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# Multivariate Analysis - PowerPoint PPT Presentation

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

• 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.

• _______-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

• Variance partitioned into:

• 1. Systematic variance:

• the result of the influence of the Ivs

• 2. Error variance:

• the result of unknown factors

• 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

Non-significant:

Within > Between

Significant:

Between > Within

• 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)

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

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

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

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

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

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

• 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

• No main effects or interactions

• Main effects of color only

• Main effects for motion only

• Main effects for color and motion

• Interactions

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Total variation =

Main effect variable 1 +

Main effect variable 2 +

Interaction +

Residual (within)

SourceSSdfMSF

Main effect 1

Main effect 2

Interaction

Within

Total

• 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.

• 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!

• 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!

• 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.

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

• 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

E(y)

Regression line

Intercept

0

Slope 1

is positive

x

E(y)

Regression line

Intercept

0

Slope 1

is 0

x

Types of Regression Models

• 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

• 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

Multivariate model