PSY 203

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PSY 203

• ANOVA 4 Complex Designs and Interactions

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Overview

• More complex designs
• More complex analyses
• More complex results

But its mostly an extension of what you already
know.
Except when multiple factors interact with each
other to give us
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Results
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Two Main effects
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Sleep Score
x
20
Conditions a b
x
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Cond1 Cond2
Both time of day and exercise type seem to be
influential
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Interactions
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Levels of Processing

• Stimuli aspen linden scotch
  pyjama bandana bison
• Encoding Task 1997 1998 1999 2001
• Same number of vowels? 15.12 18.38 12.05
  10.5
• Rhyme? 23.47 22.84 19.16 19.9
• Same category? 33.46 34.62 33.98 33.5

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Multi-Factorial ANOVA

• With the Levels of Processing data we could have
  lined up the 3 conditions for 97, 98 99 2001
  and done a one way ANOVA (12 conditions)
• This would lead to a complicated set of follow up
  tests if we wanted to examine the data for
  systematic differences between years.

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Breaking down the variance into additional
components

• Instead, analysing the data by its variance
  components (again) we are able to examine the
  effects of year and experimental condition
  separately and together (called the
  interaction)
• The variance components in these data are
• Variation due to Year
• Variation due to Condition
• Variation due to the interaction of Year and
  Condition
• Variation Within (natural) groups

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Specifying the Design

• A One-way ANOVA is so called because it tests for
  the significance of differences in one way -
  across the levels of a single factor
• In the present example we could look at level of
  processing
• Or, year separately
• To do so would be crude and ignore the
  possibility that a more complex analysis would
  reveal a year by level effect (known as the
  interaction)
• A Two (or N) - way ANOVA examines the effects of
  variables in two (or N) ways across levels of the
  two factors (e.g. processing type by year is a
  3x4 design)

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Multi Factorial ANOVA also Tests for INTERACTION

• Interaction Is the effect of Factor A is
  dependent on the influence of Factor B
• Interaction between factors -
• that the difference in the means of groups for
  one factor (e.g., word type) vary as a function
  of the level of a second (or Nth) factor (e.g.,
  year).
• Interaction - do the differences between
  processing conditions depend on the year that the
  data were collected?

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Another exampleEffects of exercise on sleep

• A 2 x 2 design (simplest multi-factorial design)
• Exercise intensity 2 levels (light or heavy)
• Time of day 2 levels (morning or evening)
• We may have three effects
• effect of exercise intensity
• time of day
• interaction between time of day and type of
  exercise regime

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Example Sleep Scores and Exercise Intensity and
Time of Day
Factor B Exercise Intensity Light (b1) Heavy
(b2)
......
Morning (a1) Factor ATime of Day Evening
(a2)
a1b1
......
a2b2
.
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Interpreting Two-Way Designs IFactor A and B
significant
Factor B Exercise Intensity Light (b1) Heavy
(b2)
Row Means
Morning (a1) Factor ATime of Day Evening
(a2) Column Means
Both time of day and exercise type seem to be
influential
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Two Main effects
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Sleep Score
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Morning Evening
Both time of day and exercise type seem to be
influential
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Interpreting Two-Way Designs IINeither Factor
significant
Factor B Exercise Intensity Light (b1) Heavy
(b2)
Morning (a1) Factor ATime of Day Evening
(a2) Means
Column and row means equal but are masking a
strong effect
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Plotting the Results for each condition
Sleep Score
Morning Evening
The effect of exercise depends on the time it is
taken
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Interpreting Two-Way Designs IIIInteractions vs
Main Effects

• In Case 1 there will be a significant effect of a
  (time of day) and of b (exercise intensity)

• In Case II there is no effect of a (time of day)
  or b (exercise intensity) but a large interaction
  effect

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Other forms of interaction
1 main effect of a interaction averaging within
b will give roughly the same means for b1 b2.
Averaging within a will show the main effect of a
1 main effect of b interaction averaging
within a would give roughly same means for a1
a2 . Averaging within b will show the main effect
of b
Follow-up apriori or post hoc t-tests can be used
to identify where the interaction effect lies
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Structure of the analysis for a two-factor
analysis of variance.
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Two-Way DesignANOVA Summary Table
Three F ratios one for each effect
Effect df
Source SS df MS F
Between Factor A SSA a - 1 MSA
MSA/MSW Factor B SSB b - 1 MSB
MSB/MSW A x B SSAxB (a - 1)(b -
1) MSAxB MSAxB/MSW Within SSW
ab(ncell - 1) MSW Total SST N - 1
Where a number of levels of Factor A b
number of levels of Factor B N total number of
observations (subjects) across all cells ncell
number of observations in each cell
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Effect Size for a 2 Factor Anova

• For Factor A
• For Factor B
• For AxB

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Are all the groups different?

• The F test is an omnibus test that does not tell
  us where the differences lie
• We could compare each group with a standard
  t-test but this is inefficient and does not use
  all the data
• Usually we formally compare between individual
  groups using special types of t-tests
• Planned (a priori) and unplanned (post hoc)
  comparisons

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The Scheffe Test

• Similar to Tukeys HSD but VERY conservative.
• Step 1 Determine the SS between treatments for
  the two groups being compared
• Step 2 Take the SS for the comparison grps and
  calculate MSbetween using the df for the entire
  effect (a-1, b-1 or (a-1)(b-1)) from which the
  means have been taken note this makes the
  numerator smaller)
• Step 3 Compute F ratio and examine F table with
  appropriate degrees of freedom (a-1, b-1
  (a-1)(b-1)), whatever the DF is for the MS within
  treatments (N-k )

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An Alternative Scheffe Test

• Apply the above formula (sw2 MSwithin) from the
  ANOVA table.
• Consult F table and find critical F for dfk-1
  and dfN-k (kdf for effect)
• Calculate F (k-1) critical F
• If FgtF then reject Ho

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In Review