multivariate analysis. one-way anova tests the difference in the means of 2 or more nominal groups...

Multivariate Multivariate AnalysisAnalysis

One-way ANOVAOne-way ANOVA

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

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

ANOVAANOVA

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

Tests whether there are differences in the Tests whether there are differences in the means of IV groupsmeans of IV groups E.g.:E.g.:

Experimental vs. control groupExperimental vs. control group Women vs. MenWomen vs. Men High vs. Medium vs. Low exposureHigh vs. Medium vs. Low exposure

Logic of ANOVALogic of ANOVA

Variance partitioned into:Variance partitioned into: 1. Systematic variance:1. Systematic variance:

the result of the influence of the Ivsthe result of the influence of the Ivs 2. Error variance:2. Error variance:

the result of unknown factorsthe result of unknown factors

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

SS total = SS between + SS withinSS total = SS between + SS within

Significant and Non-Significant and Non-significant Differencessignificant Differences

Significant: Between > Within

Non-significant: Within > Between

Partitioning the Variance Partitioning the Variance ComparisonsComparisons

Total variation = score – grand meanTotal variation = score – grand mean

Between variation = group mean – grand Between variation = group mean – grand meanmean

Within variation = score – group meanWithin variation = score – group mean

Deviation is taken, then squared, then Deviation is taken, then squared, then summed across casessummed across cases Hence the term “Sum of squares” (SS)Hence the term “Sum of squares” (SS)

One-way ANOVA One-way ANOVA exampleexample

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 53Group deviation from grand mean -3 4 -1Squared deviation 9 16 1n(squared deviation) 45 80 5

Between SS = 45 + 80 + 5 = 130

Grand mean = 54


Within SS (score - group mean)A B C51 58 53

Deviation from group means -2 -21 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 The F equation for ANOVAANOVA

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

Within groups sum of squares/(N-k)Within groups sum of squares/(N-k)

N = total number of subjectsN = total number of subjects

k = number of groupsk = number of groups

Numerator = Mean square between groupsNumerator = Mean square between groups

Denominator = Mean square within groupsDenominator = Mean square within groups

F-table page 195% POINTS FOR THE F DISTRIBUTION Page 1 Numerator Degrees of Freedom * 1 2 3 4 5 6 7 8 9 10 * 1 161 199 216 225 230 234 237 239 241 242 1 2 18.5 19.0 19.2 19.2 19.3 19.3 19.4 19.4 19.4 19.4 2 D 3 10.1 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79 3 e 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96 4 n 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74 5 o m 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06 6 i 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64 7 n 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35 8 a 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14 9 t 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98 10 o r 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85 11 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75 12 D 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67 13 e 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60 14 g 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54 15 r e 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49 16 e 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45 17 s 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41 18 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38 19 o 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35 20 f 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37 2.32 21 F 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30 22 r 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32 2.27 23 e 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25 24 e 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28 2.24 25 d o 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22 26 m 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25 2.20 27 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19 28 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22 2.18 29 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16 30 35 4.12 3.27 2.87 2.64 2.49 2.37 2.29 2.22 2.16 2.11 35 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08 40 50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03 50 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99 60 70 3.98 3.13 2.74 2.50 2.35 2.23 2.14 2.07 2.02 1.97 70 80 3.96 3.11 2.72 2.49 2.33 2.21 2.13 2.06 2.00 1.95 80 100 3.94 3.09 2.70 2.46 2.31 2.19 2.10 2.03 1.97 1.93 100 150 3.90 3.06 2.66 2.43 2.27 2.16 2.07 2.00 1.94 1.89 150 300 3.87 3.03 2.63 2.40 2.24 2.13 2.04 1.97 1.91 1.86 300 1000 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89 1.84 1000

Significance of FSignificance 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 Computer Printout ExampleExample

Descriptives

GAVE 'THE FINGER' TO SOMEONE WHILE DRIVI

1462 1.7148 1.28915 .03372 1.6486 1.7809 1.00 7.00

1858 1.3660 .93491 .02169 1.3234 1.4085 1.00 7.00

3320 1.5196 1.11830 .01941 1.4815 1.5576 1.00 7.00

1.00

2.00

Total

N Mean Std. Deviation Std. Error Lower Bound Upper Bound

95% Confidence Interval forMean

Minimum Maximum

ANOVA

GAVE 'THE FINGER' TO SOMEONE WHILE DRIVI

99.536 1 99.536 81.522 .000

4051.191 3318 1.221

4150.727 3319

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

Two-way ANOVATwo-way ANOVA

ANOVA compares:ANOVA compares: Between and within groups varianceBetween and within groups variance

