More than one Factor?

Factorial ANOVA!

We can (and often do) conduct experiments or investigations in which there is more than one factor (i.e., independent or quasi-independent variable). ◦ Recall, in a One-way ANOVA, we have…◦ one Independent Variable (i.e., one manipulated by the

experimenter) or ◦ one Quazi-Independent Variable (i.e., a variable that is

accounted for by the experimenter, but is not actually manipulated).

Now, with a Factorial ANOVA, we have ◦ 2 or more independent variables, ◦ 2 or more quasi-independent variables, or ◦ a combination of each.

Factors

Just line a one-way ANOVA, each factor can have 2 or more levels◦ Gender: pre-existing group (quasi-independent

variable) with 2 levels.◦ Video-taped confession: might have 3 levels –

focused on the confessor, focused on the interrogator, or focused on both. Because the experimenter can manipulate this, this is an

independent variable.◦ Message position: might have 2 levels – agreeable

message vs. attitude inconsistent message. Because the experimenter can manipulate this, this is an

independent variable.

Levels

We might want to investigate if participants will help more or less depending on their gender and whether or not they are alone. ◦ Gender is a quasi-independent variable with 2

levels. ◦ The experimenter can manipulate whether the

participant is alone or in a group, so this is an independent variable with 2 levels.

Simplest type of Factorial ANOVA is often called a 2x2 ANOVA. This is a 2x2.

Simple example 1

We might want to investigate if participants will help more or less depending on their mood and whether or not the helping task is enjoyable. ◦ If mood is manipulated by the experimenter (e.g., have

people watch a happy, sad, or neutral video clip), mood is an independent variable with 3 levels.

◦ The experimenter can manipulate whether the helping task is enjoyable (e.g., proofreading a funny article or a dull article), so this is an independent variable with 2 levels.

This experiment has 2 factors, one with 2 levels and one with 3 levels.◦ This is a 2x3 ANOVA

Simple example 2

How many factors can we have? How many levels can we have? When does this become too much for our

brain to handle?◦ 3-4 factors is about as far as we can go before our

brains hemorrhage, so be careful . Example of 3 factors, each with 2 levels.

◦ 2 (Social Status: control vs. excluded) x 2 (Emotion: H vs. S) x 2 (Task: F vs. B).

Science is not always simple

Our factors and levels are INDEPENDENT. # conditions = (# levels in factor 1) x…x (#

levels in last factor). Each of these conditions are independent.

Things to note

Simplest factorial ANOVA is a 2x2 (2 factors each with 2 levels). ◦ Actually, a 2-factor ANOVAs are all simple relative

to 3+ factor ANOVAs.◦ So, lets there.

Start out simple

Set up: Is methamphetamine neuro-protective following ischemic stroke when compared to no-stroke conditions?◦ Neuro-protective is operationalized as low-normal

activity (damage is indicated by more activity). Design: Rats get a surgery that induces

stroke or not, then they receive an injection that contains methamphetamine or saline.◦ 2 (Surgery: stroke v. sham) x 2 (Injection: meth v.

saline)

Full Example 1: 2x2

A one-way ANOVA tests for one set of mean differences. ◦ Tests whether at least one mean differed from

another when we had 1 factor with 2 or more levels

The 2 Factor ANOVA tests 3 separate sets of mean differences.

Recall:◦ DV: Activity level◦ Factor A: Surgery (stroke v. sham)◦ Factor B: Injection (meth v. saline)

What does a 2-Factor ANOVA test?

1) Mean differences in activity level between surgery (stroke v. sham)

2) Mean differences in activity level between injection (meth v. saline)

3) Mean differences in activity level that result from a combination of surgery and injection.

So, 3 tests are combined into one analysis. We will have 3 f-ratios.

The 2 Factor ANOVA tests:

As usual, there will be variance attributable to differences between our groups (conditions) in the numerator of our F-ratio and variance attributable to chance in the denominator.

The primary difference between our 3 f-ratios is what goes into the numerator…◦ the variance due to our first factor, ◦ the variance due to our second factor, or ◦ the variance due to our the combination of our 2

factors.

What do these F-ratios look like?

