Lecture 7: One-way and two-way repeated measures ANOVA Flashcards
1
Q
Diagram of Repeated measures regression model
A
2
Q
In regression model for repeated measures ANOVA, we have - (2)
A
a model for each participant with the values of u tweaking the model to account for individual differences in the baseline mean and the change in mean associated with the predictor(s)
g denotes the condition and i is the participant
3
Q
One-way ANOVA between groups can be linked to linear model shown below..
A
- We have three means and the model accounts for three levels of a categorical variable with dummy variables
4
Q
Diagram of repreated measure design equation
A
5
Q
**
Repeated-measures is a term used when the
A
same participants participate in all conditions of an experiment
6
Q
What is the decision tree for choosing one-way repeated measures ANOVA? - (5)
A
Q: What sort of measurement? A: Continuous
Q:How many predictor variables? A: Two or more
Q: What type of predictor variable? A: Categorical
Q: How many levels of the categorical predictor? More than two
Q: Same or Different participants for each predictor level? A: Same
7
Q
The assumption of sphericity in within-subject design ANOVA can be likened to
A
the assumption of homogeneity of variance in
between-group ANOVA
8
Q
Sphericity is sometimes denoted as
A
ε or circularity
9
Q
Sphericity is
a more general condition of
A
compound symmetry
10
Q
What is compound symmetry?
A
true when both the variances across conditions are equal and the covariances between pairs of conditions are equal
11
Q
Compound symmetry holds
true when both the variances across conditions are equal and the covariances between pairs of conditions are equal
In other words, it means..
A
variation within experimental conditions is fairly similar (similar to homogenity of variance in between) and that no two conditions are any more dependent than any other two
12
Q
Sphereicty is a less restrictive form of
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compound symmetry
13
Q
What does spherecity refer to?
A
equality of variances of the differences between treatment levels.
14
Q
Spherecity means the equality of variances of the differences between treatment levels
E.g.,if you were to take each pair of treatment levels, and calculate the differences between each pair of
scores, then it is
A
necessary that these differences have approximately equal variances.
15
Q
you need at least … conditions for spherecity to be an issue
A
three
16
Q
How is sphereicty assumed in this dataset?
A
17
Q
How is spherecity calculated? - (2)
A
- Calculating differences between between pairs of scores for all treatment levels e.g., A-B, A-C , B-C
- Calculating variances of these differences e.g., variances of A-B, A-C, B-C
18
Q
What does the data from table show in terms of assumption of spherecity (calculated by hand)? - (3)
A
there is some deviation from sphericity because the variance of the differences between conditions A and B (15.7) is greater than the variance of the differences
between A and C (10.3) and between B and C (10.3).
However, these data have local circularity (or local sphericity) because two of the variances of differences are identical.
The deviation from spherecity in the data does not seem too severe (all variances roughly equal) but here assess deviation is serve to warrant an action
19
Q
How to assess the assumption of sphereicity in SPSS?
A
via Mauchly’s test
20
Q
If Mauchly’s test statistic is significant (p < 0.05) then
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variance of differences between conditions are significnatly different - must be vary of F-ratios produced by computer
21
Q
If Mauchly’s test statistisc is non significant (p > 0.05) then it is reasonable to conclude that the
A
varainces of the differences between conditions are equal and does not significantly differ
22
Q
Signifiance of Mauchly’s tes is dependent on
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sample size
23
Q
Example of signifiance of Maulchy’s test dependent on sample size - (2)
A
in big samples small deviations from sphericity can be
significant,
small samples large violations can be non-significant
24
Q
What happens if the data violates the sphereicity assumption? - (2)
A
several corrections that can be applied to
produce a valid F-ratio
or
use multivariate test statistics (MANOVA)
25
What corrections to apply to produce valid F-ratio when data violates sphereicity? - (2)
* Greenhouse-Geisser correction ε
^
* Huynh-Feldt correction
26
The Greenhouse-Geisser correction varies between
1/k (k is number of repeated measures conditions) and 1
27
The closer that Greenhouse Geisser correction is to 1, the
more homogeneous the variances of differences, and hence the closer the data are to being spherical.
