2x2 factorial Analysis of Variance ANOVA

2 x 2 factorial Analysis of Variance ANOVA- formula and example

What are 2x2 factorial designs?
The best wat to define 2x2 factorial designs is to compare them to single-factor designs. In single-factor, one-way ANOVA, two or more groups of subjects each receive one treatment consisting of a single factor. In factorial designs 4 or more groups of subjects each receive a treatment which consists of two or more factors combined.

The layout of these two designs are given below.

The layout of a single factor ANOVA

Group 1	Group 2
Subject 1 Subject 2 Subject 3 Subject 4 Subject 5	Subject 6 Subject 7 Subject 8 Subject 9 Subject 10

In this example, Group 1 received a sugar pill, Group 2 received a new drug, Drugx.It is important to note that the subjects of group 1 do not receive Drugx. Similarly the subjects if group 2 do not receive the sugar pill. This is the concept of independence, an important concept in Statistics.

The layout of a 2x2 factorial ANOVA

	B1	B2
A1	A1B1 Subject 1 Subject 2 Subject 3 Subject 4 Subject 5	A1B2 Subject 6 Subject 7 Subject 8 Subject 9 Subject 10
A2	A2B1 Subject 11 Subject 12 Subject 13 Subject 14 Subject 15	A2B2 Subject 16 Subject 17 Subject 18 Subject 19 Subject 20

What does 2x2 mean? In this example of a factorial design, we have a 2x2 (we read this as "a two by two") factorial. A two by two, meaning two factors A and B with two levels each. In another case of a 2x3 factorial design we have two factors, A and B, factor A two levels, factor B three levels.

FORMAT OF 2x2 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.


Observe that in the interaction graph the lines are not parallel.

ANOVA 2x2 factorial-practice example

An experimenter wanted to test the effect of two drugs on emotionality of male and female teenagers. He randomly selected 10 male and 10 female teenagers and randomly assigned them to 4 groups: Group 1, Group 2, Group 3, Group 4, five subjects in each group as shown in the following table.
The layout of this experiment

	Drug 1 B1	Drug 2, B2
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
Female A2	A2B1 Group3 Subject 11 Subject 12 Subject 13 Subject 14 Subject 15	A2B2 Group4 Subject 16 Subject 17 Subject 18 Subject 19 Subject 20

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.

Table showing the layout of the experiment with data

	B1	B2
A1	A1B1 11 11 13 12 10	A1B2 17 18 17 16 17
A2	A2B1 15 14 14 16 15	A2B2 20 19 18 20 18

.

It is customary to report our calculations of the ANOVA in a table.
ANOVA SUMMARY TABLE

Source	SS	df	MS	F	p
Berween A	36.45	1	36.45	41.66	<0.0001
Between B	120.05	1	120.05	137.2	<0.0001
AxB (interaction)	2.45	1	2.45	2.8	>0.95
Within	14	16	0.88
Total	172.95	19

How we read the ANOVA summary table
After we calculate the F, we go to the F tables and enter with the degrees of freedom we have, in this case 1 and 16. We first check the 0.05 level (level of significance). The F at df 1 and 16 is 4.49. In the summary table we see that the F for factor A is 41.66, This is greater than 4.49, 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 see df 1 and 16 and F=137.2. We go to the F table and enter with df 1 and 16 and find F 4.49. This is less than 137.2, 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 see df 1 and 16 and F=2.8. This is less than the F table value of 4.49 so we conclude that here we do not have significance. We formally express this as follows: p>0.05.

Step by step calculation of 2x2 ANOVA factorial

The goal of our calculations in ANOVA is to compute the F ratio. What is the F ratio? The F ratio is MS between over MS within. What is Mean Square (MS)? 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:

Step-by-step calculations
Step 1. Calculate the mean of each group.
Step 2. Subtract each score from the mean.
Step 3. Square each difference
Step 4. Add these squared differences. font red This is the Sum of Squares, the SS on the ANOVA summary table.)
Step 5. calculate the degrees of freedom df (number of scores that went into the calculation of the mean minus 1)
Step 6 Divide the SS by the df. Voila! this the MS.
Step 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, compares 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 term in factorial designs. 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.