The mean difference among levels of a factor is called a Main Effect. ◦ In determining whether there is a difference

between stroke and sham groups on activity level, we are testing for a “main effect of surgery.” Regardless of the injection rats received, is there a

difference among the levels in surgery?◦ In determining whether there is a difference

between meth and saline groups on activity level, we are testing for a “main effect of injection.” Regardless of the surgery rats received, is there a

difference among the levels in injection?

Main Effects: Tests of each Factor

For Factor A (Surgery)◦ Ho: µA1 = µA2 OR µstroke = µsham

◦ Ha: µA1 does not = µA2 OR

◦ µstroke does not = µsham

  Conceptually: F = (the variance due to difference between

the means for Surgery)/the variance due to chance

Null and Alternative Hypotheses for Main Effects

For Factor A (Injection)◦ Ho: µB1 = µB2 OR µmeth = µsaline

◦ Ha: µB1 does not = µB2 OR

◦ µmeth does not = µsaline

  Conceptually: F = (the variance due to difference between

the means for Injection)/the variance due to chance

Null and Alternative Hypotheses for Main Effects

We might have overall main effects of our factors, but we can also have variability due to the combination of our factors. ◦ That is, our 2 factors in combination can interact

to produce effects beyond those we see just by looking at main effects. An interaction is defined as – mean differences

between conditions that are different from what would be predicted from the overall main effects of the factors. In other words, observed differences beyond possible main effects.

Interactions

stroke sham

meth 3 5 4

saline 8 5 6.5

5.5 5

More than main effects

stroke sham

meth 3 8 5.5

saline 8 3 5.5

5.5 5.5

Ho: There is no interaction between mood and task. All mean differences are explained by the main effects of mood and task.

Ha: There is an interaction between mood and task. The mean differences between conditions is not what would be predicted from the overall main effects of mood and task.

Conceptually: F = (the variance not explained by main effects)/ the variance due to chance

Null and Alternative Hypotheses for Interactions

If the effect on the DV from one factor does not influence the effect on the DV from second factor, there will be no interaction.

If the effect on the DV from one factor does influence the effect on the DV from second factor another, there is an interaction.

If the effect of one factor on the DV depends on the levels of the other factor: INTERACTION.

Interactions…

We are going to calculate SSt and SSw/e and SSB the same way!

Now, we are going to break up SSb into SSfor factor A

SSfor factor B

And Ssinteraction

We are going to require data now…

Basics about computing a 2-Factor ANOVA

(Stroke v. Sham) X (Meth v Saline)

Actual research participant.

The Data Stroke and Meth (1):

Sum (X1)= 49; mean1 = 4.08; Sum (X12) = 215; n1 = 12

Stroke and Saline(2): Sum (X2)= 94; mean2 = 7.83; Sum (X2

2) = 750; n2 = 12

Sham and Meth (3): Sum (X3)= 46; mean3 = 3.83; Sum (X3

2) = 190; n3 = 12

Sham and Saline (4): Sum (X4)= 40; mean4 = 3.33; Sum (X4

2) = 142; n4 = 12

Overall values (across groups): Sum (Xoverall )= 229; Grand mean = 4.77; Sum (Xoverall

2) = 1297; N = 48; k = ??

…then break it into two components just like before:

SSTOTAL =

= (1297-[2292/48]) = 204.479

SStotal = SSbetween + SSwithin

  Again, dftotal = N – 1 = 48– 1 = 47.

1) Find the total variability

TOT

TOTTOT

N

XX

22 )(

Again, SSwithin = SSerror =

(215 – [492]/12)+(750– [942]/12)+(190– [462]/12)+(142– [402]/12) = 50.917

Again, dfwithin/error = N – k = 48 – 4 = 44

OK, so, SSb = 204.479 – 50.917 = 153.562

2) Find the within group variability

k

kk n

XX

n

XX

n

XX

22

2

222

21

212

1

)((...)

)(()

)((

Again, SSbetween/group =

([492]/12)+([942]/12)+([462]/12)+([402]/12)-([2292]/48) = 153.562!!!

Again, dfbetween/group = k-1 =4-1 = 3

OK, so, NOW we need to do something NEW!!◦ Break up SSb =into its constituent parts.

