Lecture 13 - ANOVA 2 Flashcards
Between-group ANOVA (reminder)
ANOVA is based on:
The variance explained by the experiment (the effect)
The residual (remaining) variance that cannot be explained (noise)
For between-groups design, the variance comes from only two sources:
(1) Variance between groups (explained)
(2) Variance within groups (unexplained)
Repeated measures vs between-group
For repeated-measures designs, there are three possible sources of variance:
(1) Variance between conditions
(2) Variance (within a condition) between subjects (individual differences) – can control for
(3) Residual (unexplained) variance
In the between-groups study, the variance between subjects fell under the category ‘unexplained’
We want to know about the ratio of the effect to noise (blue area divided by red area)
Explicitly calculated in the ANOVA
The ratio is larger after removing the inter-subject variance
F-ratio
F = MSexplained / MSunexplained
MSexplained is the variance between conditions
MSunexplained is the remaining variance after accounting for individual differences:
MSunexplained = MStotal - MSexplained - MSind diffs (total = variance of every participants’ score in every condition)
MSind diffs is the variance between subjects within a condition
MStotal is the variance of all subjects in all conditions
Example (repeated measures)
Objective: we want to know how well children transfer learning in a task
Experiment: we create three versions (A, B, C) of a (related) problem-solving task and see how long they take to solve it
Measures:
time taken in first task as baseline
time taken to solve after explaining the task (teaching phase)
time taken on the third version with no further coaching (transfer)
Note: counterbalance tasks to make sure there is no difference in task difficulties
Possible outcomes
No significant difference between conditions would mean that children did not learn
Significant difference caused by short times in the teaching phase would suggest learning
Significant difference caused by short times in both teaching and transfer phase would imply good transfer of learning
We mostly care about the comparison between the baseline and final (transfer phase): we conduct an ANOVA on the whole set and then a single, planned t-test comparing baseline and transfer (so don’t need Bonferroni corrections)
SPSS example
One row for each participant in the study
Counterbalancing: 6 ways to arrange ABC. Exactly equal number of participants were run each way
Plotting the data: graphs -> error bar plot (simple plot: 1 IV, summaries of variables: want to compare different columns of data)
Running the ANOVA: analyse -> general linear model -> repeated measures (type a name for the factor and how many levels then hit ‘Add’, then hit ‘Define’, move baseline, teaching and transfer into ‘stage’ variables then hit ‘OK’)
Output
Sphericity assumed = think we’ve got similar variance in each level of the variable
Other rows are corrections if data doesn’t have equal variance
The ANOVA showed there was an effect of learning through the stages (because not all times were the same)
We can now run a within-subjects t-test to see whether the final stage was different to the first (that would imply transfer from the teaching phase):
Reporting results
Figure 1 = error bar plot
“Figure 1 shows the mean time taken by the participants to complete the task in the three phases of the experiment (error bars indicate 1 SEM). The data were analyzed with a one-way ANOVA, which showed that performance was dependent on which stage the child was in (F2,22=29.1, p<.001).
A single planned t-test showed that, in particular, children were faster in the final stage of the experiment than the first, indicating significant transfer of learning (t11=5.25, p<.001).”
Multi-factorial ANOVA
Similar in spirit to repeated-measures ANOVA
Factors can all be within-subjects, all between-group or a ‘mixed’ design
We can have a ‘main’ effect or a variety of ‘interactions’
Main effect = one of the factors (IVs) consistently effects the DV in the same way
Interaction = the effect of one factor depends on the presence of another
2x2 ANOVA
The multi-factorial ANOVA is a single test (two IVs – things being manipulated)
It returns multiple F values (one for each main effect to be checked and one for the interaction)
With only two levels, there is no need for post-hoc tests
So 2x2 is just a single test (no family-wise error)
The equivalent full comparisons using t-tests would be to check all 6 combinations of the 4 conditions (AB, AC, AD, BC, BD, CD) – would have reduced power
More factors
In the real world many things often vary together
ANOVA would seem a convenient way to take all factors into account at once: it yields an F-ratio and a significance value for any interactions you may be interested in
But interpreting a 4-way interaction with 3 levels in each factor is hard
There is a trade-off between being realistic and being able to interpret data
Maths part is easy, interpretation part is difficult
The General Linear Model (most general form of ANOVA)
Enables the analysis of any number of factors (IVs) and their interactions
Works by comparing variance explained with variance unexplained
Requires parametric data
