Sociology%20680

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Sociology 680

• Multivariate Analysis
• Analysis of Variance

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A Typology of Models
IV DV Category Quantity
Quantity
Category
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
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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
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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

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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)
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An Example of Simple Regression

• Given the following information, what would
  you expect a students 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)
• 70 75
• Sx 4 Sy 8  
• r 0.60

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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 58
  significantly heavier than those who are 54).
  That difference hypothesis could just as easily
  be recast as Are taller people significantly
  heavier than shorter people, where taller and
  shorter connote categories.

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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

Y1
Y2
X (Height)
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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 (Ys) of the categories in
  relation to the variation within those
  categories, or
• Variance Between Categories
• Variance Within Categories

Y (Weight)
.
.. ... .
... . .. ..
. ... . ...
. .. .. . ... .... ... ..
. Short Med Tall

Y
Y1
Y2
Y2
X (Height)
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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)

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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
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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 CH
115 125 135
135 145 155
140 150 160
145 155 165
165 175 185
140 150 160
Grand Mean 150
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Example of a simple ANOVA (cont.)
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.
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Example of a simple ANOVA (cont.)
In Class At Home Both CH
115 125 135
135 145 155
140 150 160
145 155 165
165 175 185
Bet SS ((5(140-150)2 5(150-150)2 5
(160-150)2)) 1000 Bet df 3-1 2 W/in SS
(115-140)2 (135-140)2 (140-140)2
(145-140)2 (165-140)2 (125-150)2 (145-150)2
(150-150)2 (155-150)2 (175-150)2
(135-160)2 (155-160)2 (160-160)2 165-160)2
(185-160)2 3900 W/in df 15 3 12
Source SS df MS F
Bet 1000 2 500 1.54
Within 3900 12 325
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SPSS Input for One-way ANOVA
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SPSS Output from a simple ANOVA
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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.

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Example of a Factorial ANOVA
In Class At Home Both CH
115 125 135
135 145 155
140 150 160
145 155 165
165 175 185
Working
135
Not Working
160
Means 140 150 160
150
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SPSS Input for 2x3 Factorial ANOVA
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SPSS Output from a 2x3 ANOVA