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10tutorial-ANOVAcalcs.pdf
How to Calculate ANOVA
CNSL 5302 (Research Methods)
What is ANOVA?
ANOVA stands for “Analysis of Variance”
It is a procedure for comparing means of three or more groups
Procedures for Computing a Simple ANOVA
Code/Table the data
Compute basic ratios
Compute sums of squares (SS)
Compute mean squares (MS)
Compute F ratios
Evaluate F ratios for statistical significance
Example ANOVA-
One Way CRD
A1 A2 A3 A4
13 9 11 10
8 8 7 4
7 5 5 4
9 6 7 3
6 8 8 2
8 3 1 5
13 12 12 11
9 11 11 10
8 9 9 8
11 10 7 4
8 7 8 8
4 5 7 5
14 11 11 12
13 9 9 6
10 11 9 9
10 8 10 7
7 10 5 7
9 6 4 6
167 148 141 121
9.3 8.2 7.8 6.7
Column totals TAj
Column means
Here is a simple one-way ANOVA with one factor and four levels. By way of example, let’s say the factor is the number of ‘sexting’ events that occurred at a particular high school, and the levels are grades- 1=Fr, 2=So, 3=Jr, 4=Sr Is there a significant difference between these column means?? In other words, does age have an effect on whether or not a student sends a ‘sexting’ message? If so, where are the differences (what ages)?
Grand mean- 8.01
Example ANOVA-
One Way CRD
an
X 2
A1 A2 A3 A4
13 9 11 10
8 8 7 4
7 5 5 4
9 6 7 3
6 8 8 2
8 3 1 5
13 12 12 11
9 11 11 10
8 9 9 8
11 10 7 4
8 7 8 8
4 5 7 5
14 11 11 12
13 9 9 6
10 11 9 9
10 8 10 7
7 10 5 7
9 6 4 6
167 148 141 121
9.3 8.2 7.8 6.7
2
X
Column totals TAj
Column means
Basic Ratios-
Grand mean- 8.01
a = number of levels of factor A…which is 4
(1) =
(2) =
(3) =
n = number of scores in group Aj … which is 18
n
T jA 2
Obtained by squaring each score and adding them up…5191
Obtained by squaring each column total and adding them up, then dividing by n, which is 4684.17
Obtained by each score and then squaring the total…5772
Example ANOVA-
One Way CRD
an
X 2
A1 A2 A3 A4
13 9 11 10
8 8 7 4
7 5 5 4
9 6 7 3
6 8 8 2
8 3 1 5
13 12 12 11
9 11 11 10
8 9 9 8
11 10 7 4
8 7 8 8
4 5 7 5
14 11 11 12
13 9 9 6
10 11 9 9
10 8 10 7
7 10 5 7
9 6 4 6
167 148 141 121
9.3 8.2 7.8 6.7
2
X
Basic Ratios-
(1) =
(2) =
(3) =
n
T jA 2
(577)2
(4) (18) = = 4624.01
=
= 132 + 82 + … + 62 = 5191
1672 + 1482 + 1412 + 1212
18 = 4684.17
These numbers are used as indicators in the equations on the next slide.
Example ANOVA-
One Way CRD
A1 A2 A3 A4
13 9 11 10
8 8 7 4
7 5 5 4
9 6 7 3
6 8 8 2
8 3 1 5
13 12 12 11
9 11 11 10
8 9 9 8
11 10 7 4
8 7 8 8
4 5 7 5
14 11 11 12
13 9 9 6
10 11 9 9
10 8 10 7
7 10 5 7
9 6 4 6
167 148 141 121
9.3 8.2 7.8 6.7
Sums of Squares-
SStotal = (2) – (1) = 5191 – 4624.01 = 566.99 SSA = (3) – (1) = 4684.17 – 4624.01 = 60.16 SSerror = (2) – (3) = 5191 – 4684.17 = 506.83
For ease of interpretation, we make a summary table.
These are the numbers from
the previous slide.
It is VERY important that you understand the relationships in the table.
Example ANOVA-
One Way CRD
A1 A2 A3 A4
13 9 11 10
8 8 7 4
7 5 5 4
9 6 7 3
6 8 8 2
8 3 1 5
13 12 12 11
9 11 11 10
8 9 9 8
11 10 7 4
8 7 8 8
4 5 7 5
14 11 11 12
13 9 9 6
10 11 9 9
10 8 10 7
7 10 5 7
9 6 4 6
167 148 141 121
9.3 8.2 7.8 6.7
ANOVA Summary Table
Source SS df MS F
A 60.16 a-1=3 20.05 2.69
Error 506.83 a(n-1)=68 7.45
Total 566.99 an-1=71
Note that the columns are additive.
60.16 + 506.83 = 566.99
3 + 68 = 71
Note that each MS is its
SS divided by its df-
20.05 = 60.16 3
7.45 = 506.83 68
Note that the F is the ratio
of MSA/MSE-
20.45 7.45 = 2.69
Results
If you compare your calculated F of 2.69 to a tabled value (from an F table) with 3 and 68 degrees of freedom, you find the tabled value (2.75) is larger than the calculated value. Therefore, you fail to reject the null hypothesis and conclude there is no significant difference in the number of ‘sexting’ incidents as the students get older. (Even though 9.3 looks bigger than 6.7, it is not statistically different.)