28
How to calculate lower-bound estimate fo spherecity for Greenhouse-Geisser correction when there is 5 conditions? - (2)
Limit of f ε^ is 1/k (number of repeated-measures conditions)
so... 1/(5-1) = 1/4 = 0.25
29
Huynh and Feldt (1976) reported that when the
Greenhouse-Geisser correction is too conservative
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Huynh-Feldt correction is less conservative than
Greenhouse-Geisser correction
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when estimates of sphericity are greater than 0.75 (1/k) then the
Huynh–Feldt
correction should be used,
32
when sphericity estimates are less than 0.75 (1/k) or nothing is
known about sphericity at all, then
the Greenhouse–Geisser correction should be used instead
33
Why is MANOVA used when data that violates spherecity?
MANOVA is not dependent upon the assumption of sphericity
34
In repeated measures ANOVA, the effect of our experiment is shown up in within participant variance than
between group variance
35
In independent ANOVA, the within-group variance is our.... and it is not contaimed by... - (2)
residual variance (SSR) = variance produced by individual differences in performance
SSR is not contaimined by experimental effect as study carried out by different people
36
In repeated-measures ANOVA, the within-participant variability is made up of
the effect of experimental manipulation and individual differences in performance (random factors outside of our control) - this is error SSR
37
Similar to independent ANOVA, repeated-measures ANOVA uses F-ratio to - (2)
compares the size of the variation due to our experimental
manipulations to the size of the variation due to random factors
has same type of variances in independent - total sum of squares (SST), model sum of squares (SSM) and a residual sum of squares (SSR)
38
What is the differences between independent ANOVA and repeated-measures design ANOVA?
repeated-measures ANOVA the model and residual sums of squares are both part of the within-participant variance.
39
In repeated-measures ANOVA
If the variance due to our manipulations is big relative to
the variation due to random factors, we get a .. and conclude - (2)
big value of F ratio
we can conclude that the observed results are unlikely to have occurred if there was no effect in the population.
40
To compute F-ratios we first compute the sum of squares which is the following... - (5)
* SST
* SSB
* SSW
* SSM
* SSR
41
How is SST calculated in one-way repeated measures ANOVA?
SST = grand variance (N-1)
42
What is the DF's of SST?
N-1
43
The SSW (within-participant) sum of squares is calculated in one-way repeated ANOVA by...
square of the standard deviation of each participant’s scores multiplied by the number of conditions minus 1, summed over all participants.
44
What is the DF of SSW of one-way repeated ANOVA? - (2)
DF = N(n-1)
number of participants multiplied by the number of conditions minus 1;
45
How is SSM calculated in one-way repeated ANOVA? - (2)
square of the differences between the mean of the participant scores for each condition and the grand mean multiplied by the number of participants tested, summed over all conditions.
do this for each condition grp
46
What is the DF of SSM in one-way repeated ANOVA? - (2)
DF = n-1
n is number of conditions
47
How is SSR calculated in one-way repeated ANOVA?
the difference between the within-participant sum of squares and the sum of squares for the model.
48
What is the DF for SSR in one-way repeated ANOVA?
DF of SSW minus DF of SSM
49
How do we calculated mean squares (MS) and mean residuals (MR) to acalculate F-ratio in one-way repeated ANOVA?
50
SSM tells us how much variation the
model (e.g. the experimental manipulation) explains
51
SSR tells us how much variation
n is due to extraneous factors
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MSM represents the average amount of variation explained by
the model (e.g. the systematic variation),
53
MSR is a gauge of the average amount of variation explained by
extraneous variables (the unsystematic variation).
54
The F-ratio is a measure of the ratio of the varation
explained by the model and the variation explained by unsystematic factors.
55
Calculating F-ratio is the same for one-way repeated-measures design and independent design as..
