2 – 2 pints, 3 – 4 pints.

Enter Dependent Variable and FactorsEnter Dependent Variable and Factors

Options ButtonOptions Button

Check homogeneity of variance if you have a between subjects factor.

Choose the Sidak post hoc test.

PlotsPlots

1. Enter Gatorade on horizontal axis, Gender for Separate Lines.

2. Click Add Button, then Continue Buttton.

Method 1 for Simple Method 1 for Simple EffectsEffects

Click Paste, then Window to view Syntax Window

UNIANOVA VO2 BY Gender Gatorade /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /PLOT=PROFILE(Gatorade*Gender) /EMMEANS=TABLES(OVERALL) /EMMEANS=TABLES(Gender) COMPARE ADJ(SIDAK) /EMMEANS=TABLES(Gatorade) COMPARE ADJ(SIDAK) /EMMEANS=TABLES(Gender*Gatorade) COMPARE(Gender) ADJ(SIDAK) /EMMEANS=TABLES(Gatorade*Gender) COMPARE(Gatorade) ADJ(SIDAK) /PRINT=OPOWER ETASQ HOMOGENEITY DESCRIPTIVE /CRITERIA=ALPHA(.05) /DESIGN=Gender Gatorade Gender*Gatorade.

Enter the first interaction term in the Compare ( ).Then switch the order.

Method 2 for Simple EffectsMethod 2 for Simple Effects

MANOVAVO2 BY Gender(0 1) Gatorade(1 3)/Design = Gender within Gatorade(1) Gender WITHIN Gatorade(2) Gender Within Gatorade(3)/Design = Gatorade Within Gender(1) Gatorade Within Gender(2)/print CELLINFO SIGNIF( Univ MULTIV AVERF HF GG).

Output: DescriptivesOutput: Descriptives

The groups have equal variance, Levine’s test F(5,42) = 1.53, p = .20

Check homogeneity of variance if you have a between subjects factor. The null hypothesis is that the groups have equal variance. In this case you retain the null. You don’t want this to be significant, if it is significant you are violating an assumption of ANVOA: homogeneity of variance.

See page 405 of Field for an additional test to check for homogeneity of variance.

No main effect for Gender F(1,42) = 2.032, p = .161.Sig. main effect for Gatorade F(2,42) = 20.065, p = .000Sig. interaction between Gender and Gatorade dose F(2,42) = 11.911, p = .000

ANOVA ResultsANOVA Results

Male Female Gatorade Mean

None 66.88 ± 10.33 60.62 ± 4.96 63.75 ± 8.47

2 pints 66.87 ± 12.52 62.50 ± 6.55 64.69 ± 9.91

4 pints 35.63 ± 10.84 57.50 ± 7.07 46.56 ± 14.34

Gender Mean 56.46 ± 18.50 60.21 ± 6.34 58.33 ± 13.81

Gender F(1,42) = 2.032, p = .161 Gatorade F(2,42) = 20.065, p=.000

Gender * Gatorade F(2,42) = 11.91, p = .000

This slide indicates which means are being compared by each F ratio.

Post hoc for Gender Main EffectPost hoc for Gender Main Effect

Gender F(1,42) = 2.032, p = .161

Post hoc for Gatorade Main EffectsPost hoc for Gatorade Main Effects

4 pints was significantly different from none and 2 pints.

Simple Effects Testing 2 StepsSimple Effects Testing 2 Steps

Male Female

None 66.88 ± 10.33

60.62 ± 4.96

2 pints

66.87 ± 12.52

62.50 ± 6.55

4 pints

35.63 ± 10.84

57.50 ± 7.07

Male Female

None 66.88 ± 10.33

60.62 ± 4.96

2 pints

66.87 ± 12.52

62.50 ± 6.55

4 pints

35.63 ± 10.84

57.50 ± 7.07

Compare gender at each level of gatorade. Are males diff from females for none? Are males diff from females for 2 pints?

Are males diff from females for 4 pints?

Compare the dose of gatorade for each level of gender. For males is there a difference between none, 2 pints, 4 pints?

For females is there a difference between none, 2 pints, 4 pints?

Difference in Gender at each Gatorade LevelDifference in Gender at each Gatorade Level

Males are significantly different from females for 4 pints of Gatorade.

Difference in Gatorade Difference in Gatorade at each Gender Levelat each Gender Level

For males, 4 pints is significantly different from none and 2 pints.

HomeworkHomework

Analyze the Task 1 the book, see page 455.

Do a Sidak post hoc test instead of the planned contrast suggested in the book.

Use simple effects testing for a significant interaction.

Use the Sample Methods and Results section as a guide to write a methods and results section for your homework.