1. Statistics for Political Science Levin and Fox Chapter Seven
Analysis of Variance
Statistics for Political Science
Levin and Fox
Chapter Seven

2. Analysis of Variance
Sometimes it is necessary to make comparisons over three or more groups.
In these instances we use a series of t ratios to make the comparisons.
If we try to pair-wise all possible combinations, we increase the likelihood of making a Type I error (rejecting the null hypothesis when we should retain it).

3. Variation between groups
Just a bit of discussion on the subject: The Logic of Analysis of Variance
The distance or deviation of raw scores from their group mean (variation within groups) and the distance or deviation of group means from one another (variation between groups).
Variation between groups
Variation within groups

4. Analysis of Variance
The analysis of variance yields an F ratio (which we will cover a little bit later) which indicates the size of the groups relative size of the variation within each group.
The larger the F ratio, the greater the probability of rejecting the null hypothesis and accepting the research hypothesis.

5. A Note on Process: The Analysis of Variance is a multi-step process.
Sum of Squares
Mean Square
F Ratio
The F ratio in Table D is the final step in the analysis of variance.

6. Sum of Squares Sum of Squares: The sum of squares is simply:
Squaring the deviations from the mean of the distribution and
Adding them up.

7. Now we will work to understand the components of the analysis of variance:
The Sum of Squares: Found by squaring the deviations from the mean of a distribution and adding these squared distributions together.
The general equation you must know to calculate the different types of sum of squares.

8. Let’s look at each one separately…
Sum of Squares
Comparing Groups:
When groups are compared, there are more than one type of sum of squares.
Total Sum of Squares (SS total)
Between Groups Sum of Squares (SS between)
Within Groups Sum of Squares (SS within)
Each type represents the sum of squared deviations from a mean.
Let’s look at each one separately…

9. Sum of Squares
Total Sum of Squares: The sum of the squared deviation of every raw score from the total mean
Where X = any raw score
= total mean for all groups combined
* Subtract the total mean from each raw score, square the deviations that result, and then sum them.

10. Within Group Sum of Squares: The sum of the squared deviations of every raw score from its group mean
Where X = any raw score
= mean of the group containing the raw score
*Subtract the group mean from each of the raw score square deviations that result, and then sum.

11. Ngroup = number of scores in any group = mean of any group
Between Group Sum of Squares: The sum of the squared deviations of every group mean from the total mean
Ngroup = number of scores in any group
= mean of any group
= mean of all groups combined
*Determine the difference between each group mean and the total mean, square this deviation, multiply by the number of scores in that group, and add these quantities.

12. We will use THESE formulas for computation: The Computational Formulas for Sum of Squares
All the scores squared and then summed
Total mean of all groups combined
Total number of scores in all groups combined
Mean of any group
Number of scores in any group

13. Before applying the formulas we have to find sum of scores (1), sum of squared scores (2), number of scores (3), and mean (4) 1) 2) 3) 4)

14. Next we calculate the following sums of squares:

15. Mean Square (MS)
Mean Square (MS):
The value of the sum of squares becomes larger as variation increases.
The sum of squares also increases with sample size. Because of this, the SS cannot be viewed as a true measure of variation.
Another measure of variation that we can use is the Mean Square.

16. Calculating the Mean Square for within and between groups:
MSbetween = between group mean square
SSbetween = between group sum of squares
Dfbetween = between group degrees of freedom
MSwithin = within group mean square
SSwithin = within group sum of squares
Dfwithin = within group degrees of freedom
Use the following equations to obtain the correct degrees of freedom:
k = number of groups

17. Calculating the mean Square (using Table 8.2 data)

18. The F Ratio The analysis of variance yields an F ratio.
The F ratio is the variance between groups and variation within groups compared.
The larger our calculated F ratio, the increased likelihood that we will have a statistically significant result.
Go to Table D in Appendix B.
Use the dfbetween (the numerator) across the top of the table.
Use the dfwithin (the denominator) along the side of the table.

19. Example: does family size vary by religious affiliation?

20. Step 1: Find the mean for each sample
Step 2: Cal. (1) Sum of scores, (2) sum of sq. scores, (3) number of subjs., (4) and mean 1) 2) 3) 4)

25. Finding: Reject Null Hypothesis: Family size does vary by religion
To reject the null hypothesis at the .05 significance level with 2 and 12 degrees of freedom, our calculated F ratio must exceed 3.8. Our obtained an F ratio of 8.24, we must reject the null hypothesis.

26. Requirements for using the F ratio:
1) Must be a comparison between three or more means.
2) Must be working with interval data.
3) Our sample must have been collected randomly from the research population.
4) We can/must assume that the sample characteristics are normally distributed.
5) We must assume that the variance between samples are all equal.