56
We don't need to use SSB (between-subject variation )to calculate F-ratio in
one-way repeated ANOVA
57
What does SSB represent in one-way ANOVA?
individual differences between cases
58
Not only does sphereicity produces problems for F in repeated measures ANOVA but causes complications for
post-hoc tests
59
When spereicity is violated in one-way repeated ANOVA , what post-hoc test to use and why - (2)
Bonferroni method seems to be generally the
most robust of the univariate techniques,
especially in terms of power and control of the Type I error rate.
60
When sphereicity is not violated in one-way repeated ANOVA, then what post-hoc tests to use?
Tukey can be used
61
In either case where sphereicity is violated or not in one-way repeated ANOVA, a post-hoc test called - (2)
Games–Howell procedure, which uses a pooled error term,
it is more preferable to Tukey’s test.
62
Due to complications of sphereicity in one-way repeated ANOVA,
standard post hoc tests used for independent designs not avaliable for repeated measure designs
63
Why is repeated contrast useful in repeated-measures design especially one-way repeated measures?
evels of the independent variable have a meaningful order e.g., meausred DV at successive time points or adminstered increasing doses of a drug
64
When should Sidack correction as post hoc be selected for one-way repeated ANOVA?
concerned about the loss
of power associated with Bonferroni corrected values.
65
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
what does these SPSS outputs show? - (2)
* Left shows variables represent each level of IV which is animal
* Right shows descriptive statistics - higher mean time to retch when celebrity eating stick insect (8.12)
66
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub)
What does this Mauchly's Test of Spherecity show? - (2)
* P-value is 0.047 which is less than 0.05
* Thus, reject the assumption of spherecity that variances of the differences between levels are equal
67
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub)
What to do if this Mauchly's Test of Spherecity shows assumption of sphereicity is violated..? - (3)
* Since there are 4 conditions, lower limit of ε^ is 1/(4-1) = 0.333 (lower-bound estimate in table)
* SPSS Output 13.2 shows that the calculated value of ε
^ is 0.533.
* 0.533 is closer to the lower limit of 0.33 than it is to the upper limit of 1 and it therefore represents a substantial deviation from sphericity
68
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What does this main ANOVA table show in terms of spherecity assumed? - (2)
- The value of F = 3.97 which is compared against a critical value for 3 and 21 DF and p-value is 0.026
- conclude there is significant difference between 4 animals in their capacity to induce retching when eaten
69
As a reminder, ANOVA never tells us where the and just tells us - (2)
group differences lie
just tells the differences between grps is significant
70
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What has changed and kept the same in the table? - (2)
* The F-ratios are the same across the rows
* the D.F is changed as well as critical value the F-statistic is compared with
71
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
How is adjustments made to DF?
* Adjustment made by multiplying the DF by the estimate of spherecity.
72
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What does the results show in terms of Greenhouse-Geisser correction and Huynh-Fedt..? - (3)
* Observed F statistic not significant using Greenhouse-Geisser ( p> 0.05)
* Greenhouse-Geisser is quite conservative and miss true effects that exist
* Thus, Huynh-Feldt showend F-statistic is still significant as p-value of 0.048
73
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What happens if Greenhouse Geisser is not-significant (p>0.05) and Huynh-Feldt is significant in this example? - (2)
* Taking average of two significant values e.g., 0.063+ 0.048/2 = 0.056
* Thus, go with Greenhouse-Geisser correction and conclude F ratio is non-significant
74
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
What happens if two corrections - Greenhouse and Felt give same conclusion then you can choose which one to
report
75
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
Important to use valid critical value of F - choosing which p-value to report as it potentially makes a difference between making a
Type 1 error (False positive) or not
76
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
This MANOVA table shows (since data violates spherecity) - (2)
MANOVA multivariate tests is significant , p = 0.002 which is less than 0.05
