Two-Way ANOVA Overview & SPSS interpretation

Two independent variables

• Often, we wish to study 2 (or more) factors in a single experiment– Compare two or more treatment protocols

– Compare scores of people who are young, middle-aged, and elderly

• The baseline experiment will therefore have twofactors as Independent Variables– Treatment type

– Age Group

Factorial (Two or more way) ANOVA

• One dependent variable

interval or ratio with a normal distribution

• Two independent variables

nominal (define groups), and independent of each other

• Three hypothesis tests:

Test effect of each independent variable controlling for the effects of the other independent variable

One: H0: Treatment type has no impact on Outcome

Two: H0: Age Group has no impact on Outcome

Three: Test interaction effect for combinations of categoriesH0: Treatment and Age Group interact in affecting Outcome

First stage

• Identical to independent samples ANOVA

• Compute SSTotal, SSBetween treatments and

SSWithin treatments

Second stage

• Partition the SSBetween treatments into three

separate components, differences attributable

to Factor A, to Factor B, and to the AxB

interaction

The validity of the ANOVA presented in this chapter depends on three assumptions common to other hypothesis tests

1. The observations within each sample must be independent of each other

2. The populations from which the samples are selected must be normally distributed

3. The populations from which the samples are selected must have equal variances (homogeneity of variance)

Total Variability

Between Treatments

Factor A

Factor B Interaction

Within Treatments

N

GXSStotal

22

treatment each insidetreatmentswithin SSSS

N

G

n

TSS treatmentsbetween

22

Factorial designs

• Consider more than one factor

• Joint impact of factors is considered.

Three hypotheses tested by three F-ratios

• Each tested with same basic F-ratio structure

effect treatment no withexpected es)(differenc variance

treatments between es)(differenc varianceF

• Factor 1 (independent variable, e.g. type of crop)

• Always nominal or ordinal (it defines distinct groups)

• Factor 2 (independent variable, e.g., fertilizer)

• Always nominal or ordinal (it defines distinct groups)

• Outcome (dependent variable, e.g. yield)

• Always interval or ratio

• Mean Outcomes of the groups defined by Factor 1 and Factor 2 are being compared.

Mean differences among levels of one factor

• Differences are tested for statistical significance

• Each factor is evaluated independently of the

other factor(s) in the study

21

21

:

:

1

0

AA

AA

H

H

21

21

:

:

1

0

BB

BB

H

H

Not the same as experimental control.

Statistical control: we look for the effect of one independent variable within each group of the other dependent variable.

This removes the impact of the other independent variable.

Sometimes a variable which showed nosignificant effect in a Oneway ANOVA becomes significant if another effect is controlled.

The mean differences between individuals

treatment conditions, or cells, are different

from what would be predicted from the

overall main effects of the factors

H0: There is no interaction between

Factors A and B

H1: There is an interaction between

Factors A and B

• First:• Does Factor 1 have any

impact on the Outcome?

• Null: The groups defined by Factor 1 will have the same Mean Outcome.

• Second:• Does Factor 2 have any impact on the Outcome?

• Null: The groups defined by Factor 2 will have the same Mean Outcome.

• Third:• Do Factor 1 and Factor 2 interact in influencing

Outcome?

• Null: No combination of Factor 1 and Factor 2 produces unusually high or unusually low mean Outcome scores.

The equations come later!

From one-way to two-way designs:

• Often, we wish to study 2 (or more) factors in a single experiment– Compare a new and standard style of noise filter (inside

a muffler) on a car

– The size of the car might also be an important factor in noise level.

• The baseline experiment will therefore have twofactors as Independent Variables– Type of noise filter (Octel vs Standard)

– Size of car (Small, Midsize, Large)

Standard Filter Octel Filter

Type of Noise Filter

760

770

780

790

800

810

820

830

840

850

860

No

ise L

eve

l R

ea

din

g

Group Statistics

18 815.56 32.217

18 804.72 25.637

Type of Noise Filter

Standard Filter

Octel Filter

Noise Level Reading

N Mean Std. Deviation

First Variable: Filter Type

• Nominal – Dichotomy

• Dependent variable isnoise level (ratio level)

Test: Two-Sample t

• Compare means (above)

• View boxplot (at right)

• t (34)=1.116, p = .272

RETAIN H0

Type of filter does not cause a significant difference in noise.

Second Variable: Car Size

• Nominal – 3 groups

• Dep.Var: noise level (ratio)

Test: Oneway ANOVA

• Compare means (above)

• View boxplot (at left)

• F (2,33) =112.44, p < .0005

REJECT H0

Size of car is related to a significant difference in noise.

Noise Level Reading

12 824.17 7.638

12 833.75 13.505

12 772.50 10.335

36 810.14 29.216

Small

Mid-Size

Large

Total

N Mean Std. Dev'n

Small Mid-Size Large

Size of Car

760

780

800

820

840

860

No

ise L

evel R

ead

ing

Multiple Comparisons

Dependent Variable: Noise Level Reading

Tukey HSD

-9.583 4.394 .089

51.667* 4.394 .000

9.583 4.394 .089

61.250* 4.394 .000

-51.667* 4.394 .000

-61.250* 4.394 .000

(J) Size of Car

Mid-Size

Large

Small

Large

Small

Mid-Size

(I) Size of Car

Small

Mid-Size

Large

Mean

Difference

(I-J) Std. Error Sig.

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

ANOVA is significant, so we need Post-hoc Tests.

Groups: Same Size so Test: Tukey HSD- Small vs Large = Sig.

- Midsize vs Large = Sig.

- Small vs Midsize = n.s.

