Chapter 14: Repeated-Measures Flashcards
1
Q
Repeated Measures Design (GLM4)
A
- when the same entities participate in all conditions of an experiment or provide data at multiple time points
2
Q
What assumption is violated in Repeated Measures Design?
A
- assumption that scores in different conditions are independent is violated
- scores taken under different experimental conditions are likely to be related because they come from the same entities
—> As such, conventional F-test will lack accuracy
3
Q
What is the new assumption in Repeated Measures Design?
A
- new assumption: the relationship between pairs of experimental conditions is similar [sphericity]
4
Q
Mixed Design: Sphericity Assumption
A
- can be linked to homogeneity of variance in between-groups ANOVA
- denoted by ε
- sometimes referred to as circularity
- general condition (less restrictive form) of compound symmetry
- assume that the variation within experimental conditions is fairly similar and that no two conditions are any more dependent than any other two
- compound symmetry not a necessary condition
- if you were to take each pair of treatment levels, and calculate the differences between each pair of scores, then it is necessary that these differences have approximately equal variances
- at least 3 conditions for sphericity to be an issue
Example:
Condition A
Condition B
Condition C
Variance (A-B)~Variance (B-C)~Variance (A-C)
5
Q
Mixed Design: Sphericity Assumption
- Assessing the severity of departures from sphericity
A
- use Mauchly’s test
6
Q
Mauchly’s Test
A
- tests hypothesis that variances of differences between conditions are equal
- if Mauchly’s test statistic is significant (p
7
Q
What is the effect of violating the assumption of sphericity?
A
- sphericity creates a loss of power and a test statistic that doesn’t have the distribution that it’s supposed to have (F- distribution)
- complications for post hoc tests
—> Bonferroni method: most robust of the univariate techniques, especially in terms of power & control of Type I error
—> When sphericity is definitely not violated, Tukey’s test can be used
8
Q
What do you do if you violate sphericity?
A
- adjust the degrees of freedom for any F-ratios affected by violation
- if data spherical, sphericity: 1
- if data non-spherical, sphericity: less than 1
- multiply df with sphericity estimate
- Smaller degrees of freedom make p- associated with F-ratio less significant
- by adjusting df, we make F-ratio more conservative when sphericity is violated
9
Q
What to use to adjust the degrees of freedom when sphericity is violated?
A
- Greenhouse–Geisser estimate varies between 1/(k − 1) and 1
- denoted by epsilon with cap
- k: no. of Repeated measures conditions
- lower bound = 1/(k-1)
- when Greenhouse-Geisser estimate is greater than .75 the correction is too conservative
- when estimates of sphericity are greater than .75, Huynh–Feldt estimate should be used
- when Greenhouse–Geisser estimate of sphericity is less than .75 or nothing is known about sphericity at all Greenhouse–Geisser correction should be used
10
Q
Other option when sphericity is violated
A
- multivariate test: MANOVA (not dependent on sphericity)
- tradeoff in power when using multivariate tests
- complex way: multilevel model
11
Q
Power in ANOVA and MANOVA:
A
- when you have a large violation of sphericity (ε < .7) and your sample size is greater than a + 10 then multivariate procedures are more powerful
- a: no. of levels in repeated measures
- with small sample sizes or when sphericity holds (ε >.7) the univariate approach is preferred
12
Q
Theory of One-Way repeated-measures ANOVA
A
- within participant
- within-participant variance will be made up of two things: effect of manipulation & individual differences in performance (SSR)
- use an F-ratio that compares size of variation due to our experimental manipulations to size of variation due to random factors
13
Q
Repeated Measures: SST
A
- same formula as in independent ANOVA
- SST=grand variance x (N-1)
- dfT= N-1
- Treat data as a single group
14
Q
Repeated Measures:
- Within-participant Sum of Squares [SSW]
A
- represents individual differences within participants
