Chapter 11Complex ANOVA Designs

Part III: Additional Hypothesis Tests

Renee R. Ha, Ph.D.James C. Ha, Ph.DIntegrative Statistics for the Social & Behavioral Sciences

What is a Factorial Design?

The simplest factorial design is a 2 × 2 design which involves two independent variables, each with two levels. E.g.: gender (male vs. female) on two levels of a treatment drug (low vs. high).

What is a Factorial Design?

Factorial designs can involve more levels of each independent variable and thus become a 2 × 3 design, a 3 × 3 design, and so on.

Definitions

Main effect: Influence of your independent variable on your dependent variable.

Definitions

Interaction: The combined effect of both (or all) independent variables over and above the separate effects of each variable alone.

Figure 11.1

Examples of Main Effects and Interactions in Graphical Form

Linear or Additive Equation

Score = + IV1 + IV2 + (IV1)(IV2) + E Where: = The grand mean starting point IV1 = Effect of the first independent variable (sex) IV2 = Effect of the second independent variable (ethnic group) (IV1)(IV2) = Interaction effect of the independent variables (sex * group) E = Unexplained error

Table 11.3

Source SS df S2 (MS) F obt.

Row (sex)        

Column (group)        

Row * Column (sex * group)

       

Within (within a cell)        

Total    

Two-Way ANOVA Summary Table (Hesitation Time Example)

Types of Variance

sw2 = within-cell variance (reflects natural variation in the population,

thus it is an estimate of σ2).Within-cell variance could also be called “error” (E) because it is variability that we cannot control, but occurs naturally.

sR2 = between-row variance (reflects natural variation in the

population AND any variability due to your row variable). sC

2 = between-column variance (reflects natural variation in the population AND any variability due to your column variable).

sRC2 = row * column variance (reflects natural variation in the population AND any variability due to the interaction between your row and column variables).

F-obtained Formulas

F-obt for Row Factor (sex) = 2

2

w

R

s

s

F-obt for Column Factor (group) = 2

2

w

C

s

s

F-obt for Row*Column (interaction) = 2

2

w

RC

s

s

Figure 11.2

Craik and Lockhart (1972) Memorization Experiment

Results if you use Microsoft Excel to calculate the Two-way ANOVA

Anova: Two-Factor

ANOVA

Source of Variation SS Df MS F P-value F crit

Sample 249.64 1 249.64 35.73091603 4.4838E-08 3.946865945

Columns 1476.66 4 369.165 52.83850191 7.85493E-23 2.472930305

Interaction 195.26 4 48.815 6.986879771 6.04944E-05 2.472930305

Within 628.8 906.986666

7

Total2550.3

6 99        

Results if you use SPSS to calculate the ANOVA (or F test)

Tests of Between-Subjects EffectsDependent Variable: SCORE

SourceType III Sum of

Squaresdf Mean Square F Sig.

Model 15331.200 10 1533.120 219.435 .000

AGE 249.640 1 249.640 35.731 .000

TECH 1476.660 4 369.165 52.839 .000

AGE * TECH 195.260 4 48.815 6.987 .000

Error 628.800 90 6.987 628.800

Total 15960.000 100

Figure 11.3

Results if you use Microsoft Excel to calculate the Two-way ANOVA

ANOVA

Source of Variation SS df MS F P-value F crit

Sample 30.33333 2 15.16667 1.95279 0.153715 3.20432

Columns 254.3333 2 127.1667 16.37339 4.54E-06 3.20432

Interaction 13.33333 4 3.333333 0.429185 0.786769 2.578737

Within 349.5 45 7.766667

Total 647.5 53        

Results if you use SPSS to calculate the ANOVA (or F test)

Tests of Between-Subjects EffectsDependent Variable: EYEGLAN

SourceType III Sum of Squares df Mean Square F Sig.

Model 298.000 8 37.250 4.796 .000

DISCTYP 30.333 2 15.167 1.953 .154

PAIRGEN 254.333 2 127.167 16.373 .000

DISCTYP * PAIRGEN 13.333 4 3.333 .429 .787

Error 349.500 45 7.767

Total 701.000 54

When to Use Two-Way ANOVA?

1. You have two independent variables and a between-groups (independent groups) design, and

2.The sampling distribution is normally distributed, and

3.The dependent variable is on an interval or ratio scale (typically), and

4. When the variances of the groups are the same, or are homogeneous.

Multifactorial ANOVA

Also called MANOVA

Score = + IV1 + IV2 + IV3 + (IV1)(IV2) + (IV1)(IV3) +(IV2)(IV3) + (IV1)(IV2)(IV3) + E Where = The grand mean IV1 = Effect of the first independent variable IV2 = Effect of the second independent variable IV3 = Effect of the third independent variable (IV1)(IV2) = Interaction effect of the first two independent variables (IV1)(IV3) = Interaction effect of the first and third independent variables (IV2)(IV3) = Interaction effect of the second and third independent variables (IV1)(IV2)(IV3) = Interaction effect of all three independent variables E = Unexplained error

Repeated-Measures ANOVA

Linear or additive equation Score = + IV1 + Subj + E

When to Use a RM ANOVA?

1. You have one independent variable with more than two levels and a within-groups design.

2. The sampling distribution is normally distributed.

3. The dependent variable is on an interval or ratio scale (typically).

Overview of Single-Sample, Two-Sample, and Three or More Sample Tests