3) Find the between-group variability

TOT

TOT

k

k

N

X

n

X

n

X

n

X 22

2

22

1

21 )()(

...)()(

SSfactor A (Surgery) =

([49+94]2/24)+([46+40]2/42)-([2292]/48) = 67.687

df factor A (Surgery) = (Levels in factor A) – 1 = 1

4) Find the between-group variability for Factor A (Surgery).

TOT

TOT

aa N

X

n

Xa

n

Xa 2

2

22

1

21 )()()(

SSfactor B (Injection) =

([49+46]2/24)+([94+40]2/42)-([2292]/48) = 31.687

df factor B (Injection) = (Levels in factor B) – 1 = 1

5) Find the between-group variability for Factor B (Injection).

TOT

TOT

bb N

X

n

Xb

n

Xb 2

2

22

1

21 )()()(

SSinteraction = SSB–(SSfactor A + SSfactor B )= 54.188

([49+40]2/24)+([46+94]2/42)-([2292]/48) = 54.18

Similarly, dfA x B interaction = dfbetween - dffactor

A - dffactor B

◦ dfA x B interaction = 3 – 1 – 1 = 1

6) Find the between-group variability for the Interaction.

TOT

TOT

baba N

X

n

XbXa

n

XbXa 2

12

212

21

221 )()()(

MSfactor A = SSfactor A/ dffactor A= 67.687 MSfactor B = SSfactor B/ dffactor B = 31.687 MSA x B interaction = SSA x B interaction / dfA x B

interaction= 54.188

MSwithin = SSwithin / dfwithin = 50.917/447 =1.157

7&8) Find the MS for your 2 main effects, interaction, and MSw.

Ffactor A= MSfactor A/ MSwithin = 67.687/1.157 = 58.493

Ffactor B = MSfactor B/ MSwithin = 31.687/1.157 = 27.383

FA x B interaction = MSA x B interaction / MSwithin = 54.188/1.157 = 46.827

Weeeeeeeee! Let’s check out SPSS

9) Compute Fs!!!

Condition

Main Effect ofSurgery

Main Effect of Injection

Interaction

Surgery Injection

1 Stroke Meth 1 -1 -1

2 Stroke Saline 1 1 1

3 Sham Meth -1 -1 1

4 Sham Saline -1 1 -1

Looking at Contrasts: What are the Fs testing?

Stroke Sham0

20

40

60

80

100

MethSaline

This is the same experiment, same data, same everything. But, I have recorded Gerbil gender as a quazi-independent variable.

What is our design now?

2x2x2 = gender x surgery x injection 3 factors:

◦ A = Gender◦ B = Surgery◦ C = Injection

OK, lets add a Factor…Gender!This also Applies to adding levels.

Calculations of:◦ SStot, SSbetween/group, SSwithin/error are all the same.

What is different?◦ We must break up SSbetween/group into more parts.

◦ What are they?

What is the same?

Female

X1 X12 Mea

nn Male X1 X1

2 Mean

n

StrokeMeth (1)

24 106 4 6 StrokeMeth (5)

25 109 4.17 6

StrokeSaline (2)

51 435 8.5 6 StrokeSaline (6)

43 315 7.17 6

Sham Meth(3)

23 93 3.83 6 Sham Meth(7)

23 97 3.83 6

Sham Saline(4)

20 74 3.33 6 Sham Saline(8)

20 68 3.33 6

Data broken up by Gender

Main effects◦ SSsurgery, SSinjection like before, but also…

◦ Ssgender now because we have an additional factor. ◦ Ok, that covers the Main effects. What else is there?

2-way Interactions (involving just 2 factors).◦ SSsxi (as before), but also: SSgxs and SSgxi

New: 3-way interaction (all 3 factors) SSgxsxi

Breaking up SSbetween/group

…then break it into two components just like before:

SSTOTAL =

= (1297-[2292/48]) = 204.479

SStotal = SSbetween + SSwithin

  Again, dftotal = N – 1 = 48– 1 = 47.

1) Find the total variability

TOT

TOTTOT

N

XX

22 )(

Again, SSwithin = SSerror =

But, now we have 8 (not 4) groups◦ (106– [242]/6)+(435– [512]/6)+(93– [232]/6)+(74– [202]/6)+

(109– [252]/6)+(315– [432]/6)+(97– [232]/6)+(68– [202]/6) = 45.5

◦ This is a different number than we had before; why?