Adds a second IV to one-way ANOVAAdds a second IV to one-way ANOVA 2 IV and 1 DV2 IV and 1 DV

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

Graphs of potential Graphs of potential outcomesoutcomes

No main effects or interactionsNo main effects or interactions

Main effects of color onlyMain effects of color only

Main effects for motion onlyMain effects for motion only

Main effects for color and motionMain effects for color and motion

InteractionsInteractions

Graphs Graphs

Color B&W

x Motion

* Still

AROUSAL

No main effects for No main effects for interactions interactions

Color B&W

x Motion

* Still

AROUSAL

No main effects for No main effects for interactions interactions

Color B&W

x Motion

* Stillx x* *

AROUSAL

Main effects for color Main effects for color onlyonly

Color B&W

x Motion

* Still

AROUSAL

Main effects for color Main effects for color only only

Color B&W

x Motion

* Still

x

x

*

*

AROUSAL

Main effects for motion Main effects for motion onlyonly

Color B&W

x Motion

* Still

AROUSAL

Main effects for motion Main effects for motion only only

Color B&W

x Motion

* Still

x x

* *

AROUSAL

Main effects for color and Main effects for color and motionmotion

Color B&W

x Motion

* Still

AROUSAL

Main effects for color and Main effects for color and motion motion

Color B&W

x Motion

* Still

x

x*

*

AROUSAL

Transverse interactionTransverse interaction

Color B&W

x Motion

* Still

AROUSAL

Transverse interaction Transverse interaction

Color B&W

x Motion

* Still

x

x*

*

AROUSAL

Interaction—color only Interaction—color only makes a difference for makes a difference for

motionmotion

Color B&W

x Motion

* Still

AROUSAL

Interaction—color only Interaction—color only makes a difference for makes a difference for

motionmotion

Color B&W

x Motion

* Still

x

x**

AROUSAL

Partitioning the variance Partitioning the variance for Two-way ANOVAfor Two-way ANOVA

Total variation = Total variation =

Main effect variable 1 +Main effect variable 1 +

Main effect variable 2 +Main effect variable 2 +

Interaction +Interaction +

Residual (within)Residual (within)

Summary Table for Two-Summary Table for Two-way ANOVAway ANOVA

SourceSource SSSS dfdf MSMS FF

Main effect 1Main effect 1

Main effect 2Main effect 2

InteractionInteraction

WithinWithin

TotalTotal

Printout ExamplePrintout ExampleTests of Between-Subjects Effects

Dependent Variable: MARIJUANA USE SHOULD BE LEGALIZED

74.465a 7 10.638 3.392 .001

5889.077 1 5889.077 1877.565 .000

13.191 1 13.191 4.205 .040

19.048 3 6.349 2.024 .108

.560 3 .187 .060 .981

10366.297 3305 3.137

31942.000 3313

10440.762 3312

SourceCorrected Model

Intercept

SEX

RACE2

SEX * RACE2

Error

Total

Corrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .007 (Adjusted R Squared = .005)a.

Printout plotPrintout plot

Estimated Marginal Means of MARIJUANA USE SHOULD BE LEGALIZED

SEX OF RESPONDENT

2.001.00

Estimated Marginal Means

3.2

3.0

2.8

2.6

2.4

2.2

RACE OF RESPONDENT(W

1.00

2.00

3.00

4.00

Scatter Plot of Price and Scatter Plot of Price and AttendanceAttendance

Price is the average seat price for a single regular season game in today’s Price is the average seat price for a single regular season game in today’s dollarsdollars

Attendance is total annual attendance and is in millions of people per annum.Attendance is total annual attendance and is in millions of people per annum.

2 3 4 5 6Price

0.5

1

1.5

2

2.5

Attendance

Is there a relation Is there a relation there?there?

