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# Repeated Measures ANOVA Two-Factor ANOVA - PowerPoint PPT Presentation

Repeated Measures ANOVA Two-Factor ANOVA. Introduction to Statistics Chapter 14 Apr 16-21, 2009 Classes #25-26. Recalling between vs. within subject distinction…. Difference between?

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Repeated Measures ANOVATwo-Factor ANOVA

Introduction to Statistics

Chapter 14

Apr 16-21, 2009

Classes #25-26

• Difference between?

• When we have a within subjects design (e.g., repeated measures, matched subjects), can’t use one-way ANOVA from last chapter

• That ANOVA assumes each observation is independent

• Within-subjects design = non-independent observations (e.g., same person answering questions over time  each response from same person will be related)

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

Main effects – the effect of 1 IV on 1 DV Interactions – the effect of multiple IV’s on 1 DV

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

increased power - due to decreased variance

can use smaller sample sizes

allows for study over time

carryover effects - early treatments affect later ones.

practice effects - subjects’ experience with test can influence their score on the DV.

fatigue

sensitization - subjects’ awareness of treatment is heightened due to repeated exposure to test.

• Key to statistics = see where variability is coming from

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

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

• Still involves F ratio

• Still same interpretation – greater than 1 = more variance due to condition than to error

• But variance due to participant is removed from the model

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

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

• = MS between conditions / MS error

Step 1:Set Hypothesis

• STEP 1: State the Hypothesis

• H0: m1 = m2 = m3 = m4

• HA : m1≠ m2≠m3≠ m4

• See p. 533

• a = .01 (boldface) or .05 (lightface)

• Numerator = df between treatments

• Denominator = df error

• SSbetween treatments = (T2/n) – (G2/N)

• SSwithin treatments = SS inside each treatment = (SS1+SS2+SS3+...+SSk)

• SStotal = X2 – (G2/N) or SSbetween + SSwithin

n = # of scores in a tx condition

N = total # of scores in whole study

T = sum of scores for each tx condition

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

• Calculate dfbetween treatments = k – 1

• Calculate dfwithin treatments = dfinside each treatment

• Calculate dftotal = N-1

• dfbetween + dfwithin = dftotal

k = number of factor levels

n = number of scores in a treatment condition

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

T = sum of scores for each treatment condition

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

• SSbetween subjects =

p2 - G2

k N

• SSerror = SSwithin treatments - SSbetween treatments

• dfbetween subjects = n – 1

• dferror = dfwithin treatments - dfbetween subjects

Step 3: STAGE 3F-ratio calculation

• For numerator:

• MSbetween treatments =

SS between treatments

df between treatments

STEP 3: STAGE 3F-ratio calculation

• For denominator:

• MSerror =

SS error

df error

STEP 3: STAGE 3F-ratio calculation

• Fcalculated = MS between treatments

MS error

Source SS df MS F

Between treatments

Within treatments

Between Subjects

Error

Total

• Effect size:

• measured by eta squared, still captures proportion of variability that’s explained by condition

n2= SS between treatments

between treatments + SS error

• To report in APA style:

• F (df numerator, df denominator) = value, p information

• df numerator = number conditions – 1

• df denominator = error df = df total – df numerator

n2 = SS between treatments = SS between treatments

SS between treat + SS within treat SS total

• Still have options such as Tukey and Scheffé

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

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

• We will now be looking at three types of mean differences within this analysis

• The mean differences that might among the levels of each factor (IV)

• The mean differences of the rows (IV1)

• The mean differences of the columns (IV2)

• Null Hyp:H0: µA1 = µA2

• Alternative Hyp: HA: µA1≠ µA2

• Null Hyp:H0: µB1 = µB2

• Alternative Hyp: HA: µB1≠ µB2

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

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

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

• This will allow us to determine how many groups we have and where different participants fall within these groups

• For an example see page 401

• dfbetween treatments = number of cells – 1

• dfwithin = dfeach treatment

• dftotal = N-1

STEP 5: Stage 1: Analyses

• Total Variability:

• SStotal = X2 – (G2/N)

• Between Treatments Variability

• SSbetween treatments = (T2/n) – (G2/N)

• Within Treatments Variability

• SSwithin treatments = SSeach treatment

STEP 5: Stage 2: Analyses

• Factor A:

• SSfactor A = (T2ROW/nROW) – (G2/N)

• dffactor A = number of rows – 1

• Factor B:

• SSfactor B = (T2COL/nCOL) – (G2/N)

• dffactor B = number of columns – 1

• A X B Interaction:

• SSAXB = SSbetween treatments – SSfactor A – SSfactor B

STEP 6: Calculations for MS and F

• MSwithin treatments = SSwithin treat/dfwithin treat

• MSA = SSA/dfA

• MSB = SSB/dfB

• MSAXB = SSAXB/dfAXB

STEP 6: Calculations for MS and F

• FA = MSA/MSwithin treatments

• FB = MSB/MSwithin treatments

• FAXB = MSAXB/MSwithin treatments

Source SS df MS F

Between treatments

Factor A

Factor B

A X B Interaction

Within treatments

Total

• For Factor A:

• n2 = SSA__________________

SS total – SSB - SSAXB

• For Factor B:

• n2 = SSB__________________

SS total – SSA - SSAXB

• For AXB:

• n2 = SSB__________________

SS total – SSA - SSB