Repeated Measures ANOVARepeated Measures ANOVA

Two-Factor ANOVATwo-Factor ANOVA

Introduction to StatisticsIntroduction to StatisticsChapter 14Chapter 14

Apr 16-21, 2009Apr 16-21, 2009Classes #25-26Classes #25-26

Recalling between vs. within Recalling between vs. within subject distinction…subject distinction…

► Difference between?Difference between?► When we have a within subjects design (e.g., When we have a within subjects design (e.g.,

repeated measures, matched subjects), can’t repeated measures, matched subjects), can’t use one-way ANOVA from last chapteruse one-way ANOVA from last chapter That ANOVA assumes each observation is That ANOVA assumes each observation is

independentindependent Within-subjects design = non-independent Within-subjects design = non-independent

observations (e.g., same person answering observations (e.g., same person answering questions over time questions over time each response from each response from same person will be related)same person will be related)

Repeated Measures Repeated Measures ANOVAANOVA

► Reaction Time Study - could be modified Reaction Time Study - could be modified into RM design by having one group of into RM design by having one group of subjects perform under all conditionssubjects perform under all conditions

►Main effectsMain effects – the effect of 1 IV on 1 DV – the effect of 1 IV on 1 DV InteractionsInteractions – the effect of multiple IV’s on 1 – the effect of multiple IV’s on 1 DVDV

Repeated Measures Repeated Measures ANOVAANOVA

►““Within-subjects” design - subjects are Within-subjects” design - subjects are measured on DV more than once as in measured on DV more than once as in training studies or learning studiestraining studies or learning studies

►AdvantagesAdvantages increased power - due to decreased increased power - due to decreased

variancevariance can use smaller sample sizescan use smaller sample sizes allows for study over timeallows for study over time

Repeated Measures ANOVARepeated Measures ANOVA

►DisadvantagesDisadvantages carryover effectscarryover effects - early treatments affect - early treatments affect

later ones.later ones. practice effectspractice effects - subjects’ experience with - subjects’ experience with

test can influence their score on the DV.test can influence their score on the DV. fatiguefatigue sensitizationsensitization - subjects’ awareness of - subjects’ awareness of

treatment is heightened due to repeated treatment is heightened due to repeated exposure to test.exposure to test.

The strength of a within The strength of a within subjects designsubjects design

► Key to statistics = see where variability Key to statistics = see where variability is coming fromis coming from

► Less noise have to deal with, the better – Less noise have to deal with, the better – easier for the picture of the data to easier for the picture of the data to come throughcome through

► Same person answers questions multiple Same person answers questions multiple times times variability that’s unique to that variability that’s unique to that person is constant across each person is constant across each participant participant less noise in the data less noise in the data

Comparing repeated ANOVA Comparing repeated ANOVA with one-way ANOVAwith one-way ANOVA

► Still involves F ratioStill involves F ratio► Still same interpretation – greater than 1 Still same interpretation – greater than 1

= more variance due to condition than to = more variance due to condition than to errorerror

► But variance due to participant is removed But variance due to participant is removed from the modelfrom the model

► F = variance between conditions, F = variance between conditions, divided by amount of variance would divided by amount of variance would expect by chance (excluding variance due expect by chance (excluding variance due to individual differences – error variance, to individual differences – error variance, or residual variance)or residual variance)

In other wordsIn other words

►F captures how much signal there is F captures how much signal there is (how much difference across (how much difference across conditions) compared to noise (how conditions) compared to noise (how much variability there is that can’t be much variability there is that can’t be explained by individual differences)explained by individual differences)

►= MS between conditions / MS error= MS between conditions / MS error

Step 1:Step 1:Set HypothesisSet Hypothesis

► STEP 1STEP 1: : State the HypothesisState the Hypothesis HH00: : 4 4

HHAA : : ≠ ≠ ≠ 44

Step 2: Determining F-Step 2: Determining F-criticalcritical

►See p. 533 See p. 533 ►= .01 (boldface) or .05 (lightface)= .01 (boldface) or .05 (lightface)►Numerator = Numerator = df df between treatmentsbetween treatments

►Denominator = Denominator = df df errorerror

Step 3: STAGE 1Step 3: STAGE 1

SSSSbetween treatmentsbetween treatments = = (T(T22/n) – (G/n) – (G22/N)/N)

