Calculating Factorial ANOVA

The basic logic of a Factorial ANOVA (e.g. 2x2 ANOVA) is the same as the One-way ANOVA.

You calculate an F-ratio and this represents the contrast of Between Groups variance / Within Subjects variance.

If the F ratio is sufficiently high, then at least one mean is significantly different from at least one other mean.

Calculating Factorial ANOVA

F=Between Groups variance / Within Subjects variance. If the F ratio is sufficiently high, then at least one mean is significantly different from at least one other mean.

Calculating Factorial ANOVA

There are 3 F ratios for Two-way ANOVA:

F for the column main effect

F for the row main effect

F for the Interaction (Row x Column)

Social Support

No Social Support

Unimpaired 30 25 28

Impaired/

Disabled

20 10 15

25 18

Calculating Factorial ANOVA

There are 3 F ratios for Two-way ANOVA:

F for the column main effect

F for the row main effect

F for the Interaction (Row x Column)

0

5

10

15

20

25

30

35

Support NoSupport

Unimpaired

Disabled

Calculating Factorial ANOVA

There are 3 F ratios for Two-way ANOVA:

F for the column main effect

Social Support

No Social Support

Unimpaired 30 25 28

Impaired/

Disabled

20 10 15

25 18

Calculating Factorial ANOVA

There are 3 F ratios for Two-way ANOVA:

F for the column main effect

0

5

10

15

20

25

30

35

Unimpaired Disabled Average

Support

NoSupport

Calculating Factorial ANOVA

There are 3 F ratios for Two-way ANOVA:

F for the row main effect

Social Support

No Social Support

Unimpaired 30 25 28

Impaired/

Disabled

20 10 15

25 18

Calculating Factorial ANOVA

There are 3 F ratios for Two-way ANOVA:

F for the row main effect

0

5

10

15

20

25

30

35

Support NoSupport Average

Unimpaired

Disabled

Calculating Factorial ANOVA

There are 3 F ratios for Two-way ANOVA:

F for the interaction

Social Support

No Social Support

Unimpaired 30 25 28

Impaired/

Disabled

20 10 15

25 18

Calculating Factorial ANOVA

The Within Group Variance for these F ratios is calculated from the variability within each respective set of cells.

Social Support

No Social Support

Unimpaired 30 ± 4 25 ± 4 28

Impaired/

Disabled

20 ± 5 10 ± 6 15

25 18

Calculating Factorial ANOVAThe Grand Mean (GM): The mean of all your scores.

The deviation of an individual score from the GM is composed of the following:

1. The score’s deviation from the mean of its cell (Within Group variance)

2. The score’s row mean from the grand mean (Row Between Group variance)

3. The score’s column mean from the grand mean (Column Between Group variance)

4. After you subtract the (Within+Row+Column) from the total variance, you are left with the Interaction Between Group variance

Factorial ANOVA Example

A researcher was interested in men’s and women’s ability to navigate using two different kinds of directions: maps versus routes. Navigation ability was measured by the time to reach the destination (in minutes).

At the p < .05 level, were there any main effects or an interaction?

MEN WOMEN

MAP 30 47

35 43

33 34

42 36

Route 18 15

27 19

32 15

23 15

Steps of Hypothesis Testing

Step 1: Restate research questionMain effect of gender

H0: men = women

H1: men women

Main effect of direction type

H0: route = map

H1: route map

Interaction of gender and direction type

H0: men,route - women,route = men,map - women,map

H1: men,route - women,route men,map - women,map

Steps of Hypothesis Testing

Step 2: Determine the comparison distribution– Three F distributions– dfwithin = dfeach cell = 3 + 3 + 3 + 3 = 12

– dfgender = Ngenders – 1 = 2 – 1 = 1

– dfdirection type = Ndirection types – 1 = 2 – 1 = 1

– dfinteraction = Ncells – dfgender – dfdirection type – 1 = 4 – 1 – 1 – 1 = 1

Step 3: Determine the cutoffs– p < .05– F ratio with dfnumerator = 1, dfdenominator = 12– Fcutoff = 4.75

Steps of Hypothesis Testing

Step 4: Determine the sample scores

Download the excel spreadsheet Gender & Directions Review the findings.

