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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 anova two factor anova
Repeated Measures ANOVATwo-Factor ANOVA

Introduction to Statistics

Chapter 14

Apr 16-21, 2009

Classes #25-26

recalling between vs within subject distinction
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
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 anova1
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 anova2
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
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
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
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:Set Hypothesis
  • STEP 1: State the Hypothesis
    • H0: m1 = m2 = m3 = m4
    • HA : m1≠ m2≠m3≠ m4
step 2 determining f critical
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
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 11
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
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 3 f ratio calculation
Step 3: STAGE 3F-ratio calculation
  • For numerator:
  • MSbetween treatments =

SS between treatments

df between treatments

step 3 stage 3 f ratio calculation1
STEP 3: STAGE 3F-ratio calculation
  • For denominator:
  • MSerror =

SS error

df error

step 3 stage 3 f ratio calculation2
STEP 3: STAGE 3F-ratio calculation
  • Fcalculated = MS between treatments

MS error

step 4 summary table and decision
Step 4: Summary table and decision

Source SS df MS F

Between treatments

Within treatments

Between Subjects

Error

Total

the similarity in logic continues
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
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
Effect size

n2 = SS between treatments = SS between treatments

SS between treat + SS within treat SS total

still need post hoc tests
Still need post hoc tests
  • Still have options such as Tukey and Scheffé
two factor anova
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
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
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
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
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
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
STEP 4: Determine df
  • dfbetween treatments = number of cells – 1
  • dfwithin = dfeach treatment
  • dftotal = N-1
step 5 stage 1 analyses
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
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
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 f1
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
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 size1
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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