- SSW=s^2 (entity1) x (n1-1)+…s^2 (entity i) x (ni-1)
—> n: number of repeated conditions
- dfW= N x (n-1)
15
Q
Repeated Measures:
- The model sum of squares [SSM]
A
- same formula as independent ANOVA
- SSM=sum[nk(condition mean - grand mean)^2]
- dfM=k-1
- k=no. of conditions
16
Q
Repeated Measures:
- Residual sum of squares [SSR]
A
~ SSR= SSW- SSM
~ dfR= dfW- dfM
17
Q
Repeated Measures:
- Mean Squares
A
- eliminate bias
- MSM: average amount of variation explained by model (systematic variation)
—> MSM= SSM/dfM - MSR: average amount of variation explained by extraneous variables (unsystematic variation)
—> MSR= SSR/dfR
18
Q
Repeated Measures:
- F-ratio
A
- F=MSM/MSR
- F>1
- F(dfM, dfR)
19
Q
Repeated Measures:
- Between-participants sum of squares [SSB]
A
- represents individual differences between cases
—> e.g: different participants have different tolerances - SST = SSB + SSW
- SSB = SST - SSW
20
Q
Repeated Measures: Assumptions
A
- Linearity
- Normality
- Independence of Errors
- Sphericity
- if violated: one-way repeated ANOVA: use Friedman’s ANOVA
- no non-parametric version for factorial repeated measures
- follow ch.5 for correcting for bias
21
Q
One-way repeated-measures ANOVA: SPSS
A
I. Analyze
II. General Linear Model
III. Repeated Measures
- Within Subject Factor Name: (no spaces in the name)
- Number of levels
- Click Add
- Click Define
- Transfer levels to ‘Within Subjects Variables’
22
Q
One-way repeated-measures ANOVA: SPSS
- Defining Contrasts
A
- not possible to define planned contrasts for repeated measures
- use standard contrasts
23
Q
One-way repeated-measures ANOVA: SPSS
- Post-Hoc tests
A
- When sphericity is definitely not violated: Tukey’s test can be used
- if sphericity can’t be assumed: use Games–Howell procedure
- sphericity-related complications: post hoc tests seen for independent ANOVA not available for repeated-measures analyses
- Some Post-Hoc tests can be done by Clicking ‘Options’
24
Q
One-way repeated-measures ANOVA: SPSS
- Post-Hoc tests using ‘Options’
A
- Click ‘Options’
- Drag repeated-Measures variable to ‘Display Means for’
- Compare main effects
- Confidence adjustment: Use Bonferroni or Sidák
25
Output: One-Way repeated-measures ANOVA
- Descriptives and other diagnostics
- This box: useful to check that the variables were entered in correct order
- look at means
- These mean values are useful for interpreting any effects that may emerge from the main analysis
26
Assessing and correcting for sphericity: Mauchly’s test
- to assume sphericity Mauchly’s test should be non-significant
- If Greenhouse–Geisser estimate is closer to its lower limit than to upper
limit of 1: substantial deviation from sphericity
- when adjusting - F-ratio itself remains unchanged, but its df and p-value are adjusted
- df are adjusted by multiplying them by the estimate of sphericity (dfM)
27
Output: One-Way repeated-measures ANOVA
- Main ANOVA
- if sphericity is violated: use either Greenhouse Geisser
- Stevens: average p. value: [p(s.assumed)+p(greenhouse)]/2
- effect size is not affected by sphericity corrections
28
Output: One-Way repeated-measures ANOVA
- Contrasts
- don’t look at contrasts if no main effect of repeated-measures variable was found
29
Output: One-Way repeated-measures ANOVA
- Post-Hoc tests
- don’t look at Post-Hoc tests if no main effect of repeated-measures variable was found
30
Effect sizes for repeated-measures ANOVA
- best measure of the overall effect size is omega squared (ω^2)
31
Reporting one-way repeated-measures ANOVA
- Mauchly’s test indicated that the assumption of sphericity had been violated, χ2(5) = 11.41, p = .047, therefore Greenhouse–Geisser corrected tests are reported (ε = .53). The results show that the time to retch was not significantly affected by the type of animal eaten, F(1.60, 11.19) =
3. 79, p = .063, ω2 = .24.
- Mauchly’s test indicated that the assumption of sphericity had been violated, χ2(5) = 11.41, p = .047, therefore degrees of freedom were corrected using Huynh–Feldt estimates of sphericity (ε = .67). The results show that the time to retch was significantly affected by the type of animal eaten, F(2, 13.98) = 3.79, p = .048, ω2 = .24.
