Chapter 17 Two-way Anova

8/6/2019 Chapter 17 Two-way Anova

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Chapter 17: Two-way ANOVA

In the lecture on one-way ANOVA, we discussed the following experiment:

Observe an emergency alone

Observe an emergency w/ 1 other

Observe an emergency w/ 2 others

P + 2 people P + 1

person

P + 0 people

X 2= 9 X 1= 8 X 0= 3

In this experiment, we have ONE IV: the # of bystanders present

What if we wanted to look at other IVs too?

Chapter 17: Page 1

IV = Number of people present

DV = Time it takes (in secs.) to call for help


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EX: Would the amount of time people wait to help be affected by

the gender of the person in need of help?

Chapter 17: Page 2


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Lets do an experiment where we stage an emergency situation:

IV1: The victim of the emergency is either male OR female

IV2: The P witnesses the emergency alone, w/ 1 other, OR 2 others

These two IVs are crossed, meaning that each level of one IV ispaired with all levels of the other IV

Put another way, we have all possible combinations of the IVs

P + 2 people P + 1

person

P + 0 people

Male victim X = 9 X = 8 X = 3

Female victim X = 4 X = 4 X = 2

Chapter 17: Page 3


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Chapter 17: Page 4


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VOCABULARY

Factor = Independent variable

Two-factor ANOVA / Two-way ANOVA: an experiment with 2

independent variables

Levels: number of treatment conditions (groups) for a specific IVNOTATION

3 X 2 factorial = experiment w/ 2 IVs: one w/ 3 levels, one w/ 2

levels

2 X 2 factorial = experiment w/ 2 IVs: both w/ 2 levels

3 X 2 X 2 = ????

Chapter 17: Page 5


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Why do a two-factor (two-way) ANOVA?

1. Greatergeneralizability of results

--EX: If experiment is only done with a male victim, we dont

know if the results are also true for female victims

2. Allows one to look forinteractions--The effect of one IV depends on the level of the other IV

--EX: Sample of patients who have an infection:

get antibiotics and are not allergic

dont get antibiotics and are not allergic get antibiotics and are allergic

dont get antibiotics and are allergic

--Measure how well the patients feel the next day

Chapter 17: Page 6


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3

5

3

1

0

1

2

3

4

5

6

No Yes

Receive Anti-biotic

How

doyoufeelrightnow?

No

Yes

This illustrates an interaction

Chapter 17: Page 7

Allergic?


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3

5

3

5

0

1

2

3

4

5

6

No Yes

Receive anti-biotic

How

doyoufeelrightnow?

No

Yes

If there were no interaction, then the graph would have looked

like this

Chapter 17: Page 8

Allergic?


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Lets return to our bystander intervention experiment:

P + 2 people P + 1

person

P + 0 people

Male victim X = 9 X = 8 X = 3

Female victim X = 4 X = 4 X = 2

When we do a two-way ANOVA, we will obtain three different

statistical tests:

1. Main effect of IV1: Gender of Victim

2. Main effect of IV2: # of Bystanders Present

3. Interaction b/n the two IVs (gender & # of bystanders)

Chapter 17: Page 9


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Each is a hypothesis test:

Gender Main Effect:

H0: all levels of gender have the same mean

H1: all levels of gender do not have the same mean

Bystander Main Effect:H0: all levels of bystander have the same mean

H1: all levels of bystander do not have the same mean

Interaction:

H0: there is no interaction between the factorsH1: there is an interaction between the factors

Chapter 17: Page 10


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Main Effects

Defined: The effect of ONE IV on the DV averaged across the levels of theother IV

In our example:

--Main effect of gender: Is there a difference in response time for male

versus female victims, averaging over the number of bystanders present?

That is: Ignoring the number of bystanders, does response time differ for

male versus female victims?

--Main effect of # of bystanders: Is there a difference in response time when

there is 1 versus 2 versus 3 bystanders present, averaging over the victimsgender?

That is: Ignoring the gender of the victim, does response time differ based

on the number of bystanders present?

Chapter 17: Page 11


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One way to understand main effects is to examine something

called marginal means

P + 2 people P + 1 person P + 0 people Marginal

Means

Male victim

X = 9 X = 8 X = 3 X = 6.67Female victim

X = 4 X = 4 X = 2 X = 3.33

Marginal MeansX = 6.5 X = 6 X = 2.5

Chapter 17: Page 12

Response time

for 3bystanders,

averaging over

gender

(9 + 4) / 2 = 6.5

Response time

for 2bystanders,

averaging over

gender

(8 + 4) / 2 = 6

Response timefor 1 bystander,

averaging over

gender

(3 + 2) / 2 = 2.5

Response time for

male victims,

averaging over # of

bystanders

(9 + 8 + 3) / 3 =

6.67

Response time for

female victims,

averaging over # of

bystanders

(4 + 4 + 2) / 3 =

3.33


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A main effect of gender asks:

Do the marginal means of 6.67 (male victims) and 3.33 (female

victims) differ?

