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Multivariate Analysis. One-way ANOVA. Tests the difference in the means of 2 or more nominal groups E.g., High vs. Medium vs. Low exposure Can be used with more than one IV Two-way ANOVA, Three-way ANOVA etc. ANOVA. _______-way ANOVA Number refers to the number of IVs

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### Multivariate Analysis

One-way ANOVA

- Tests the difference in the means of 2 or more nominal groups
- E.g., High vs. Medium vs. Low exposure

- Can be used with more than one IV
- Two-way ANOVA, Three-way ANOVA etc.

ANOVA

- _______-way ANOVA
- Number refers to the number of IVs

- Tests whether there are differences in the means of IV groups
- E.g.:
- Experimental vs. control group
- Women vs. Men
- High vs. Medium vs. Low exposure

- E.g.:

Logic of ANOVA

- Variance partitioned into:
- 1. Systematic variance:
- the result of the influence of the Ivs

- 2. Error variance:
- the result of unknown factors

- 1. Systematic variance:
- Variation in scores partitions the variance into two parts by calculating the “sum of squares”:
- 1. Between groups variation (systematic)
- 2. Within groups variation (error)

- SS total = SS between + SS within

Significant and Non-significant Differences

Non-significant:

Within > Between

Significant:

Between > Within

Partitioning the Variance Comparisons

- Total variation = score – grand mean
- Between variation = group mean – grand mean
- Within variation = score – group mean
- Deviation is taken, then squared, then summed across cases
- Hence the term “Sum of squares” (SS)

One-way ANOVA example

Total SS (deviation from grand mean)

Group A Group B Group C

49 56 54

52 57 52

52 57 56

53 60 50

49 60 53

Mean = 51 58 53

Grand mean = 54

One-way ANOVA example

Total SS (deviation from grand mean)

Group A Group B Group C

-5 25 2 4 0 0

-2 4 3 9 -2 4

-2 4 3 9 2 4

-1 1 6 36 -4 16

-5 25 6 36 -1 1

Sum of squares = 59 + 94 + 25 = 178

One-way ANOVA example

Between SS (group mean – grand mean)

A B C

Group means 51 58 53

Group deviation from grand mean -3 4 -1

Squared deviation 9 16 1

n(squared deviation) 45 80 5

Between SS = 45 + 80 + 5 = 130

Grand mean = 54

One-way ANOVA example

Within SS (score - group mean)

A B C

51 58 53

Deviation from group means -2 -2 1

1 -1 -1

1 -1 3

2 2 -3

-2 2 0

Squared deviations 4 4 1

1 1 1

1 1 9

4 4 9

4 4 0

Within SS = 14 + 14 + 20 = 48

The F equation for ANOVA

F = Between groups sum of squares/(k-1)

Within groups sum of squares/(N-k)

N = total number of subjects

k = number of groups

Numerator = Mean square between groups

Denominator = Mean square within groups

Significance of F

F-critical is 3.89 (2,12 df)

F observed 16.25 > F critical 3.89

Groups are significantly different

-T-tests could then be run to determine which groups are significantly different from which other groups

Two-way ANOVA

- ANOVA compares:
- Between and within groups variance

- Adds a second IV to one-way ANOVA
- 2 IV and 1 DV

- Analyzes significance of:
- Main effects of each IV
- Interaction effect of the IVs

Graphs of potential outcomes

- No main effects or interactions
- Main effects of color only
- Main effects for motion only
- Main effects for color and motion
- Interactions

Interaction—color only makes a difference for motion

A

R

O

U

S

A

L

x

x Motion

* Still

*

x

*

Color

B&W

Partitioning the variance for Two-way ANOVA

Total variation =

Main effect variable 1 +

Main effect variable 2 +

Interaction +

Residual (within)

Scatter Plot of Price and Attendance

- Price is the average seat price for a single regular season game in today’s dollars
- Attendance is total annual attendance and is in millions of people per annum.

Is there a relation there?

- Lets use linear regression to find out, that is
- Let’s fit a straight line to the data.
- But aren’t there lots of straight lines that could fit?
- Yes!

Desirable Properties

- We would like the “closest” line, that is the one that minimizes the error
- The idea here is that there is actually a relation, but there is also noise. We would like to make sure the noise (i.e., the deviation from the postulated straight line) to be as small as possible.

- We would like the error (or noise) to be unrelated to the independent variable (in this case price).
- If it were, it would not be noise --- right!

Scatter Plot of Price and Attendance

- Price is the average seat price for a single regular season game in today’s dollars
- Attendance is total annual attendance and is in millions of people per annum.

Simple Regression

The simple linear regression MODEL is:

y = 0 + 1x +

describes how y is related to x

0 and 1 are called parameters of the model.

is a random variable called the error term.

x

y

e

Simple Regression

- Graph of the regression equation is a straight line.
- β0 is the population y-intercept of the regression line.
- β1 is the population slope of the regression line.
- E(y) is the expected value of y for a given x value

Types of Regression Models

Regression Modeling Steps

- 1. Hypothesize Deterministic Components
- 2. Estimate Unknown Model Parameters
- 3. Specify Probability Distribution of Random Error Term
- Estimate Standard Deviation of Error

- 4. Evaluate Model
- 5. Use Model for Prediction & Estimation

Linear Multiple Regression Model

- 1. Relationship between 1 dependent & 2 or more independent variables is a linear function

Population Y-intercept

Population slopes

Random error

Dependent (response) variable

Independent (explanatory) variables

Multiple Regression Model

Multivariate model

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