Again, dfwithin/error = N – k = 48 – 8 = 40

OK, so, SSb = 204.479 – 45.5 = 158.979

2) Find the within group variability

k

kk n

XX

n

XX

n

XX

22

2

222

21

212

1

)((...)

)(()

)((

Again, SSbetween/group =

Again, we have 8 groups now, so…◦ ([242]/6)+([512]/6)+([232]/6)+([212]/6)+([252]/6)+

([432]/6)+([232]/6)+([202]/6)-([2292]/48) = 158.9799!!!

Again, dfbetween/group = k-1 =8-1 = 7

OK, so, NOW we need to…◦ Break up SSb =into its constituent parts.

3) Find the between-group variability

TOT

TOT

k

k

N

X

n

X

n

X

n

X 22

2

22

1

21 )()(

...)()(

SSfactor A (Gender) =

([24+51+23+20]2/24)+([25+43+23+20]2/24)-([2292]/48) = 1.021

df factor A (gender) = (Levels in factor A) – 1 = 1

4) Find the between-group variability for Factor A (GENDER).

TOT

TOT

aa N

X

n

Xa

n

Xa 2

2

22

1

21 )()()(

SSfactor B (Surgery) =

([24+51+25+43]2/24)+([23+20+23+20]2/24)-([2292]/48) = 67.687

df factor B (Surgery) = (Levels in factor B) – 1 = 1

5) Find the between-group variability for Factor B (Surgery).

TOT

TOT

bb N

X

n

Xb

n

Xb 2

2

22

1

21 )()()(

SSfactor C (Injection) =

([24+23+25+23]2/24)+([51+20+43+20]2/42)-([2292]/48) = 31.687

df factor C (Injection) = (Levels in factor C) – 1 = 1

6) Find the between-group variability for Factor C (Injection).

TOT

TOT

cc N

X

n

Xc

n

Xc 2

2

22

1

21 )()()(

SSaxb = SSAB–(SSfactor A + SSfactor B )◦ SSAB is looking at 4 groups collapsing across Injection.

SSAB =

([24+51]2/12)+([23+20]2/12)+([25+43]2/12)+([23+20]2/12)-([2292]/48) = 69.72◦ So, SSaxb = 69.72- (1.021+67.688) = 1.021

dfA x B interaction = (dffactor A )( dffactor B)◦ dfA x B interaction = 1x1= 1

7a) Find the between-group variability for the Interactions.

TOT

TOT

bababa N

X

n

bXa

n

bXa

n

bXa

bna

bXa 22

222

122

21

21

211 )()()()()(

221221

SSaxc = SSAC–(SSfactor A + SSfactor C ) SSAC is looking at 4 groups collapsing across Surgery.

SSAC =

dfA x C interaction = (dffactor A )( dffactor C)◦ dfA x C interaction = 1x1= 1

7b) Find the between-group variability for the Interactions.

TOT

TOT

cacaca N

X

n

cXa

n

cXa

n

cXa

cna

cXa 22

222

122

21

21

211 )()()()()(

221221

SSbxc = SSBC–(SSfactor B + SSfactor C ) SSBC is looking at 4 groups collapsing across Gender.

SSBC =

dfB x C interaction = (dffactor B )( dffactor C)◦ dfB x C interaction = 1x1= 1

7c) Find the between-group variability for the Interactions.

TOT

TOT

cbcbcb N

X

n

cXb

n

cXb

n

cXb

cnb

cXb 22

222

122

21

21

211 )()()()()(

221221

SSaxbxc = SSBetween – (SSaxb + SSaxc + SSbxc + SSfactor A + SSfactor B + SSfactor C )

dfA x B x C interaction = (dffactor A )(dffactor B )( dffactor C)

◦ dfAxBxC interaction = 1x1x1= 1

8) Find the b-g variability for the 3-way Interaction.

TOT

TOT

cbcbcb N

X

n

cXb

n

cXb

n

cXb

cnb

cXb 22

222

122

21

21

211 )()()()()(

221221

Main effects◦ MSfactor A = SSfactor A/ dffactor A

◦ MSfactor B = SSfactor B/ dffactor B

◦ MSfactor C = SSfactor C/ dffactor C

2-way Interactions◦ MSA x B interaction = SSA x B interaction / dfA x B interaction

◦ MSA x C interaction = SSA x C interaction / dfA x C interaction

◦ MSB x C interaction = SSB x C interaction / dfB x C interaction

3-way Interaction MSAxBx C interaction = SSAxBx C interaction / dfAxBx C interaction

MSwithin = SSwithin / dfwithin

9) Find the MSs

F = MSb/MSw for each Main effect and interaction.

