multivariate analysis
Download
Skip this Video
Download Presentation
Multivariate Analysis

Loading in 2 Seconds...

play fullscreen
1 / 45

Multivariate Analysis - PowerPoint PPT Presentation


  • 208 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Multivariate Analysis' - bridget


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
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 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 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 6 36 -4 16

-5 25 6 36 -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 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 example3
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
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

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

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

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

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

ad