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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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