Download the file genderdir.sav. Run the same analysis with SPSS

Steps of Hypothesis Testing

Step 5: Conclude– There was no main effect of gender

On average, men and women took the same amount of time to navigate

– There was a main effect of direction type Map directions took longer to follow compared with route

directions

– There was an interaction between gender and direction type

The difference in direction type was smaller for men compared to women

Men were faster than women with map directions, but women were faster than men with route directions

Assumptions of Two-Way ANOVA

Populations follow a normal curve

Populations have equal variances

Assumptions apply to the populations that correspond to each cell

Assumptions of Two-Way ANOVA

Effect Size and Two-Way ANOVA

R2 The proportion of variance accounted for

also called eta squared or correlation ratioThis is the proportion of the total variation of scores from

the Grand Mean that is accounted for by the variation between the means of the groups.

R2Columns= SSColumns /(SSTOTAL-SSRows-SSInteraction)

R2Rows= SSRows / (SSTOTAL -SSColumns-SSInteraction)

R2Interaction=SSInteraction /(SSTOTAL-SSRows-SSColumns)

Small = .01; Medium = .06; Large = .14Compute this for the Gender x Directions Study

Power and Two-Way ANOVAProbability of finding an effect when it is present.All effects in a 2x2 table:

N per cell Small (.01) Medium(.06) Large (.14)

10 .09 .33 .68

20 .13 .60 .94

30 .19 .78 .99

40 .24 .89 .99

50 .29 .94 .99

<Graph these in Excel>

Sample Size and Factorial ANOVASample size needed for 80% power, Using a 2x2 or 2x3

ANOVA, p < .05.

Small (.01) Medium(.06) Large (.14)

2x2 All Effects

197 33 14

2x3 Two-level main effect

132 22 9

2x3 Three level main effect and interaction

162 27 11

<Graph these in Excel>

Sample Size and Factorial ANOVA

Use SPSS to analyze the Gender x Directions study. Include Options such that SPSS generates the effect size and power for the analysis. How large are the effects?

Double the sample size and examine what happens to the power calculation. Do power and effect size change?

Run correlations between the variables and compare the Pearson r to the effect size.

Complex Factorial ANOVAMultiple Levels of Factors

A study can have as many contrasts as there are variables

A study can have multiple dependent measures

A study can have repeated measures

2x2x2

2x4x8x2x4

Complex Factorial ANOVAMultiple Levels of Factors

A study can have as many contrasts as there are variables

Let’s do a complex analysis with 3 factors

Download salary.sav

Analyze Gender, Education and Job Category, dependent measure is current salary. Run the basic analysis first and then set Options for effect size and power calculations.

Complex Factorial ANOVADichotomizing Numerical Variables

A common practice is to simplify an analysis by dichotomizing or categorizing a continuous variable and then using the variable as a factor in ANOVA.

This results in reduced information and lower power. When the resolution of measurement is lowered then it is harder to find a significant effect when it is present.

Categorizing variables also lowers the effect size.

Calculate correlations with salary.sav data set. Categorize the work experience variable and observe the change in correlations.

Complex Factorial ANOVA

Repeated Measures ANOVA

ANOVA may also be used to examine a within-subjects factor. This factor represents another variable measured on the same subjects. The measures are therefore correlated, or dependent on each other.

The repeated measures ANOVA is logically the same as the paired, or dependent measures t-test.

ExamplesSingle Factor:Change in illness before and after

treatmentMultiple Factor: Do Males and Females differ in change

in illness before and after treatmentGender x PrePost Change

Repeated Measures Factorial ANOVATraumatic Brain Injury

Repeated Measures Factorial ANOVA

Traumatic Brain InjuryPathologyDirect ImpactShearing InjuryHematoma

Increased Intracranial Pressure

Disruption of Neurotransmitter Systems

<Play Stone in Head movie>

Repeated Measures Factorial ANOVA

Traumatic Brain Injury

Coma Level:0 = Unresponsive1 = Responds only to pain2 = Responds to pain and verbal command with

nonspecific response3 = Responds to pain or verbal command with

meaningful response4 = Somnolent (falls asleep)5 = Awake and alert

Cycloserine for Traumatic Brain Injury

Cycloserine for Traumatic Brain Injury

The NMDA receptor may be important for memory. Cycloserine facilitates NMDA receptor function

Cycloserine for Traumatic Brain InjuryRepeated Measures ANOVA: Basic Study Design

Dependant Measure: Memory ScoresIndependent Variables: Treatment/Control, Coma LevelRepeated Measures: Pre and Post Drug Treatment

Analyze the data set trauma.sav Analyze Main effects of Independent Variables and

dependent w/o repeated measures.Analyze Treatment/Control x PrePost repeated

measures. -> Plot resultsAnalyze Coma Level x PrePost repeated measures ->

Plot resultsCalculate an average of Pre+Post/2 and then analyze the

new variable for a main effect for Coma Level.