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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

Advantages

increased power - due to decreased variance

can use smaller sample sizes

allows for study over time

Repeated Measures ANOVA

Disadvantages

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 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

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