The results supports a decision to conclude that there are significant differences between the time taken to retch after eating different animal
77
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
what does this repeated contrast table show? - (6)
* Contratss of 0 means grp not included in a contrast
* Contrast 1 (labelled Level 1 vs Level 2) ignores the fish eyeball and witchetty grub (as labelled 0 throughout column
* Grps with positive wieght is compared with grps with negative weight
* E.g., first contrast (Level 1 vs Level 2) compares the stick insect (1)with the kangaroo testicle (-1)
* E.g., second contrast (Level 2 vs Level 3) compares kangaroo testicle (1) with fish eye (-1)
* E.g., third contrast (Level 3 vs 4) compared fish eyeball (1) with witchetty grub
78
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
what does this summary table of repeated contrasts show? - (3)
Level 1 vs 2 is stick insect vs kangaroo testicle
Level 2 vs 3 is kangaroo testicle vs fish eyeball
Level 3 vs 4 is fish eyeball vs witchetty grub
* celebrities took significantly longer to retch after
eating the stick insect compared to the kangaroo testicle (Level 1 vs. Level 2) - p-value of 0.002
* Time taken to retch was not significantly different in Level 2 vs 3 and Level 3 vs 4
79
Researcher measures mean time taken for celebrities to retch for each animal (sticky insect, kangaroo testicle, fish eye, witchey grub) - one-way repeated ANOVA
If main effect is not significant in main ANOVA table for this data then significant contrasts in table below should be ... but if MANOVA was significant then... - (2)
ignored
inclined to conclude main effects of animal was significant and proceed with further tetss like contrasts
80
What IV, DV , design and test to use for this research scenario? - (4)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
81
What does LSD correction (post-hoc option in SPSS)
does not actually make any adjustments to p-value in terms of critical value as what post-hoc test should do
82
What does output show? - (3)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
* sales are increasing across the weeks
* Week 1 start at 427.93 and gradually rise by week 4 to 642,28 pounds
* looks like incentives are having an effect and seem to generate higher sales
83
What does this output show in terms of Maulchys Test of Spherecity? - (2)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
* P-value is not significant ( p = 0.080)
* Assumption of spherecity is satisfied so we got equal variances between differences across conditions
84
If Maulchy's test of spherecity is not significant in one-way repeated ANOVA, then which line do we use in main ANOVA table?
85
If Maulchy's test of spherecity is significant in one-way repeated ANOVA, then which line do we use in main ANOVA table?
86
What does this main ANOVA table show? - (2)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
* DF for week is 3 and 5
* Week: F(3,57) = 26.30, p < 0.001 (p = 0.000), eta-squared is 0.58 - large effect
87
What does this Sidmak correction table and table of means show you in this output? - (6)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
* No sig difference betwen W1 and W2
* Sig difference between W1 and W3 = ihigher sales in W3 (538.570) compared to W1 (427.933)
* Sig difference between W1 and W4 = ihigher sales in W3 (642.284) compared to W1 (427.933)
*Not sig diff with W2 and W3
* Sig difference between W2 and W4 , higher sales in W4 (642.284) than W2 (481.388)
* Sig difference between W3 and W4 , higher sales in W4 (642.284) than W3 (538.570)
88
What does this output show in terms of repeated contrasts? - (3)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
* - Did sales increase from W1 to W2? = p = 0.010 significant
- Did sales increase from W2 to W3? = p = 0.030
- Did sales increase from W3 to W4? = p = 0.008
89
What happens if post hoc and contrasts are telling a different story? - contrasts says weekly increase e.g. W1 to W2 increase, W2 to W3 increase , W3 to W4 increase but post-hoc W1 to W3 was increased sig, W1 to W4 was sig increase but W2 to W3 was not - (2)
- Repeated measures design
- One IV (Incentive) , four conditions (week 1, week 2, week 3, week 4)
- One DV (Sales Generated)
- One-way repeated ANOVA
* Post hoc has lack of power due to many multiple comparisons
* By limiting comparisons in contradt we get around problem
90
What effect sizes for one-way repeated ANOVA can be used? - (3)
* Omega squared
* R
* Partial-eta squared
91
How to interpret partial-eta squared? - (3)
η2 = 0.01 indicates a small effect.
