ANOVA Analysis of Variance

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ANOVA(Analysis of Variance)

• Essentially you have already encountered two
  types of test (relationships/correlations and
  tests of differences.)
• T-test is a parametric test of difference
  allowing the statistical comparison of two
  conditions of groups. It meets the criteria for
  parametric analysis. Do you remember what they
  are?

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Criteria for parametric analysis

• Real numbers (interval level, continuous scale)
• Normal distribution (test with a histogram)
• Homogeneity of variance (test with Levenes test
  or Box M test)

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A reminder on Design Issues

• Between Subjects- Different groups of people
  take part in separate conditions (also called
  independent groups sometimes)
• Within Subjects- The same people take part in
  more than one condition (also called dependent
  groups)

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Terminology

• IV /DV - the influence on, and outcome of, an
  experimental design respectively
• Effect- a difference/divergence or strength of
  relationship
• Treatment effect- the effect associated with the
  condition(s) administered the IV(s)
• Random error- chance fluctuations due to
  measurement, individual differences, etc.

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Basic ideas Central tendency and variability

• Median 1, 3, 5, 7, 9 5
• Mean 13579 25/N 5
• Mode most frequent, may be bi-modal or
  multi-modal
• Range 1,3,5,7,9 diff btwn 1 9 8
• Variance S2 mean of squared deviation scores
• Standard Deviation square root of the variance
  / deviation of scores around the mean

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ANOVA (analysis of variance)

• Questions of difference for more than 2 conditions

ANOVA
t-test
Allows comparison of 3 or more conditions
Condition 1
Condition 2
Condition 3
Think of the levels of the IV of age you might
use in a study
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Single variables
Age
Young
Middle
Old
What other independent variable might you be
interested in comparing with age?
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Multiple variables
How many IVs are there? What are the levels of
each?
Age
A multivariate design 2 x 3
Young
Middle
Old
Male
Gender
Female
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A multivariate design
Age
Young
Middle
Old
Time of day 1-4
Male
Gender
Female
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Testing differences with multiple
condition/variables
Parametric
T-test
ANOVA 1 - way
ANOVA 2-way
Nonparametric
There isnt a comparable non-parametric test for
two IVs
Kruskal wallis/ Friedman
Mann-Whitney/ Wilcoxon
Study on Environment and innate intelligence
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Study looking at the effect of environment on
different strains of rats 2 way independent
ANOVA

• IV 1 type of environment
• Levels of IV1 free and restricted
• IV 2 strain of rat
• Levels of IV 2 dim, bright and mixed

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The final result
F - ratio
Variance due to effect
Variance due to error
Degrees of freedom
Probability
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The final result
Variance due to error
F - ratio
Variance due to effect
Probability
Degrees of freedom
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Homogeneity of variance

• ANOVA (parametric) depends upon variance to tell
  us about difference between means so similarity
  of variances in conditions is important
• A Box M test is used for multiple variable
  designs and we look for (the opposite of a normal
  inferential test) a non sig. result no diff in
  variances

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Post hoc tests

• Multiple testing is a possibility e.g. t-test
• E.g. Compare A with B A with C B with C three
  comparisons each at .05 .15 of finding a sig.
  Result
• Aim of post hocs is to identify the differences
  between conditions/groups more conservatively and
  avoid multiple testing

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Repeated MeasuresOne Way ANOVA

• Because the participants in each condition are
  the same people this reduces some of the
  unsystematic variation that exists in between
  participants designs.
• This makes these tests more sensitive and
  powerful.
• However as well as the homogeneity of variance
  assumption we have the criteria of SPHERICITY
  that has to be met.

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Sphericity

• The variance of the difference between pairs of
  scores are equal for all groups (i.e. A-B A-C
  B-C) Read Field p324 for more details.
• It is tested for by Mauchlys test which (like
  Levenes) you dont want to be significant.
• However if it is significant then there is a
  correction called the Greenhouse-Geisser and it
  is this row in the output that you should use.

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Descriptive statistics
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2-way repeated measures ANOVA

• Imagine that we are investigating the effect of 2
  IVs on 1 DV
• e.g. the effect of alcohol and also eating a
  kebab on the DV on vomiting at end of night
• Or the effect of gender and also the amount of
  violence in a film on the DV of how much the film
  was liked

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Mean interaction plot for drinking and kebab
eating.
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ANOVA summary table
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Mean interaction plot for gender and violence on
liking a film
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ANOVA summary table
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Mixed design ANOVAs

• This is when one or more variables are between
  subjects AND
• One or more variables is within subjects
• Any combination of between and within subjects
  variables is termed mixed

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Mixed Anova

• In this type of Anova one of the IVs is between
  participants and the other is a
  within-participant variable.
• For example you could investigate the effects of
  alcohol on driving but want to know if gender was
  also an important factor.
• In this case gender would be a between-participant
  IV (Obviously!) but alcohol could be a
  within-participant variable (in this case with 3
  levels sober, a bit drunk and completely psed)
• This would be a 2 x 3 mixed design.

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Some reminders

• ANOVA tests multiple conditions/ IVs
• All IVs between subjects between subjects ANOVA
• All IVs within subjects within subjects ANOVA
• Some of each mixed ANOVA design
• Different sources of variance are analysed to see
  if IVs have affected the DV

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• Interaction effects one of the key advantages
  of ANOVAs
• Sphericity repeated measures or within subjects
  designs (only with 3 or more levels)
• Homogeneity of variance (Levenes) for between
  subjects designs
• Type I error
• Type II error
• Avoiding family-wise error using post hocs