Repeated Measures ANOVA Two-Factor ANOVA

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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

Recalling between vs. within subject distinction…
• 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)
Repeated Measures ANOVA

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

Repeated Measures ANOVA

“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

Repeated Measures ANOVA

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.

The strength of a within subjects design
• 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
Comparing repeated ANOVA with one-way ANOVA
• 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)
In other words
• 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
Step 2: Determining F-critical
• See p. 533
• a = .01 (boldface) or .05 (lightface)
• Numerator = df between treatments
• Denominator = df error
Step 3: STAGE 1
• 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)

Step 3: STAGE 1
• 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)

Step 3: STAGE 2
• 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

Step 4: Summary table and decision

Source SS df MS F

Between treatments

Within treatments

Between Subjects

Error

Total

The similarity in logic continues…
• Effect size:
• measured by eta squared, still captures proportion of variability that’s explained by condition

n2= SS between treatments

between treatments + SS error

The similarities continue…
• 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
Effect size

n2 = SS between treatments = SS between treatments

SS between treat + SS within treat SS total

Still need post hoc tests
• Still have options such as Tukey and Scheffé
Two-Factor ANOVA
• 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
Main Effects
• 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)
STEP 1: State the Hypotheses for the Main Effects
• Null Hyp:H0: µA1 = µA2
• Alternative Hyp: HA: µA1≠ µA2
• Null Hyp:H0: µB1 = µB2
• Alternative Hyp: HA: µB1≠ µB2
Interactions
• This occurs when the mean differences between individual treatment conditions (cells) are different than what would be predicted from the overall main effects
STEP 2: State the Hypotheses for 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 matrix
• 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
STEP 4: Determine df
• 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
Step 7: Summary table and decision

Source SS df MS F

Between treatments

Factor A

Factor B

A X B Interaction

Within treatments

Total

Effect Size
• 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