Two-Factor ANOVATwo-Factor ANOVA

OutlineOutline

Basic logic of a two-factor ANOVABasic logic of a two-factor ANOVA Recognizing and interpreting main & Recognizing and interpreting main &

interaction effects interaction effects F-ratiosF-ratios How to compute & interpret a two-How to compute & interpret a two-

way ANOVAway ANOVA AssumptionsAssumptions Extension of Factorial ANOVAExtension of Factorial ANOVA

Factorial DesignsFactorial Designs

Move beyond the one-way ANOVA to Move beyond the one-way ANOVA to designs that have 2+ IVsdesigns that have 2+ IVs

The variables can have unique effects The variables can have unique effects or can combine with other variables to or can combine with other variables to have a combined effecthave a combined effect

Why Should We Use a Why Should We Use a Factorial Design?Factorial Design?

We can examine the influence that We can examine the influence that each factor by itself has on a each factor by itself has on a behaviour, behaviour, as well asas well as the influence the influence that combining these factors has on that combining these factors has on the behaviourthe behaviour

Can be efficient and cost-effectiveCan be efficient and cost-effective

Interpretation of Factorial Interpretation of Factorial DesignsDesigns

Two Kinds of Information:Two Kinds of Information:

1.1. Main effect of an IVMain effect of an IV– Effect that one IV has independently Effect that one IV has independently

of the effect of the other IVof the effect of the other IV– Design with 2 IVs, there are 2 main Design with 2 IVs, there are 2 main

effects (one for each IV):effects (one for each IV): Main Effect of Factor A (1st IV): Overall difference among the levels of A collapsing Main Effect of Factor A (1st IV): Overall difference among the levels of A collapsing

across the levels of B. across the levels of B.

Main Effect of Factor B (2nd IV): Overall difference among the levels of B Main Effect of Factor B (2nd IV): Overall difference among the levels of B collapsing across the levels of A.collapsing across the levels of A.

Interpretation of Factorial Interpretation of Factorial DesignsDesigns

Two Kinds of Information:Two Kinds of Information:

2.2. InteractionInteraction– Represent how independent variables work Represent how independent variables work

together to influence behaviortogether to influence behavior– The relationship between one factor and The relationship between one factor and

the DV change with, or depends on, the the DV change with, or depends on, the level of the other factor that is presentlevel of the other factor that is present

– The influence of changing one factor is NOT The influence of changing one factor is NOT the same for each level of the other factorthe same for each level of the other factor

Two-Way ANOVATwo-Way ANOVA

F= variance between groupsvariance within groups

In a 2-way ANOVA, there are 3 F-ratios:

1.Main effect for Factor A

2.Main effect for Factor B

3.Interaction A x B

Guidelines for the Analysis of a Guidelines for the Analysis of a Factorial DesignFactorial Design

First determine whether the interaction between First determine whether the interaction between the independent variables is statistically the independent variables is statistically significant.significant.– If the interaction is statistically significant, If the interaction is statistically significant,

identify the source of the interaction by identify the source of the interaction by examining the simple main effects examining the simple main effects

– Main effects should be interpreted cautiously Main effects should be interpreted cautiously whenever an interaction is present in an whenever an interaction is present in an experimentexperiment

Then examine whether the main effects of each Then examine whether the main effects of each independent variable are statistically significant.independent variable are statistically significant.

Analysis of Main EffectsAnalysis of Main Effects

When a statistically significant main When a statistically significant main effect has only 2 levels, the nature of effect has only 2 levels, the nature of the relationship is determined in the the relationship is determined in the same manner as for the independent same manner as for the independent samples t-testsamples t-test

When a main effect has 3 or more When a main effect has 3 or more levels, the nature of the relationship levels, the nature of the relationship is determined using a Tukey HSD testis determined using a Tukey HSD test

Effect Size Effect Size Three different values of ŋThree different values of ŋ22 are computed are computed

ŋŋ22 for Factor A = for Factor A = SSA_______ SSA_______ SStotal – SSB - SSAxB SStotal – SSB - SSAxB ŋŋ22 for Factor B = for Factor B = SSB_______ SSB_______ SStotal – SSA - SSAxB SStotal – SSA - SSAxB ŋŋ22 for Factor AxB = for Factor AxB = SSAxB______ SSAxB______

SStotal – SSA - SSB SStotal – SSA - SSB

Effect Size – alternate Effect Size – alternate formulasformulas

within

AxBAxB

SSSS

SS

AxB

2within

AxBAxB

SSSS

SS

AxB

2

within

AA

SSSS

SS

A

2within

AA

SSSS

SS

A

2

within

BB

SSSS

SS

B

2within

BB

SSSS

SS

B

2

AssumptionsAssumptions The observations within each sample The observations within each sample

must be independentmust be independent DV is measured on an interval or ratio DV is measured on an interval or ratio

scalescale The populations from which the The populations from which the

samples are selected have must have samples are selected have must have equal variances equal variances