A main effect of # of bystanders asks:

Do the marginal means of 6.5 (3 bystanders), 6 (2 bystanders), &

2.5 (1 bystander) differ?

Chapter 17: Page 13


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Calculations

To perform all the calculations for Sums of Squares for a two-way

anova by hand is time and labor intensive

These are almost exclusively done using the aid of a statistical

package, like SPSS

Thus, Ill just explain the calculations conceptually

Chapter 17: Page 14


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Calculations: Main Effects Sums of Squares

Calculations for main effects SS in a two-way ANOVA are very similar to thecalculations we used in one-way ANOVA

Conceptually, to calculate the SS for a main effect, one is comparing each

marginal mean to the overall (grand) mean

P + 2 people P + 1 person P + 0 people Marginal Means

Male victimX = 9 X = 8 X = 3 X = 6.67

Female victimX = 4 X = 4 X = 2 X = 3.33

Marginal MeansX = 6.5 X = 6 X = 2.5

Overall: 5

For a main effect of # of bystanders, one is taking the squared difference b/n:

The mean from 3 bystanders & the overall mean (6.5 5)The mean from 2 bystanders & the overall mean (6-5)

The mean from 1 bystander & the overall mean (2.5 5)

These squared differences are multiplied by the # of scores per mean

Chapter 17: Page 15


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For a main effect of gender, one is taking the squared difference b/n:

The mean from male victims & the overall mean (6.67 5)The mean from female victims & the overall mean (3.33-5)

These squared differences are multiplied by the # of scores per mean

Calculations: Sums of SquaresTotal

The Total SS is calculated by summing the squared deviations

between each score and the overall grand mean

Chapter 17: Page 16


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Calculations: Sums of SquaresCells

The SS Cells is calculated by summing the squared deviations

between each cell mean and the overall grand mean

Each deviation is weighted by the number of observations in that

cell

Why would we want this?

Cell means could differ b/c:

1. They come from different levels of gender

2. They come from different levels of # of bystanders

3. There is an interaction b/n gender & # of bystanders

Chapter 17: Page 17


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SS cells is made up of 3 parts:

SS gender

SS # of bystanders

SS interaction

We can calculate SS gender & SS # of bystanders directly

Then, to find SS interaction:

SS interaction = SS cells SS gender SS # of bystanders

Chapter 17: Page 18


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Calculations: Sums of SquaresError

The SS error is whats left over

Of the total SS, we know what is due to gender (SS gender), what

is due to the # of bystanders (SS bystanders) and what is due

to the interaction (SS interaction). Thus:

SSerror= SStotal (SSgender + SSbystanders + SSinteraction)

Chapter 17: Page 19


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ANOVA Table

Source df SS MS FGender

# of Bystanders

Interaction (G*B)

ErrorTotal

Chapter 17: Page 20


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Calculations: Degrees of Freedom

dftotal = N 1

(where N is the total sample size of the entire experiment)

dfgender = k 1 (where k is the # of gender levels)

dfbystanders = k 1 (where k is the # of bystander levels)

dfinteraction = dfgender * dfbystanders

(Product of the df for the two IVs)

dferror= dftotal - dfgender - dfbystanders - dfinteraction

Chapter 17: Page 21


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Calculations: Mean Squares

To find any MS, take the SS & divide by its corresponding df

MSGender = GenderGender

df

SS

MSBystanders = dersBysdersBys

df

SS

tan

tan

MSInteraction = nInteractionInteractio

df

SS

MSError =Error

Error

df

SS

Chapter 17: Page 22


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Calculations: F statistics

To find the 3 F statistics for our tests, take the MS for the 2 IVs &the MS for the interaction & divide them by the MSError

FGender =Error

Gender

MS

MS

FBystanders =Error

dersBys

MS

MStan

FInteraction= ErrornInteractio

MS

MS

Chapter 17: Page 23


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Calculations: Critical Values

To determine if a particular F statistic is statistically significant,

you obtain the appropriate critical value from Table E.3 or E.4

For each F, you have two df: one for the corresponding factor and

the dfErrorAs usual, if the obtained F value equals or exceeds the critical

value, then we reject the null hypothesis and conclude that we

have a statistically significant effect

Chapter 17: Page 24


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Tests of Between-Subjects Effects

Dependent Variable: How long do P wait to help?