SSSSwithin treatmentswithin treatments = = SS inside each treatment SS inside each treatment

==(SS(SS11+SS+SS22+SS+SS33+...+SS+...+SSkk))

SSSStotaltotal = = XX22 – (G – (G22/N) /N) oror SS SSbetweenbetween + SS + SSwithinwithin

n = # of scores in a tx conditionn = # of scores in a tx conditionN = total # of scores in whole study N = total # of scores in whole study T = sum of scores for each tx conditionT = sum of scores for each tx conditionG = sum of all scores in the study (Grand Total)G = sum of all scores in the study (Grand Total)

Step 3: STAGE 1Step 3: STAGE 1

• Calculate dfCalculate dfbetween treatmentsbetween treatments = k – 1 = k – 1

• Calculate dfCalculate dfwithin treatmentswithin treatments = = dfdf inside each treatmentinside each treatment

• Calculate dfCalculate dftotaltotal = N-1 = N-1

• dfdfbetweenbetween + df + dfwithinwithin = df = dftotaltotal

k = number of factor k = number of factor levelslevels

n = number of scores in n = number of scores in a treatment conditiona treatment condition

N = total number of N = total number of scores in whole study scores in whole study (N = nk)(N = nk)

T = sum of scores for T = sum of scores for each treatment each treatment conditioncondition

G = sum of all scores in G = sum of all scores in the study (Grand the study (Grand Total)Total)

Step 3: STAGE 2Step 3: STAGE 2

• SSSSbetween subjects = between subjects =

pp22 - - GG22

k Nk N

• SSSSerror = error = SSSSwithin treatmentswithin treatments - SS - SSbetween treatmentsbetween treatments

• dfdfbetween subjects = n – 1between subjects = n – 1

• dfdferror = error = dfdfwithin treatmentswithin treatments - - dfdfbetween subjectsbetween subjects

Step 3: STAGE 3Step 3: STAGE 3F-ratio calculationF-ratio calculation

►For numerator:For numerator:►MSMSbetween treatments = between treatments =

SS SS between treatmentsbetween treatments

df df between treatmentsbetween treatments

STEP 3: STAGE 3STEP 3: STAGE 3F-ratio calculationF-ratio calculation

►For denominator:For denominator:►MSMSerror = error =

SS SS errorerror

df df errorerror

STEP 3: STAGE 3STEP 3: STAGE 3F-ratio calculationF-ratio calculation

►FFcalculatedcalculated = = MS between treatments MS between treatments

MS errorMS error

Step 4: Summary table and Step 4: Summary table and decisiondecision

SourceSource SS SS dfdf MSMS F F

Between treatmentsBetween treatments

WithinWithin treatments treatments

Between SubjectsBetween Subjects

ErrorError

TotalTotal

The similarity in logic The similarity in logic continues…continues…

►Effect size: Effect size: measured by eta squared, still captures measured by eta squared, still captures

proportion of variability that’s explained proportion of variability that’s explained by conditionby condition

nn22 = = SS between treatmentsSS between treatments

between treatments + SS errorbetween treatments + SS error

The similarities The similarities continue…continue…

►To report in APA style:To report in APA style: FF (df numerator, df denominator) = value, (df numerator, df denominator) = value,

pp information information

►df numerator = number conditions – 1df numerator = number conditions – 1►df denominator = error df = df total – df denominator = error df = df total –

df numeratordf numerator

Effect sizeEffect size

nn2 2 = = SS between treatmentsSS between treatments = = SS between SS between treatmentstreatments

SS between treat + SS within treatSS between treat + SS within treat SS total SS total

Still need post hoc testsStill need post hoc tests

►Still have options such as Tukey and Still have options such as Tukey and ScheffSchefféé

Two-Factor ANOVATwo-Factor ANOVA

► In Chapter 13, we looked at ANOVAs In Chapter 13, we looked at ANOVAs with several levels of one IVwith several levels of one IV

►Here, we are looking at ANOVAs with Here, we are looking at ANOVAs with several levels of more two IVsseveral levels of more two IVs

►We will now be looking at three types We will now be looking at three types of mean differences within this of mean differences within this analysisanalysis

