1. Multivariate Analysis: Analysis of Variance
Sociology 680
Multivariate Analysis:
Analysis of Variance

2. 1) Analysis of Variance Models 2) Structural Equation Models
A Typology of Models IV DV
Category
Quantity
1) Analysis of Variance Models
(ANOVA)
2) Structural Equation Models
(SEM)
Linear Models
3) Log
Linear Models
(LLM)
4) Logistic Regression
Models
(LRM)
Category Models

3. Examples of the Four Types
1. The effects of sex and race on Income
2. The effects of age and education on income
3. The effects of sex and race on union membership
4. The effects of age and income on union membership

4. The General Linear Model
Recall that the bi-variate Linear Regression model focuses on the prediction of a dependent variable value (Y), given an imputed value on a continuous independent variable (X).
The variation around the mean of Y less the variation around the regression line (Y’) is our measure of r2
Y (Weight) .. .…
. . ….
. … . … ….
…. .
.. … .
... .
X (Height) Y’

5. The General Linear Model (cont.)
Fixing a value of (X) and predicting a value of (Y) allows us to use the layout of points, under an assumption of linearity, to determine the effect of the IV on the DV. We do this by calculating the Y’ value in conjunction with the standard error of that value (Sy’) Where:
and
Y (Weight) Y’ .. .…
. . ….
. … . …
. .. . ….
…. .
.. … .
... . { Y’
X (Height)

6. An Example of Simple Regression
Given the following information, what would you expect a student’s score to be on the final examination, if his score on the midterm were 62? Within what interval could you be 95% confident the actual score on the Final would fall (i.e. what is the standard error)?
Midterm (X)   Final (Y)
= = 75
Sx = Sy = 8  
r = 0.60
Y’ = 75 + (0.6)(8/4)(62-70) = 65.4
= 8 (.8) = 6.4

7. The Test of Differences
But now assume that the goal is not prediction, but a test of the difference in two predictions (e.g. “are people who are 5’8” significantly heavier than those who are 5’4”). That difference hypothesis could just as easily be recast as “Are taller people significantly heavier than shorter people, where taller and shorter connote categories.
Y (Weight) .. .…
. . ….
. … . … ….
…. .
.. … .
... .
X (Height) Y’
Y’1
Y’2

8. The t-test
If there are simply two categories, we would be doing an ordinary t-test for the difference of means where:
Y (Weight) Y’ ….
... …. … .. .
Shorter Taller
| |
Y’1
Y’2
X (Height)

9. Analysis of Variance
If we were to have three categories, the test of significance becomes a simple one-way analysis of variance (ANOVA) where we are assessing the variance between means (Y’s) of the categories in relation to the variation within those categories, or:
Variance Between Categories
Variance Within Categories
Y (Weight) ….
... ….
.... .. .
Short Med Tall
| | | Y’
Y’1
Y’2
Y’2
X (Height)

10. Three Types of Analysis of Variance
One Way Analysis of Variance - ANOVA (Factorial ANOVA if two or more - IVs)
Analysis of Covariance - ANCOVA (Factorial ANCOVA if two or more - IVs)
Multiple Analysis of Variance (MANOVA) (Factorial MANOVA if two or more 2IVs)

11. Simple One Way ANOVA
Concept: When two or more categories of a non-quantitative IV are tested to see if a significant difference exists between those category means on some quantitative DV, we use the simple ANOVA where we are essentially looking at the ratio of the variance between means / variance within categories. As an F-ratio:
F-ratio = Bet SS/df divided by Within SS/df.
As a formula it is:

12. Example of a simple ANOVA
Suppose an instructor divides his class into three sub-groups, each receiving a different teaching strategies (experimental condition). If the following results of test scores were generated, could you assume that teaching strategy affects test results?
In Class
At Home
Both C+H
115
125
135
145
155
140
150
160
165
175
185
140
150
160
Grand Mean = 150

13. Example of a simple ANOVA (cont.)
Step 1: State hypotheses: Ho: 1 = 2 = 3;
Step 2: Specify the distribution: (F-distribution)
Step 3: Set alpha (say .05; therefore F = 3.68)
Step 4: Calculate the outcome:
Step 5: Draw the conclusion: Retain or Reject Ho:
Type of instruction does or does not influence test scores.

14. Example of a simple ANOVA (cont.)
In Class
At Home
Both C+H
115
125
135
145
155
140
150
160
165
175
185
Bet SS = ((5( )2 + 5( )2 +5 ( )2)) = 1000
Bet df = 3-1 = 2
W/in SS = ( )2 + ( )2 + ( )2 + ( )2+ ( )2 + ( )2 + ( )2 + ( )2 + ( )2 + ( )2 + ( )2 + ( )2 + ( ) )2 + ( )2 = 3900
W/in df = 15 – 3 = 12
Source SS df MS F
Bet
1000 2
500
1.54
Within
3900 12
325

15. SPSS Input for One-way ANOVA

16. SPSS Output from a simple ANOVA

17. Two Way or Factorial ANOVA
Concept: When we have two or more non-quantitative or categorical independent variables, and their effect on a quantitative dependent variable, we need to look at both the main effects of the row and column variable, but more importantly, the interaction effects.

18. Example of a Factorial ANOVA
In Class
At Home
Both C+H
115
125
135
145
155
140
150
160
165
175
185
Working
135
Not Working
160
Means

19. SPSS Input for 2x3 Factorial ANOVA

20. SPSS Output from a 2x3 ANOVA