Filters – Octel vs Standard• Independent sample t-test

• No significant differences

Size of Car – Small, Midsize, Large• ANOVA

• Significant differences

• Large cars are significantly more quiet

BUT – is it possible that the Octel filter might work better with just one of the types of cars?

Is car size related to noise level, ifeffect of filter type is controlled?

Is filter type related to noise level, if effect of size of car is controlled?

Is there a combination of Size of Car and Noise Filter Type that is especially loud, or especially soft?

• called an INTERACTION effect.

Multiple comparison tests

Small Mid-Size Large

Size of Car

760

780

800

820

840

860

No

ise

Lev

el R

ea

din

g (

De

cib

els

)

Type of Noise Filter

Standard Filter

Octel Filter

Factorial (Two or more way) ANOVA• One dependent variable

interval or ratio

normal distribution

• Two independent variables nominal (define groups)

independent of each other

• Test effect of each I.V. controlling for the effects of the other I.V.

• Test interaction effect for combinations of categories

SIZE effect is still significant

TYPE effect is significant when size is controlled

INTERACTION effect is significant• There is a combination which shows more than the combined

impact of SIZE and TYPE

Tests of Betw een-Subjects Effects

Dependent Variable: Noise Level Reading (Decibels)

23655612.5a 6 3942602.083 60269.076 .000

26051.389 2 13025.694 199.119 .000

1056.250 1 1056.250 16.146 .000

804.167 2 402.083 6.146 .006

1962.500 30 65.417

23657575.0 36

Source

Model

size

type

size * type

Error

Total

Type III Sum

of Squares df Mean Square F Sig.

R Squared = 1.000 (Adjusted R Squared = 1.000)a.

Means of each combination of Size & Type

INTERACTION: Whenever lines not parallel

Manufacturers of the new Octel noise filter claim that it

reduces noise levels in cars of all sizes. In a Two-Way

ANOVA, this claim proved to be true. The Size of Car

effect was significant (F(2,36) = 199.119, p < .001). When

the impact of size was controlled, the Filter Type effect

was also significant (F(1,36) = 16.146, p < .001), with the

Octel Filter having lower noise levels than standard filters.

The Interaction effect was also significant (F(2,30) =

6.146, p = .006). For Small cars, the noise difference

between filter types was 3.33; for Large cars it was 5.000,

but Midsize cars with the Octel filter averaged 24.166

points lower on the Noise Level scale.

A complete report would include the Mean and SD of each cell where a

significant difference occurred, either in a table or in narrative. It

would include effect size (η2) for significant effects.

A table or graph of group means

A report of the three hypothesis tests:• One for Factor A

• One for Factor B

• One for the interaction of A with B

Asterisks often used to report hypothesis test results* = significant with alpha = .05** = significant with alpha = .01*** = significant with alpha = .001

If there are more than two factors, there will be more hypothesis tests for factors, and more interactions.

Total Variability

Between Treatments

Factor A

Factor B Interaction

Within Treatments

Three distinct tests

• Main effect of Factor A

• Main effect of Factor B

• Interaction of A and B

A separate F test is conducted for each

Notation describes procedureTables usually used to present

the results Group means (cell)Row means Column means

Each factor is operationalized by one or more variables (measures)

Images from Trochim’s Research Methods Knowledge Base at http://www.socialresearchmethods.net/kb/index.php

Plot the means of each group (defined as a combination of Factor 1 and Factor 2)

If all the null hypotheses are true, all the points will have about the same Mean Outcome level.

The two row means are the same

The two column means are the same

All groups have the same mean score

Neither factor had any effect

Images from Trochim’s Research Methods Knowledge Base at http://www.socialresearchmethods.net/kb/index.php

Row means: the same

Column means: differ

No score especially high or especially low

Row means: differ

Column means: the same

No score especially high or especially low

Images from Trochim’s Research Methods Knowledge Base at http://www.socialresearchmethods.net/kb/index.php

The mean differences between individuals

treatment conditions, or cells, are different

from what would be predicted from the

overall main effects of the factors

H0: There is no interaction between

Factors A and B

H1: There is an interaction between

Factors A and B

Row means differ Column means differ One group is different Others are the same

Row means the same Column means the same Graph shows that pattern

in one factor depends on the

status of the otherImages from Trochim’s Research Methods Knowledge Base at http://www.socialresearchmethods.net/kb/index.php

Dependence of factors

• The effect of one factor depends on the level

or value of the other

Non-parallel lines (cross or converge) in a

graph

• Indicate interaction is occurring

Typically called the A x B interaction

Total Variability

Between Treatments

Factor A

Factor B Interaction

Within Treatments

We will compute problems by hand to gain understanding, but not on a test

N

GXSStotal

22

treatment each insidetreatmentswithin SSSS

N

G

n

TSS treatmentsbetween

22

Total Variability

Between Treatments

Factor A

Factor B Interaction

Within Treatments

dftotal = N – 1

dfwithin treatments = Σdfinside each treatment

dfbetween treatments = k – 1

dfA = number of rows – 1

dfB = number of columns– 1

dfAxB = dfbetween treatments – dfA – dfB

reatmentst within

reatmentst withinreatmentst within

df

SSMS

AxB

AxBAxB

B

BB

A

AA

df

SSMS

df

SSMS

df

SSMS

within

AxBAxB

within

BB

within

AA

MS

MSF

MS

MSF

MS

MSF

η2, is computed as the percentage of

variability not explained by other factors.

treatments withinA

A

AxBBtotal

AA

SSSS

SS

SSSSSS

SS

2

treatments withinB

B

AxBAtotal

BB

SSSS

SS

SSSSSS

SS

2

treatments withinAxB

AxB

BAtotal

AxBAxB

SSSS

SS

SSSSSS

SS

2

Two independent variables