Multivariate analysis
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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

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Multivariate analysis

Multivariate Analysis


One way anova

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

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


Logic of anova

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

  • 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

Significant and Non-significant Differences

Non-significant:

Within > Between

Significant:

Between > Within


Partitioning the variance comparisons

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

One-way ANOVA example

Total SS (deviation from grand mean)

Group AGroup BGroup 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 example1

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 1 636 -416

-5 25 636 -1 1

Sum of squares = 59 + 94 + 25 = 178


One way anova example2

One-way ANOVA example

Between SS (group mean – grand mean)

A B C

Group means515853

Group deviation from grand mean-3 4-1

Squared deviation 916 1

n(squared deviation)4580 5

Between SS = 45 + 80 + 5 = 130

Grand mean = 54


One way anova example3

One-way ANOVA example

Within SS (score - group mean)

ABC

515853

Deviation from group means -2 -21

1-1-1

1-1 3

2 2-3

-2 2 0

Squared deviations4 4 1

111

119

449

440

Within SS = 14 + 14 + 20 = 48


The f equation for anova

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

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


Computer printout example

Computer Printout Example


Two way anova

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

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


Graphs

Graphs

A

R

O

U

S

A

L

x Motion

* Still

Color

B&W


No main effects for interactions

No main effects for interactions

A

R

O

U

S

A

L

x Motion

* Still

Color

B&W


No main effects for interactions1

No main effects for interactions

A

R

O

U

S

A

L

x Motion

x

x

* Still

*

*

Color

B&W


Main effects for color only

Main effects for color only

A

R

O

U

S

A

L

x Motion

* Still

Color

B&W


Main effects for color only1

Main effects for color only

A

R

O

U

S

A

L

*

x

x Motion

* Still

*

x

Color

B&W


Main effects for motion only

Main effects for motion only

A

R

O

U

S

A

L

x Motion

* Still

Color

B&W


Main effects for motion only1

Main effects for motion only

A

R

O

U

S

A

L

x

x

x Motion

* Still

*

*

Color

B&W


Main effects for color and motion

Main effects for color and motion

A

R

O

U

S

A

L

x Motion

* Still

Color

B&W


Main effects for color and motion1

Main effects for color and motion

A

R

O

U

S

A

L

x

x Motion

x

* Still

*

*

Color

B&W


Transverse interaction

Transverse interaction

A

R

O

U

S

A

L

x Motion

* Still

Color

B&W


Transverse interaction1

Transverse interaction

A

R

O

U

S

A

L

x

*

x Motion

* Still

x

*

Color

B&W


Interaction color only makes a difference for motion

Interaction—color only makes a difference for motion

A

R

O

U

S

A

L

x Motion

* Still

Color

B&W


Interaction color only makes a difference for motion1

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

Partitioning the variance for Two-way ANOVA

Total variation =

Main effect variable 1 +

Main effect variable 2 +

Interaction +

Residual (within)


Summary table for two way anova

Summary Table for Two-way ANOVA

SourceSSdfMSF

Main effect 1

Main effect 2

Interaction

Within

Total


Printout example

Printout Example


Printout plot

Printout plot


Scatter plot of price and attendance

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

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

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 attendance1

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

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 regression1

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


Simple regression2

Simple Regression

E(y)

Regression line

Intercept

0

Slope 1

is positive

x


Simple regression3

Simple Regression

E(y)

Regression line

Intercept

0

Slope 1

is 0

x


Types of regression models

Types of Regression Models


Regression modeling steps

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

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

Multiple Regression Model

Multivariate model


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