η2 = 0.06 indicates a medium effect.
η2 = 0.14 indicates a large effect.
92
Diagram of writing up one-way repeated ANOVA
93
How is omega-squared interpreted? - (2)
Values of omega squared range from 0 to 1.0 and can be interpreted semantically as the percentage of variation in the dependent variable attributable to the independent variable.
If calculation yields a negative number, it is interpreted as 0.
94
How to interpret r effect size?
95
Two-way repeated ANOVA involves
more than one IV
96
What does four-way ANOVA mean?
4 different IV
97
What does 2x3 ANOVA means? - (2)
* IV with 3 levels
* IV with 2 levels
98
What design, IV, DV and test would you to to investigate the follow scenario? - (4)
* Repeated measures design
* Two IVs: alcohol (3 conditions) and sleep (2 conditions)
* DV: Reaction Times
* Two-way repeated measures ANOVA
99
What does this two-way repeated ANOVA SPSS output show? - (2)
* Repeated measures design
* Two IVs: alcohol (3 conditions) and sleep (2 conditions)
* DV: Reaction Times
* Two-way repeated measures ANOVA
* large number for RT means slower RT
* Alcohol seem to have an effect on RT but particularly for 2 pints + no sleep
100
What does this two-way repeated ANOVA SPSS output show for Mauchly's Test of Sphericity? - (2)
* Repeated measures design
* Two IVs: alcohol (3 conditions) and sleep (2 conditions)
* DV: Reaction Times
* Two-way repeated measures ANOVA
* Two p-values: alcohol ( p = 0.00) and alcohol * sleep [ interaction effect] (p = 0.00) -- > sig so assumption of spherecity is violated so report Grenhouse-Geisser values from main ANOVA table
* No p-value for sleep as only 2 conditions and test of sphericity need more than 2
101
What does this two-way repeated ANOVA main table show? - (3)
* Repeated measures design
* Two IVs: alcohol (3 conditions) and sleep (2 conditions)
* DV: Reaction Times
* Two-way repeated measures ANOVA
* Error DF was 38.
* Test of Spherecity was sig --> assumption violated
* Main sig effect of alcohol: F(1.16,22.06) = 51.38, p < 0.001, partial eta-squared = 0.73
* Main sig effect of sleep: F(1,19) = 88.61, p < 0.001, partial-eta-squared = 0.82
* Interaction effect: F(1.15,21.91) = 23.36, p < 0.001, partial-eta squared = 0.55
102
What does this two-way repeated ANOVA output show in post hocs? - Sidmak correction - (4)
* Repeated measures design
* Two IVs: alcohol (3 conditions) and sleep (2 conditions)
* DV: Reaction Times
* Two-way repeated measures ANOVA
* condition 1 and condition 2 which was significant
* Condition 1 vs 3 which was significant
* Condition 2 with Condition 3 was significant
* So all groups differing significantly from each other so interpret from that higher does of alcohol has more impact on RT
103
What does this two-way repeated ANOVA interaction plot show? - (3)
* Repeated measures design
* Two IVs: alcohol (3 conditions) and sleep (2 conditions)
* DV: Reaction Times
* Two-way repeated measures ANOVA
* Interaction effect is there = as line continue they cross
* Most pronouned effect was in alcohol grp 3 (2 pints)
* When alcohol grp 3 had full nights sleep (2), impairs their RT very slightly
* When alcohol grp 3 had sleep deprivation (1) in combination with 2 pints, it impairs RT by a lot --> use simple effect analysis as well as two-way independent ANVOA to see if difference in grp 3 of blue and green line is sig
104
Effect sizes for factorial two-way repeated ANOVA - (2)
* R
* Partial-eta-squared
105
What happens when assumptions vilated in repeated-measures ANOVA? - (2)
Can do non-parametric test called Friedman's ANOVA if only one IV
There is no non-parametric counterpart for more than one IV in repeated design