Main EffectsMain Effects

►The mean differences that might The mean differences that might among the levels of each factor (IV)among the levels of each factor (IV) The mean differences of the rows (IVThe mean differences of the rows (IV11)) The mean differences of the columns (IVThe mean differences of the columns (IV22))

STEP 1: State the Hypotheses STEP 1: State the Hypotheses for the Main Effectsfor the Main Effects

► Null Hyp: H0: µA1 = µA2

► Alternative Hyp: HA: µA1 ≠ µA2

► Null Hyp: H0: µB1 = µB2

► Alternative Hyp: HA: µB1 ≠ µB2

InteractionsInteractions

►This occurs when the mean differences This occurs when the mean differences between individual treatment between individual treatment conditions (cells) are different than conditions (cells) are different than what would be predicted from the what would be predicted from the overall main effectsoverall main effects

STEP 2: State the Hypotheses STEP 2: State the Hypotheses for the Interactionsfor the Interactions

► H0: There is no interaction between factors A and B.

► HA: There is an interaction between factors A and B.

STEP 3: Draw a matrixSTEP 3: Draw a matrix

►This will allow us to determine how This will allow us to determine how many groups we have and where many groups we have and where different participants fall within these different participants fall within these groupsgroups For an example see page 401 For an example see page 401

STEP 4: Determine STEP 4: Determine dfdf

►dfdfbetween treatmentsbetween treatments = number of cells – 1 = number of cells – 1

►dfdfwithinwithin = = dfdfeach treatmenteach treatment

►dfdftotaltotal = N-1 = N-1

STEP 5: STEP 5: Stage 1: AnalysesStage 1: Analyses

►Total Variability:Total Variability: SSSStotaltotal = = XX22 – (G – (G22/N)/N)

►Between Treatments VariabilityBetween Treatments Variability SSSSbetween treatmentsbetween treatments = = (T(T22/n) – (G/n) – (G22/N) /N)

►Within Treatments VariabilityWithin Treatments Variability SSSSwithin treatments within treatments = = SSSSeach treatmenteach treatment

STEP 5: STEP 5: Stage 2: AnalysesStage 2: Analyses

►Factor A:Factor A: SSSSfactor Afactor A = = (T(T22ROWROW/n/nROWROW) – (G) – (G22/N)/N)

dfdffactor A factor A = number of rows – 1= number of rows – 1

►Factor B:Factor B: SSSSfactor Bfactor B = = (T(T22COLCOL/n/nCOLCOL) – (G) – (G22/N)/N)

dfdffactor B factor B = number of columns – 1= number of columns – 1

►A A XX B Interaction: B Interaction: SSSSAXBAXB = SS = SSbetween treatmentsbetween treatments – SS – SSfactor Afactor A – SS – SSfactor Bfactor B

STEP 6: STEP 6: Calculations for MS and FCalculations for MS and F

►MSMSwithin treatmentswithin treatments = SS = SSwithin treatwithin treat/df/dfwithin treatwithin treat

► MSMSAA = SS = SSAA//dfdfAA

►MSMSBB = SS = SSBB//dfdfBB

►MSMSAXBAXB = SS = SSAXBAXB//dfdfAXBAXB

STEP 6: STEP 6: Calculations for MS and FCalculations for MS and F

►FFAA = MS = MSAA/MS/MSwithin treatmentswithin treatments

►FFBB = MS = MSBB/MS/MSwithin treatmentswithin treatments

►FFAXBAXB = MS = MSAXBAXB/MS/MSwithin treatmentswithin treatments

Step 7: Summary table and Step 7: Summary table and decisiondecision

SourceSource SS SS dfdf MSMS F F

Between treatmentsBetween treatments

Factor AFactor A

Factor B Factor B

A X B InteractionA X B Interaction

WithinWithin treatments treatments

TotalTotal

Effect SizeEffect Size

► For Factor A:For Factor A:► nn2 2 = = SSSSA__________________A__________________

SS total – SS total – SSSSB B - - SSSSAXBAXB

► For Factor B:For Factor B:► nn2 2 = = SSSSB__________________B__________________

SS total – SS total – SSSSA A - - SSSSAXBAXB

► For AXB:For AXB:► nn2 2 = = SSSSB__________________B__________________

SS total – SS total – SSSSA A - - SSSSBB