The populations for which the samples The populations for which the samples are selected must be normally are selected must be normally distributed distributed

Calculating 2 Factor Between Calculating 2 Factor Between Subjects Design ANOVA by Subjects Design ANOVA by

handhand Influence of a specific hormone on eating Influence of a specific hormone on eating

behaviourbehaviour IV (A): GenderIV (A): Gender

– MalesMales– FemalesFemales

IV (B): Drug DoseIV (B): Drug Dose– No drugNo drug– Small doseSmall dose– Large doseLarge dose

DV: Eating consumption over a 48-hour periodDV: Eating consumption over a 48-hour period

The Data ….The Data ….

No drugNo drug Small doseSmall dose Large doseLarge dose

MaleMale

FemalFemalee

Factor B – Amount of drug

Facto

r A

-

Gen

der

1

6

1

1

1

7

7

11

4

6

3

1

1

6

4

0

3

7

5

5

0

0

0

5

0

0

2

0

0

3

Homogeneity of varianceHomogeneity of variance

= = ss22 largest largest = =

ss22 smallest smallest

Satisfied or violated???Satisfied or violated???

Step 1: State the Step 1: State the HypothesesHypotheses

Main Effect for Factor AMain Effect for Factor A

Main Effect for Factor BMain Effect for Factor B

Step 1: State the Step 1: State the HypothesesHypotheses

Interaction between dosage & Interaction between dosage & gendergender

Step 2: Compute dfStep 2: Compute df

Double Check:

dftotal= dfbetween + dfwithin

Step 3: Determine F-criticalStep 3: Determine F-critical

Use the F distribution table Use the F distribution table

F F Critical Critical (df effect, df within)(df effect, df within) Using Using = .05 = .05

Step 4: Calculate SSStep 4: Calculate SS

SSSSTOTALTOTAL = = 22 – – GG22

NN

Step 4: Step 4: Calculate SSCalculate SS

SSSSBETWEEN TxBETWEEN Tx = = TT22 – – GG22

nn NN

Step 4: Calculate SSStep 4: Calculate SS

SSSSWITHIN TXWITHIN TX = = SS SS inside each treatmentinside each treatment

Double Check SSDouble Check SS

TxTx SSSSSS BetweenWithinTotal TxTx SSSSSS BetweenWithinTotal

SS for Factor ASS for Factor A

SSSS A A = = TTrowrow22 – – GG22

nnrowrow NN

SS for Factor BSS for Factor B

SSSS B B = = TTColumnColumn22 – – GG22

nnColumnColumn NN

SS for InteractionSS for Interaction

BABetweenAxB SSSSSSSS

Step 5: Calculate MS for Step 5: Calculate MS for Factor AFactor A

MSMSAA = = SSSSAA

dfdfAA

Step 5: Calculate MS for Step 5: Calculate MS for Factor BFactor B

MSMSBB = = SSSSBB

dfdfBB

Step 5: Calculate MS for Step 5: Calculate MS for InteractionInteraction

MSMSAxBAxB = = SSSSAxBAxB

dfdfAxBAxB

Step 5: Calculate MS Step 5: Calculate MS Within TreatmentsWithin Treatments

MSMSwithinwithin = = SSSSwithinwithin

dfdfwithinwithin

Step 6: Calculate F ratios – Step 6: Calculate F ratios – Factor AFactor A

w

A

ithinMS

MSF

w

A

ithinMS

MSF

Step 6: Calculate F ratios Step 6: Calculate F ratios – Factor B– Factor B

w

B

ithinMS

MSF

w

B

ithinMS

MSF

Step 6: Calculate F ratios Step 6: Calculate F ratios – Interaction– Interaction

w

AxB

ithinMS

MSF

w

AxB

ithinMS

MSF

Step 7: Summary Table Step 7: Summary Table

SourceSource SSSS dfdf MSMS FF

Between TxBetween Tx

Factor AFactor A

Factor BFactor B

InteractionInteraction

Within TxWithin Tx

TotalTotal

Extension of Factorial Extension of Factorial ANOVAANOVA

1 factor is between subject & 1 1 factor is between subject & 1 factor is within subjectfactor is within subject

e.g.: pre-post-control designe.g.: pre-post-control design– All subjects are given a pre-test and All subjects are given a pre-test and

a post-testa post-test– Participants divided into two Participants divided into two

groupsgroups– Experimental group vs. control Experimental group vs. control

groupgroup

2 x 3 mixed design

Group Time

Before After 3 mos. afterTherapy Control

Between-Subjects

Within-Subjects