2 x 3 factorial Analysis of Variance ANOVA formula and example
2 x 3 factorial Analysis of Variance ANOVA formula and example
The layout of a 2x3 factorial ANOVA
B1  B2  B3  

A1 
A1B1 Subject 1 Subject 2 Subject 3 Subject 4 Subject 5 
A1B2 Subject 6 Subject 7 Subject 8 Subject 9 Subject 10 
A1B3 Subject 11 Subject 12 Subject 13 Subject 14 Subject 15 
A2 
A2B1 Subject 16 Subject 17 Subject 18 Subject 19 Subject 20 
A2B2 Subject 21 Subject 22 Subject 23 Subject 24 Subject 25 
A2B3 Subject 26 Subject 27 Subject 28 Subject 29 Subject 30 
In this example of a factorial design, we have a 2x3 (we read this as "a two by three") factorial. Two by three, meaning two factors: A and B. "two" meaning two levels for factor A. "three" meaning three levels for B. In another case of a 3x2 factorial design we have two factors, A and B, factor A three levels, factor B two levels.
FORMAT OF 2x3 FACTORIAL ANOVA SUMMARY TABLE
Source  SS  df  MS  F  p 

Berween A  
Between B  
AxB (interaction)  
Within  
Total 
Interaction factorial designs
Note the term "Interaction" in the ANOVA summary table of the factorial design What is interaction? The best way to grasp the concept of interaction is to graph it.ANOVA 2x3 factorial practice example
An experimenter wanted to test the effect of two drugs on emotionality of male and female teenagers. He randomly selected 15 male and 15 female teenagers and randomly assigned them to 6 groups: Group 1, Group 2, Group 3, Group 4, Group 5, Group 6. five subjects in each group as shown in the following table.
Drug 1 B1  Drug 2, B2  Drug 3, B3 


Male A1 
A1B1 Group1 Subject 1 Subject 2 Subject 3 Subject 4 Subject 5 
A1B2 Group2 Subject 6 Subject 7 Subject 8 Subject 9 Subject 10 
A1B3 Group3 Subject 11 Subject 12 Subject 13 Subject 14 Subject 15 
Female A2 
A2B1 Group3 Subject 16 Subject 17 Subject 18 Subject 19 Subject 20 
A2B2 Group4 Subject 21 Subject 22 Subject 23 Subject 24 Subject 25 
A2B3 Group2 Subject 26 Subject 27 Subject 28 Subject 29 Subject 30 
The data are presented on the table below. The scores are the values recorded on a device measuring galvanic skin response, a measure of emotionality. Higher values indicate stronger emotion.
B1  B2  B3  

A1 
A1B1 11 11 13 12 10 
A1B2 17 18 17 16 17 
A1B3 27 25 24 26 27 
A2 
A2B1 15 14 14 16 15 
A2B2 20 19 18 20 18 
A2B3 30 29 29 28 31 
2x3 FACTORIAL ANOVA SUMMARY TABLE
Source  SS  df  MS  F  p 