Let’s check out SPSS

10) 7 Compute Fs!!!

The Surgery x Injection interaction is significant. Sweet. Now what?

We can “decompose this interaction in several ways to determine which means are different from which within that interaction.

Post-hoc tests: like Tukey and Sheffe, etc.◦ Generally, these are ok for exploratory purposes.

Simple effects tests Planned Comparisons/contrasts

Waa hoo! We have Interactions

One way to do this is by doing a “Simple Effect” test. This looks at the main effect of one factor at each level of the other factor.◦ These are nice because they help control Type 1

error.◦ Because the overall MSWE is in the denominator.

What does the equation look like? Any F really. Let’s break down the SxI

interaction…

Simple Effects Tests

Simple Effect of Injection within Stroke Conditions.

)(

2)(

)(2

2)(2

)(1

2)(1 )()()(

strokeTOT

strokeTOT

strokec

stroke

strokec

stroke

N

X

n

Xc

n

Xc SSfactor C/Injection (stroke only) =

([24+25]2/12)+([51+43]2/12)-([1432]/24) = 200.08+736.33-852.04 = 84.375

df factor C/Injection (stroke only) = (Levels in factor C) – 1 = 1

MSfactor C/Injection (stroke only) = SSfactor C/Injection

(stroke only) /Dffactor C/Injection (stroke only)

Ffactor C/Injection (stroke only) =MSfactor C/Injection

(stroke only) /MSW/E = 84.375/1.137 = 74.176

The factorial ANOVA is not necessarily testing the interaction patterns you are predicting.

You can test for specific, predicted, interaction patterns even if the ANOVA says an interaction is not significant.

What is it testing (at least with a 2x2x2)?

Lets think about Contrasts

Cond(F)

MEGen

ME Surg

ME Inj

GxSInt

GxIInt

SxIInt

GxSxIInt

Surg

Inj

1 Str Meth 1 1 -1 1 -1 -1 -1

2 Str Sal 1 1 1 1 1 1 1

3 Sh Meth 1 -1 -1 -1 -1 1 1

4 Sh Sal 1 -1 1 -1 1 -1 -1

Looking at Contrasts: What are the Fs testing (in a 2x2x2)?

Cond(M)

MEGen

ME Surg

ME Inj

GxSInt

GxIInt

SxIInt

GxSxIInt

Surg

Inj

1 Str Meth -1 1 -1 -1 1 -1 1

2 Str Sal -1 1 1 -1 -1 1 -1

3 Sh Meth -1 -1 -1 1 1 1 -1

4 Sh Sal -1 -1 1 1 -1 -1 1

Each contrast gives us a t-value testing for a main effect or an interaction.◦ Each t2 corresponds to an F testing for that same

thing. The if we sum all the and divide by 7 (i.e.,

take an average), we get the omnibus F-value testing for an overall difference among all the conditions.

That is, we get the F for MSbetween/MSwithin/error

Check it!

About these contrasts

The omnibus interaction is testing for two opposite crossover interactions. What if you predicted something different?

1 crossover and 1 no interaction 1 fan and one opposite fan Etc.

You predicted a different 3-way

Stroke Sham0

1

2

3

4

5

6

7

8

9

MethSaline

Females Males

Stroke Sham0

1

2

3

4

5

6

7

8

MethSaline

Cond(F)

Surg

Inj

1 Str Meth -1 0 -1

2 Str Sal 1 0 1

3 Sh Meth 1 -1 0

4 Sh Sal -1 1 0

Other Contrasts

Cond(M)

Surg

Inj

1 Str Meth 0 -1 0

2 Str Sal 0 1 0

3 Sh Meth 0 0 -1

4 Sh Sal 0 0 1

We can use d or g in comparing two means.◦ Mean difference/Grand SD

For interactions, we can use partial eta squared or Omega squared.

Effect Sizes

Omega Squared is better