Factorial ANOVA Chapter 12

Research 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 effects of testing mode (treadmill, bike) and gender (male, female) on maximum VO 2. The purpose of this experiment was to determine the effects of testing mode (treadmill, bike) and gender (male, female) on maximum VO 2. Testing mode is a within subjects factor with 2 levels Testing mode is a within subjects factor with 2 levels Gender is a between subjects factor with 2 levels Gender is a between subjects factor with 2 levels Maximum VO 2 is the dependent variable. Maximum VO 2 is the dependent variable.

A 3 x 2 Design The designs are sometimes identified by the number of factors and the levels of each factor. The designs are sometimes identified by the number of factors and the levels of each factor. The purpose of this experiment was to determine the effects of intensity (low, med, high) and gender (male, female) on strength development. All subjects experience all three intensities. The purpose of this experiment was to determine the effects of intensity (low, med, high) and gender (male, female) on strength development. All subjects experience all three intensities. A 3 x 2 factorial ANOVA was used to determine the effects of intensity (low, med, high) and gender (male, female) on strength development. A 3 x 2 factorial ANOVA was used to determine the effects of intensity (low, med, high) and gender (male, female) on strength development. Gender is a between subjects factor, intensity is a within subjects factor. Gender is a between subjects factor, intensity is a within subjects factor.

Interaction? Interaction is the combined effects of the factors on the dependent variable. Interaction is the combined effects of the factors on the dependent variable. Two factors interact when the differences between the means on one factor depend upon the level of the other factor. Two factors interact when the differences between the means on one factor depend upon the level of the other factor. If training programs affect men and women differently then training programs interact with gender. If training programs affect men and women differently then training programs interact with gender. If training programs affect men and women the same they do not interact. If training programs affect men and women the same they do not interact.

No Interactions (Parallel 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 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 Interaction Groups A1 and A2 are NOT EQUALLY affected by the levels of B.

Strong 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 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 Assumptions Between-Between designs have the same assumptions as One-way ANOVA. Between-Between designs have the same assumptions as One-way ANOVA. Dependent Variable is interval or ratio. Dependent Variable is interval or ratio. The variables are normally distributed The variables are normally distributed The groups have equal variances (for between-subjects factors) The groups have equal variances (for between-subjects factors) The groups are randomly assigned. The groups are randomly assigned. Between-Within are similar to Repeated measures ANOVA, but now sphericity must be applied to the pooled data (across groups) & the individual group, this is referred to as multisample sphericity or circularity. Between-Within are similar to Repeated measures ANOVA, but now sphericity must be applied to the pooled data (across groups) & the individual group, this is referred to as multisample sphericity or circularity. Sphericity :requires equal differences between within subjects means. In other words the changes between each time point must be equal. Sphericity :requires equal differences between within subjects means. In other words the changes between each time point must be equal.

A Between-Between Factorial ANOVA The purpose of this experiment was to determine the effects of practice (1, 3, 5 days/wk) and experience (athlete, non-athlete) on throwing accuracy. The purpose of this experiment was to determine the effects of practice (1, 3, 5 days/wk) and experience (athlete, non-athlete) on throwing accuracy. 9 athletes & 9 non-athletes were randomly assigned to the practice groups (1, 3, 5 days/wk). 9 athletes & 9 non-athletes were randomly assigned to the practice groups (1, 3, 5 days/wk). A 3 x 2 Factorial ANOVA with two between subjects factors practice (1, 3, 5 days/wk) and experience (athlete, non-athlete) was used to test the effects of practice and experience on throwing accuracy. A 3 x 2 Factorial ANOVA with two between subjects factors practice (1, 3, 5 days/wk) and experience (athlete, non-athlete) was used to test the effects of practice and experience on throwing accuracy.

ANOVA Terminology The purpose of this experiment was to compare the effects of Gender (M,F) and the dose of Gatorade (none, 2 pints, 4 pints) on VO2. Subjects were randomly assigned to Gatorade groups. The purpose of this experiment was to compare the effects of Gender (M,F) and the dose of Gatorade (none, 2 pints, 4 pints) on VO2. Subjects were randomly assigned to Gatorade groups. The independent variables Gatorade and Gender are FACTORS. The independent variables Gatorade and Gender are FACTORS. The Gatorade has 3 LEVELS (none, 2 pints, 4 pints), Gender has 2 LEVELS The Gatorade has 3 LEVELS (none, 2 pints, 4 pints), Gender has 2 LEVELS The dependent variable in this experiment is VO2 The dependent variable in this experiment is VO2 This a 2 x 3 ANOVA with two between subjects factors. This a 2 x 3 ANOVA with two between subjects factors.

The 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 Factors

Options Button Check homogeneity of variance if you have a between subjects factor. Choose the Sidak post hoc test.

Plots 1.Enter Gatorade on horizontal axis, Gender for Separate Lines. 2.Click Add Button, then Continue Buttton.

Method 1 for Simple Effects 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 Effects MANOVA VO2 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: 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) = , p =.000 Sig. interaction between Gender and Gatorade dose F(2,42) = , p =.000 ANOVA Results

MaleFemaleGatorade Mean None66.88 ± ± ± pints66.87 ± ± ± pints35.63 ± ± ± Gender Mean56.46 ± ± ± Gender F(1,42) = 2.032, p =.161 Gatorade F(2,42) = , 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 Effect Gender F(1,42) = 2.032, p =.161

Post hoc for Gatorade Main Effects 4 pints was significantly different from none and 2 pints.

Simple Effects Testing 2 Steps MaleFemale None66.88 ± ± pints66.87 ± ± pints35.63 ± ± 7.07 MaleFemale None66.88 ± ± pints66.87 ± ± pints35.63 ± ± 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 Level Males are significantly different from females for 4 pints of Gatorade.

Difference in Gatorade at each Gender Level For males, 4 pints is significantly different from none and 2 pints.

Homework 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.