200.000a 5 40.000 17.778 .000

750.000 1 750.000 333.333 .000

83.333 1 83.333 37.037 .000

95.000 2 47.500 21.111 .000

21.667 2 10.833 4.815 .017

54.000 24 2.250

1004.000 30254.000 29

SourceCorrected Model

Intercept

GENDER

BYSTAND

GENDER * BYSTAND

Error

TotalCorrected Total

Type III Sumof Squares df Mean Square F Sig.

R Squared = .787 (Adjusted R Squared = .743)a.

Interpreting SPSS Output

Chapter 17: Page 25

Between-Subjects Factors

15

15

10

10

10

f

m

Gender of

Victim

1.00

2.00

3.00

# of bystanders

present

N

Main

effect of

gender

Main effect

of

bystanders

Interaction

between

gender &

bystanders


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Estimated Marginal Means

. Gender of Victim1


.3333 .387 .2534 .4133

.6667 .387 .5867 .7466

Gender of Victim

f

m

Mean Std. Error Lower Bound Upper Bound

% Confidence Interval95

. # of bystanders present2


.2500 .474 .1521 .3479

.6000 .474 .5021 .6979

.6500 .474 .5521 .7479

# of bystanders present

.100

.200

.300



Chapter 17: Page 26

There was a

main effect

of gender, sothese means

differ

There was a

main effect

of bystander,

so there is a

difference

among these

means


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. Gender of Victim * # of bystanders present3


.2000 .671 .615 .3385

.4000 .671 .2615 .5385

.4000 .671 .2615 .5385

.3000 .671 .1615 .4385

.8000 .671 .6615 .9385

.9000 .671 .7615 .10385


.100

.200

.300

.100

.200

.300

Gender of Victim

f

m



Chapter 17: Page 27

There was a significant

interaction, so the

effect of bystander

depends on the level of

gender


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Multiple-comparison procedures: Post-hoc tests VS simple effects

Understanding Main Effects

If there is a significant main effect, & that factor has only two levels, you knowthat those two marginal means differ significantly from each other

If there is a significant main effect, & that factor has 3 or more levels, youknow that at least two of those marginal means differ. Need multiple-comparison procedures (post-hocs) to determine which ones.

Can be obtained in SPSS

Chapter 17: Page 28


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Post Hoc Tests: # of bystanders presentMultiple Comparisons


LSD

- .35000* .6708 .000 - .48845 - .21155

- .40000* .6708 .000 - .53845 - .26155

.35000* .6708 .000 .21155 .48845

-.5000 .6708 .463 - .18845 .8845.40000* .6708 .000 .26155 .53845

.5000 .6708 .463 -.8845 .18845

(J) # of bystanders

present

.200

.300

.100

.300

.100

.200

(I) # of bystanders

present

.100

.200

.300

Mean

Difference

(I-J) Std. Error Sig. Lower Bound Upper Bound


Based on observed means.

The mean difference is significant at the . level.05*.

Chapter 17: Page 29

1 bystander

differs from2

1 bystander

differs from

3

2 bystanders

DO NOT

differ from 3


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Multiple-comparison procedures: Post-hoc tests VS simple effects

Understanding Interactions

An interaction occurs when the effect of one IV depends on the level of the

other IV

If you obtain a significant interaction, you may want to examine it closely to seewhat is causing it

A useful first step is to graph the means to see the pattern of the interaction

Can be obtained in SPSS or done by hand

Chapter 17: Page 30


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You could graph the means two different ways:

Estimated Marginal Means of How long do P wait to help?

Gender of Victim

mf

EstimatedMargin

alMeans

10

8

6

4

2

0

# of bystanders pres

.100

.200

.300

Graphed this way, we see that the bystander effect seems smaller for female

victims compared to male victims

Chapter 17: Page 31


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Estimated Marginal Means of How long do P wait to hel


.300.200.100

EstimatedMarginalMeans

10

8

6

4

2

0

Gender of Victim

f

m

Graphed this way, we see that male and female victims get helped about equally

quickly with one bystander present, but when multiple bystanders arepresent, a gender difference emerges

Chapter 17: Page 32


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Understanding Interactions

Another thing to do to understand an interaction is to calculate simple effects

The effect of one IV at one level of another IV

Can be done in SPSS using Syntax

Chapter 17: Page 33


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Simple effects of gender at each level of bystander:

Univariate Tests


.2500 1 .2500 .1111 .302

.54000 24 .2250

.40000 1 .40000 .17778 .000

.54000 24 .2250

.62500 1 .62500 .27778 .000

.54000 24 .2250

Contrast

Error

Contrast

Error

Contrast

Error


.100

.200

.300

Sum of

Squares df Mean Square F Sig.

Each F tests the simple effects of Gender of Victim within each level combination of the other effects

shown. These tests are based on the linearly independent pairwise comparisons among the estimated

marginal means.

