1. Sociology 5811: Lecture 14: ANOVA 2
Copyright © 2005 by Evan Schofer
Do not copy or distribute without permission

2. Announcements Midterm next class Today: Bring a Calculator
Bring Pencil/Eraser
Today:
Wrap up ANOVA
Midterm review activities

3. Review: ANOVA ANOVA = “ANalysis Of VAriance”
“Oneway ANOVA” : The simplest form
ANOVA lets us test to see if any group mean differs from the mean of all groups combined
Answers: “Are all groups equal or not?”
H0: All groups have the same population mean
m1 = m2 = m3 = m4
H1: One or more groups differ
But, doesn’t distinguish which specific group(s) differ.

4. ANOVA: Concepts & Definitions
The grand mean is the mean of all groups
ex: mean of all entry-level workers = $8.70/hour
The group mean is the mean of a particular sub-group of the population
The effect (a) of a group is the difference between that group’s mean from the grand mean
The dependent variable is our variable of interest
We explore whether its value “depends” on a case’s group

5. Review: Sum of Squared Deviation
The total deviation can partitioned into aj and eij components:
The total variance (SStotal) is made up of:
aj : between group variance (SSbetween)
eij : within group variance (SSwithin)
SStotal = SSbetween + SSwithin

6. Review: Sum of Squared Variance
The sum of squares grows as N gets larger.
To derive a more comparable measure, we “average” it, just as with the variance: i.e, (divide) by N-1
It is desirable, for similar reasons, to “average” the Sum of Squares between/within
Result the “Mean Square” variance
MSbetween and MSwithin

7. Sum of Squared Variance
Choosing relevant denominators we get:

8. Mean Squares and Group Differences
MSbetween > MSwithin:
MSbetween < MSwithin:

9. Mean Squares and Group Differences
Question: Which suggests that group means are quite different:
MSbetween > MSwithin or MSbetween < MSwithin
Answer: If between group variance is greater than within, the groups are quite distinct
It is unlikely that they came from a population with the same mean
But, if within is greater than between, the groups aren’t very different – they overlap a lot
It is plausible that m1 = m2 = m3 = m4

10. The F Ratio
The ratio of MSbetween to MSwithin is referred to as the F ratio:
If MSbetween > MSwithin then F > 1
If MSbetween < MSwithin then F < 1
Higher F indicates that groups are more separate

11. The F Ratio The F ratio has a sampling distribution
That is, estimates of F vary depending on exactly which sample you draw
Again, this sampling distribution has known properties that can be looked up in a table
The “F-distribution”
Different from z & t!
Statisticians have determined how much area falls under the curve for a given value of F…
So, we can test hypotheses.

12. The F Ratio
Assumptions required for hypothesis testing using an F-statistic
1. J groups are drawn from a normally distributed population
2. Population variances of groups are equal
If these assumptions hold, the F statistic can be looked up in an F-distribution table
Much like T distributions
But, there are 2 degrees of freedom: J-1 and N-J
One for number of groups, one for N.

13. The F Ratio Example: Looking for wage discrimination within a firm
The company has workers of three ethnic groups:
Whites, African-Americans, Asian-Americans
You observe in a sample of 200 employees:
Y-barWhite = $8.78 / hour
Y-barAfAm = $8.52 / hour
Y-barAsianAm = $8.91 / hour

14. The F Ratio Suppose you calculate the following from your sample:
Recall that N = 200, J = 3
Degrees of Freedom: J-1 = 2, N-J = 197
If a = .05, the critical F value for 2, 197 is 3.00
See Knoke, p. 514
The observed F easily exceeds the critical value
Thus, we can reject H0
We can conclude that the groups do not all have the same population mean.

15. Critical a area: Reject H0
The F Ratio: Visually
Rough sketch of F-distribution
N = 200, J = 3; Degrees of Freedom: J-1 = 2, N-J = 197
If a = .05, the critical F value for 2, 197 is 3.00
Critical a area: Reject H0

16. Post-Hoc Tests
Limitation: ANOVA doesn’t tell you which group(s) differ from each other
Solution: “Post-Hoc Tests”:
e.g., Bonferroni, Scheffe, Tukey tests
Tests provide pair-wise comparison of all group
Ex: Bonferroni
Corrects (reduces) a for pair-wise tests, to avoid Type I error
Which one should you use?
Bonferroni & Scheffe are very conservative
For most purposes, they all produce identical results
Consult an advanced stats text to learn the subtle differences.

17. ANOVA: Example Example: GSS data: Schooling & Race
Oneway ANOVA, plus descriptives and Scheffe Post-hoc test
Note: N = 2752, grand mean = 13.36

18. ANOVA: Example Example: GSS data: Schooling & Race
Sum of Squares: Between, within, and total
Mean square variance: between and within
Compare F-value to “critical f” in F-table. Or compare p-value (“Sig.”) to a
We reject H0 because F is big, p-value is small.

19. ANOVA Example: Scheffe Test
Tests compare every group
Compare p-value (“Sig.”) to a
Note: White/Black difference is significant; White/Other difference is not

20. Comparison with T-Test
T-test strategy: Determine the width of the sampling distribution of the difference in means…
Use that info to assess probability that groups have same mean (difference in means = 0)
ANOVA strategy
Compute F-ratio, which indicates what kind of deviation is larger: “between” vs. “within” group
High F-value indicates groups are separate
Note: For two groups, ANOVA and T-test produce identical results.

21. Bivariate Analyses
Up until now, we have focused on a single variable: Y
Even in T-test for difference in means & ANOVA, we just talked about Y – but for multiple groups…
Alternately, we can think of these as simple bivariate analyses
Where group type is a “variable”
Ex: Seeing if girls differ from boys on a test…
… is equivalent to examining whether gender (a first variable) affects test score (a second variable).

22. 2 Groups = Bivariate Analysis
Group 1: Boys
Case
Score 1 57 2 64 3 48
Case
Gender
Score 1 57 2 64 3 48 4 53 5 87 6 73
Group 2: Girls
Case
Score 1 53 2 87 3 73
2 Groups = Bivariate analysis of Gender and Test Score

23. T-test, ANOVA, and Regression
Both T-test and ANOVA illustrate fundamental concepts needed to understand “Regression”
Relevant ANOVA concepts
The idea of a “model”
Partitioning variance
A dependent variable
Relevant T-test concepts
Using the t-distribution for hypothesis tests
Note: For many applications, regression will supersede T-test, ANOVA
But in some cases, they are still useful…