Lets use linear regression to find out, that Lets use linear regression to find out, that isis Let’s fit a straight line to the data.Let’s fit a straight line to the data. But aren’t there lots of straight lines that But aren’t there lots of straight lines that

could fit?could fit? Yes! Yes!

Desirable PropertiesDesirable Properties

We would like the “closest” line, that is the We would like the “closest” line, that is the one that minimizes the errorone that minimizes the error The idea here is that there is actually a relation, The idea here is that there is actually a relation,

but there is also noise. We would like to make but there is also noise. We would like to make sure the noise (i.e., the deviation from the sure the noise (i.e., the deviation from the postulated straight line) to be as small as postulated straight line) to be as small as possible.possible.

We would like the error (or noise) to be We would like the error (or noise) to be unrelated to the independent variable (in this unrelated to the independent variable (in this case case priceprice).).

If it were, it would not be noise --- right!If it were, it would not be noise --- right!

Scatter Plot of Price and Scatter Plot of Price and AttendanceAttendance

Price is the average seat price for a single regular season game in today’s Price is the average seat price for a single regular season game in today’s dollarsdollars

Attendance is total annual attendance and is in millions of people per annum.Attendance is total annual attendance and is in millions of people per annum.

2 3 4 5 6Price

0.5

1

1.5

2

2.5

Attendance

Simple RegressionSimple Regression

The The simple linear regression MODELsimple linear regression MODEL is: is:

yy = = 00 + + 11xx + +

describes how y is related to xdescribes how y is related to x 00 and and 11 are called are called parameters of the modelparameters of the model.. is a random variable called theis a random variable called the error term error term..

x y

e


Graph of the regression equation is a Graph of the regression equation is a straight line.straight line.

ββ0 is the population is the population y-y-intercept of the intercept of the regression line.regression line.

ββ11 is the population slope of the is the population slope of the regression line.regression line.

EE((yy) is the expected value of ) is the expected value of yy for a for a given given xx value value


EE((yy))

xx

Slope Slope 11

is positiveis positive

Regression lineRegression line

InterceptIntercept00


EE((yy))

xx

Slope Slope 11

is 0is 0

Regression lineRegression lineInterceptIntercept

00

Types of Types of Regression ModelsRegression Models

RegressionModels

LinearNon-

Linear

2+ ExplanatoryVariables

Simple

Non-Linear

Multiple

Linear

1 ExplanatoryVariable

RegressionModels

LinearNon-

Linear

2+ ExplanatoryVariables

Simple

Non-Linear

Multiple

Linear

1 ExplanatoryVariable

Regression Modeling Regression Modeling Steps Steps

1.1. Hypothesize Deterministic ComponentsHypothesize Deterministic Components 2.2. Estimate Unknown Model ParametersEstimate Unknown Model Parameters 3.3. Specify Probability Distribution of Specify Probability Distribution of

Random Error TermRandom Error Term Estimate Standard Deviation of ErrorEstimate Standard Deviation of Error

4.4. Evaluate ModelEvaluate Model 5.5. Use Model for Prediction & Estimation Use Model for Prediction & Estimation

Linear Multiple Linear Multiple Regression ModelRegression Model

1.1. Relationship between 1 dependent & 2 Relationship between 1 dependent & 2 or more independent variables is a linear or more independent variables is a linear functionfunction

Y X X Xi i i k ki i 0 1 1 2 2 Y X X Xi i i k ki i 0 1 1 2 2

Dependent Dependent (response) (response) variablevariable

Independent Independent (explanatory) (explanatory) variablesvariables

Population Population slopesslopes

Population Population Y-interceptY-intercept

Random Random errorerror

X2

Y

X1E(Y) = 0 + 1X 1i + 2X 2i

0

Y i = 0 + 1X 1i + 2X 2i + i

ResponsePlane

(X 1i,X 2i)

(Observed Y )

iX2

Y

X1E(Y) = 0 + 1X 1i + 2X 2i

0

Y i = 0 + 1X 1i + 2X 2i + i

ResponsePlane

(X 1i,X 2i)

(Observed Y )

i

Multiple Regression Multiple Regression ModelModel

Multivariate Multivariate modelmodel

multivariate analysis. one-way anova tests the difference in the means of 2 or more nominal groups...

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