Chapter 17: Page 34

There is a difference

between male and femalevictims in response time

when 2 or 3 bystanders

are present

There is a difference

between male and femalevictims in response time

when 2 or 3 bystanders

are present

There is no difference in

response time for male and

female victims when only

1 bystander is present


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Simple effects of bystander at each level of gender:

Univariate Tests


.13333 2 .6667 .2963 .071

.54000 24 .2250

.103333 2 .51667 .22963 .000

.54000 24 .2250

Contrast

Error

Contrast

Error

Gender of Victim

f

m

Sum of

Squares df Mean Square F Sig.

Each F tests the simple effects of # of bystanders present within each level combination of theother effects shown. These tests are based on the linearly independent pairwise comparisons

among the estimated marginal means.

Chapter 17: Page 35

There is a difference amongthe levels of bystander for

male victims. There is no

such difference for female

victims

There is a difference amongthe levels of bystander for

male victims. There is no

such difference for female

victims


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Interpretation

Reporting the results of a two-way anova is complex because there are so manytests conducted. Below is one way you might report the results of the

above analyses:

Chapter 17: Page 36


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An experiment was conducted to determine if the number of bystanders present

in an emergency situation and the gender of the victim in an emergency

situation affect the time it takes a person to help. A two-way ANOVAfound a main effect of gender,F(1,24)=37.037,p .05, indicating thatfemale victims are helped sooner than male victims. There was also a

main effect of the number of bystanders present,F(2,24)=21.111,p .05.This effect showed that a lone bystander helped much sooner than when

there were 2 or 3 bystanders present. Finally, there was an interaction

between gender and the number of bystanders present,F(2,24)=4.815,p .05. Simple effects tests showed that male and female victims receive

help equally quickly when only 1 bystander is present,F(1,24)=1.11,p> .05. However, when 2 or 3 bystanders are present, female victims were

helped more quickly than male victims,F(1,24)=17.778,p .05 for 2bystanders andF(1,24)=27.778,p .05 for 3 bystanders.

Chapter 17: Page 37


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Filling in an ANOVA Table

You should be able to complete a partially filled in ANOVA table

Lets suppose we did an experiment where we investigate the effectiveness of

advertisements. We manipulate:

IV1: Whether the ad has a celebrity spokesperson or a normal person

IV2: Whether the ad is in black/white or color

DV: How persuasive do Ps find the ad on a scale where 1 = Not at all persuaded;

7 = Extremely persuaded

There are 20 participants per cell

Chapter 17: Page 38


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ANOVA Table

Source df SS MS FSpokesperson 200

Color 150

Interaction (S*C) 50

ErrorTotal 850

We should be able to complete everything else!

Calculate df

Spokesperson has 2 levels df = (k 1) = (2 1) = 1Color has 2 levels df = (k 1) = (2 1) = 1

Interaction df is the product of the df for the 2 IVs = 1*1 = 1

Total df = (N 1) = (80-1) = 79

Error df = whats left over (79 1 1 1) = 76

Chapter 17: Page 39


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Source df SS MS FSpokesperson 1 200

Color 1 150

Interaction (S*C) 1 50

Error 76

Total 79 850

The SS for all factors and interactions should add up to the totalSS. Thus:

SSerror= SStotal SSspokesperson SScolor- SSinteraction

SSerror= 850 200 150 50 = 450

Chapter 17: Page 40


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Source df SS MS FSpokesperson 1 200

Color 1 150

Interaction (S*C) 1 50

Error 76 450

Total 79 850

MSSpokesperson = 2001200

==

onspokespers

onspokespers

df

SSMScolor= 1501

150==

color

color

dfSS

MSInteraction = 50150

==

nInteractio

nInteractio

df

SS

MSError = 92.576450

==

Error

Error

df

SS

Chapter 17: Page 41


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Source df SS MS FSpokesperson 1 200 200

Color 1 150 150

Interaction (S*C) 1 50 50

Error 76 450 5.92

Total 79 850

FSpokesperson = 7.3392.5200

==

Error

onSpokespers

MS

MSFColor = 34.2592.5

150==Error

Color

MSMS

FInteraction = 45.892.550

==

Error

nInteractio

MS

MS

Chapter 17: Page 42


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Source df SS MS FSpokesperson 1 200 200 33.78

Color 1 150 150 25.34

Interaction (S*C) 1 50 50 8.45

Error 76 450 5.92

Total 79 850

Finally, wed compare each F to the appropriate critical F value

All tests in this example have 1,76 df. Assuming = .05, from

Table E.3, well use the df that are closest to but smaller than

the actual df, b/c they are not listed

Thus, we will use the df for (1,60) = 4.00

All Fs exceed the critical value

Chapter 17: Page 43

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Chapter 17 Two-way Anova