- Pillai: V
Mauchly’s test indicated that the assumption of sphericity had been violated, χ2(5) = 11.41, p = .047, therefore multivariate tests are reported (ε = .53). The results show that the time to retch
was significantly affected by the type of animal eaten, V = 0.94, F(3, 5) = 26.96, p = .002, ω2 = .24.
32
Factorial repeated-measures designs
- incorporate a second, third or even fourth independent variable into a repeated-measures analysis
33
Main Analysis: Factorial Repeated Design
- Total Number of conditions= number of conditions for variable 1 x number of conditions for variable 2
34
Factorial Design ANOVA: SPSS
I. Analyze
2. General Linear Model
3. Repeated Measures
- Enter name of all repeated Measures and their levels
- Click Add
- Click Define
35
Factorial Design ANOVA: SPSS
- brackets
- represent levels of the factors
- (1,1): level 1 of factor 1 and level 1 of factor 2
- enter control level as either 1st of last to compute contrasts
- remaining levels can be assigned arbitrarily
36
Factorial Design ANOVA: SPSS
- contrasts
- we can NOT do simple contrasts for repeated design
| - Click ‘Contrasts’: for standard contrasts
37
Factorial Design ANOVA: SPSS
- Simple Effects Analysis
- to break down an interaction term
- looks at the effect of one independent variable at individual levels of the other independent variable
- only using syntax
38
Factorial Design ANOVA: SPSS
- Graphing Interaction
- Click ‘Plots’
- Horizontal axis
- Separate Lines
- Add
- Continue
- at your discretion
39
Factorial Design ANOVA: SPSS
- Options
- Some Post-Hoc tests
- enter variables on ‘Display Means for’
- Display Descriptive Statistics
- These tests are interesting only if the interaction effect
is not significant
40
Output: Factorial repeated-measures ANOVA
- Interaction
- to verify the interpretation of interaction: look at contrasts
- use means to make sense of interactions
41
Factorial Repeated Design ANOVA
- Simple Effects Analysis
GLM beerpos beerneg beerneut winepos wineneg wineneut waterpos waterneg waterneut /WSFACTOR=Drink 3 Imagery 3
/EMMEANS = TABLES(Drink*Imagery) COMPARE(Imagery)
42
Reporting the results from factorial repeated-measures ANOVA
- Mauchly’s test indicated that the assumption of sphericity had been violated for the main
effects of drink, χ2(2) = 23.75, p < .001, and imagery, χ2(2) = 7.42, p = .024. Therefore degrees of freedom were corrected using Greenhouse–Geisser estimates of sphericity (ε = .58 for the main effect of drink and .75 for the main effect of imagery).
- Unless otherwise stated p < .001. There was a significant main effect of the type of drink on ratings of the drink, F(1.15, 21.93) = 5.11, p = .011. Contrasts revealed that ratings of beer, F(1, 19) = 6.22, p = .022, r = .50, and wine, F(1, 19) = 18.61, r = .70, were significantly higher than water.
- There was also a significant main effect of the type of imagery on ratings of the drinks, F(1.50, 28.40) = 122.57. Contrasts revealed that ratings after positive imagery were significantly higher than after neutral imagery, F(1, 19) = 142.19, r = .94. Conversely, ratings after negative imagery were significantly lower than after neutral imagery, F(1, 19) = 47.07, r = .84.
- There was a significant interaction effect between the type of drink and the type of imagery used, F(4, 76) = 17.16. This indicates that imagery had different effects on people’s ratings depending on which type of drink was used.
- To break down this interaction, contrasts were performed comparing all drink types to their baseline (water) and all imagery types to their baseline (neutral imagery). These revealed significant interactions when comparing negative imagery to neutral imagery both for beer compared to water, F(1, 19) = 6.75, p = .018, r = .51, and wine compared to water, F(1, 19) = 26.91, r = .77. Looking at the interaction graph, these effects reflect that negative imagery (compared to neutral) lowered scores significantly more in water than it did for beer, and lowered scores significantly more for wine than it did for water.
- The remaining contrasts revealed no significant interaction term when comparing positive imagery to neutral imagery both for beer compared to water, F(1, 19) = 1.58, p = .225, r = .28, and wine compared to water, F(1, 19) = 0.24, p = .633, r = .11. However, these contrasts did yield small to medium effect sizes.