Berween A  67.5  1  67.5  62.31  <.0001 
Between B  1088.07  2  544.04  502.19  <.0001 
AxB (interaction)  3.8  2  1.9  1.75  0.1952 
Within  26  24  1.08  
Total  1185.37  29 
After we calculate the F, we go to .........Next
the F table to find the required F value for A, B, and AxB (interaction). Because,( remember?) the F ratio is A over within, B over within, AxB over within, we enter the F table with
df of A and df within , which is 1 and 24.
also B and df within, which is 2 and 24
and lastly AxB wihin., which is 2 and 24. We first choose the F tanle at 0.05 level of significance.
Factor A: The F at df 1 and 24 is 4.25. In the summary table we see that the F for factor A (rows) is 62.31, This is greater than 4.25, so we conclude that here we have significance at the 0.05 level of significance; we say p<0.05, p less than 0.05. It has been accepted among scientists that at the 0.05 level we are allowed to say that we have significance, that the finding of our experiment is reliable.
Next we look at factor B. We enter the F table with df 2 and 24 and find F=3.50. This is less than the F of our summary table 502.19, therefore we conclude that we have significance at the 0.05 level. We formally express this as follows: p<0.05.
Next we look at AxB. We enter the F table with df 2 and 24 and find F=3.50.. This is greater than the F at the summary table value of 1.75 so we conclude that here we do not have significance. We formally express this as follows: p>0.05.
Step by step calculation of 2x3 ANOVA factorial.........Next
df of A and df within , which is 1 and 24.
also B and df within, which is 2 and 24
and lastly AxB wihin., which is 2 and 24. We first choose the F tanle at 0.05 level of significance.
Factor A: The F at df 1 and 24 is 4.25. In the summary table we see that the F for factor A (rows) is 62.31, This is greater than 4.25, so we conclude that here we have significance at the 0.05 level of significance; we say p<0.05, p less than 0.05. It has been accepted among scientists that at the 0.05 level we are allowed to say that we have significance, that the finding of our experiment is reliable.
Next we look at factor B. We enter the F table with df 2 and 24 and find F=3.50. This is less than the F of our summary table 502.19, therefore we conclude that we have significance at the 0.05 level. We formally express this as follows: p<0.05.
Next we look at AxB. We enter the F table with df 2 and 24 and find F=3.50.. This is greater than the F at the summary table value of 1.75 so we conclude that here we do not have significance. We formally express this as follows: p>0.05.
Step by step calculation of 2x3 ANOVA factorial.........Next
The goal of our calculations in ANOVA is to compute the F ratio, The F ratio is MS between over MS within. Mean Square is the mean of the squared deviations (differences) of each score from the mean. These are very simple calculations involving high school mathematics. Simple as they are, they are very important concepts in data analysis and beyond, that is science in general.
You will never need to perform these calculations. There are many free Statistics calculators online. However, for the purpose of developing the concepts of ANOVA here are the steps:
1. Calculate the mean of each group.
2. Subtract each score from the mean.
3. Square each difference
4. Add these squared differences. font red This is the Sum of Squares, the SS on the ANOVA summary table.)
5. calculate the degrees of freedom df (number of scores that went into the calculation of the mean minus 1)
6 Divide the SS by the df. Voila! this the MS.
7. The last step is to calculate the F. Divide MS by the MS of the error term (which is the MS within but may be something else depending on which ANOVA design you have. )
The F ratio, as all ratios, compare two things. For example the ratio 8/4 compares 8 to 4 and finds that 8 is two times greater than 4..
Understanding the 2x2 factorial ANOVA summary table
A
Looking at the layout tables above, we see that factor A is gender. Factor B is drug, Our calculations gave a p value <0.05 meaning that factor A, gender, gave a significant difference. In other words, there is a difference in emotionality between male and female
B
.Looking at the layout tables above, we see that factor B is Drug. Our calculations gave a p value <0.05, meaning that factor B, drug, gave a significant difference. In other words, there is a difference in emotionality between subjects that received drug 1 as compared to subjects that received drug 2.
AxB
This is the interaction term. Definition of the interaction. What is interaction in factorial designs? Interaction is present if one level of one factor has a disproportionate effect on one level of the other factor.
1. Calculate the mean of each group.
2. Subtract each score from the mean.
3. Square each difference
4. Add these squared differences. font red This is the Sum of Squares, the SS on the ANOVA summary table.)
5. calculate the degrees of freedom df (number of scores that went into the calculation of the mean minus 1)
6 Divide the SS by the df. Voila! this the MS.
7. The last step is to calculate the F. Divide MS by the MS of the error term (which is the MS within but may be something else depending on which ANOVA design you have. )
The F ratio, as all ratios, compare two things. For example the ratio 8/4 compares 8 to 4 and finds that 8 is two times greater than 4..
Understanding the 2x2 factorial ANOVA summary table
A
Looking at the layout tables above, we see that factor A is gender. Factor B is drug, Our calculations gave a p value <0.05 meaning that factor A, gender, gave a significant difference. In other words, there is a difference in emotionality between male and female
B
.Looking at the layout tables above, we see that factor B is Drug. Our calculations gave a p value <0.05, meaning that factor B, drug, gave a significant difference. In other words, there is a difference in emotionality between subjects that received drug 1 as compared to subjects that received drug 2.
AxB
This is the interaction term. Definition of the interaction. What is interaction in factorial designs? Interaction is present if one level of one factor has a disproportionate effect on one level of the other factor.
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What does p mean?
What does p mean?
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