Factorial ANOVA

Lecture 9 Vincent Chap 11

Research Designs classification by the number of factors x levels

Factor x Level

Example

2 X 1 Change in BP before and after medication 2 Factors: 1) Drug-pre ; 2) Drug-post 1 Level: one group of people pre- and post- medication use

3 X 1 Comparison of 6mo weight loss with 3 different diets. 3 Factors: 1) diet A, 2) diet B, 3) diet C 1 Level: 3 groups of people drawn from the same population

2 X 2 Effect of a performance supplement and gender on strength training 2 Factor: 1) Strength-w/ supplement; 2) Strength-w/o supplement 2 Levels: 1) Males, 2) Females

3 X 2 Effect of training intensity on 100m run time in males and females 3 Factors: 1) low-intensity, 2) medium-intensity, 3) high-intensity 2 Levels: 1) Males, 2) Females

Research Designs cont.

AKA Description Example Between-between 2 (or more) between

subject factors Weight lost in one group of individuals after diet A versus another group on diet B.

Within-within 2 (or more) within subject factors

Weight lost in individuals before and after a diet

Between-within 1 (or more) between and 1 (or more) within subject factor

Weight lost in males and females before and after a diet

Interaction • 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.

• Example: – 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. *** See example 1 in spreadsheet

Visual Assessment

• Is there an effect of factor A?

• Is there an effect of factor B?

• Is there an interaction effect?

Examples from Literature

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.

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.

– Dependent Variable is interval or ratio.

– The variables are normally distributed

– The groups have equal variances (for between-subjects factors)

– The groups are randomly assigned.

• Between-Within are similar to Repeated measures ANOVA (in chapter 10), but now sphericity must be applied to the pooled data (across groups) and 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.

– We will not cover this in today’s lecture.

Example: A 3x2 Between-between Factorial ANOVA

• Research Question: What are the effects of practice (1, 3, 5 days/wk) and experience (athlete, non-athlete) on throwing accuracy?

• Experimental Design: 18 subjects (n = 9 athletes, and n = 9 non-athletes) were randomly assigned to 1 of three the practice groups (1, 3, 5 days/wk).

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

** spreadsheet example 2 for data and statistical analysis

0.00

2.00

4.00

6.00

8.00

10.00

B1 B2 B3

Thro

win

g Acc

urac

y

Practice Schedule

Effect of experience and practice on throwing accuracy

A1

A2

Main Effects Testing:

FA: Main effect of Experience F(df = 1,12) = 8.52, p = .013

FB: Main effect of Practice F(df = 2,12) = 29.35, p = .000

FAB: Interaction F(df = 2,12) = 3.44, p = .066 is N.S. α = 0.05, however, the book calls this significant for illustration of post hoc testing.

The Interaction term

1. If the interaction is NOT SIGNIFICANT, analysis stops after the assessment of the main effects

2. If the interaction IS SIGNIFICANT, you can also do

simple effects testing. Using simple ANOVA, you can:

1. Simple Effects (A) 1. F across A at B1 = 0.5, N.S. 2. F across A at B2 = 2.4, N.S 3. F across A at B3 = 13.47, p = 0.022

2. Simple Effects (B)

1. F across B at A1 = 18.81, p = 0.03, HSD* = 3.34, p < 0.05; 4.87, p < 0.01 2. F across B at A2 = 10.86, p = 0.01, HSD* = 2.21, p < 0.05; 3.22, p < 0.01

EXAMPLE in SPSS See example in Excel worksheet. Input data on Monthly Income

• Note that the categorical variables are represented by numeric values instead of strings. These values can be labeled in Variable View.

• Analysis > General Linear Model > Univariate

Subject Gender Jobcat Monthly 1 1 1 1900 2 2 1 1800 3 1 1 3199 4 2 1 1998 5 1 1 2180 6 2 1 1829 7 1 1 3299 8 2 1 2879 9 1 2 4500

10 2 2 3720 11 1 2 4211

• In the pop-up menu, move “monthly” to the dependent variable box and “gender” and “jobcat” to the Fixed Factors box.

• Click Plot to include a graph in the output

• Click Options to enable post-hoc analysis of main effects • Check “Compare main effects” then move the factors and

factor interaction to the box to Display Means.

• Check Descriptive Statistics, Estimates of Effect Size, and Homogeneity test to include in output.

• Click Continue and then OK to generate output.

Plot and Descriptive Stats

Output of Omnibus Test

• Significance values indicate that there is a significant main effect in job category, but not in gender nor in the interaction between job category and gender.

Main Effect: Job Category

There is a significant difference in monthly incomes of all paired comparisons

Main Effect: Gender

This was already deemed insignificant in the omnibus test, but we will look at it just for practice.

The example does not justify post-hoc analysis of the interaction effect, however in the case that it is needed, a pairwise comparison of the interaction effect can be enabled. HOW TO: • Adjust the syntax to enable pairwise

comparison of the interaction effect • Select Paste to view the syntax editor

In the syntax editor, change the interaction line (Line 9): /EMMEANS=TABLES(gender*jobcat) should be /EMMEANS=TABLES(gender*jobcat) COMPARE(gender) Click the green triangle to update the output.

Post-hoc analysis of Interaction Effect