PSY 203: Analysis of Variance ANOVA II

1
PSY 203 Analysis of Variance (ANOVA) II

• David Morrison
• davidm_at_psy.uwa.edu.au
• 6488-3240

2
Imagine an Experiment

• The research question is Is there a safe level
  of alcohol in the blood for driving?
• What could be a dependent variable?
• What could be the independent variable?
• Design

3
The Dependent Variable

• Actual driving behaviour
• Number of crashes or other infringements
• Observation
• Simulator performance
• Tracking tasks (driving corrections, breaking
  reaction time, attention span, memory)
• Laboratory tasks with face validity
• Short term memory
• Information encoding
• Choice Reaction times

4
Independent Variables

• Dose of Alcohol
• How many levels?
• How much (std drinks, mmg/kg)
• Ingested when
• Possible confounds
• Sex
• Weight
• age

5
What are the hypotheses?

• Where should we look for differences?
• How big do we expect the effects to be?
• Are we making a directional prediction?

6
Design

• Within or between subjects design?
• Will time between administrations will be an
  issue?
• Will Subjects level of practice/experience be an
  issue?
• Match on key variables or Randomize?
• Matching is hard but can be worth it
• Randomise condition allocation (easy but might
  need more subjects)

7
What form of Analysis?

• t, r, or ANOVA?
• Why use ANOVA
• It allows more than two groups to be used
• Its efficient (uses all the data)
• It is effective in identifying signal from noise
  (SEM for all data smaller than SEM from subgroups)

8
A visual representation of the between-treatments
variability and the within-treatments variability
that form the numerator and denominator,
respectively, of the F-ratio. In (a), the
difference between treatments is relatively large
and easy to see. In (b), the same 4-point
difference between treatments is relatively small
and is overwhelmed by the within-treatments
variability.
9
The F Ratio

• Between Groups Variance
• Within Groups Variance

10
The Test Statistic for ANOVA is F
11
The F Distribution
The distribution of F-ratios with df 2,12. Of
all the values in the distribution, only 5 are
larger than F 3.88, and only 1 are larger than
F 6.93.
12
The distribution of t statistics with df 18 and
the corresponding distribution of F-ratios with
df 1,18 Notice that the critical values for ?
.05 are t 2.101 and that F 2.1012 4.41
How to the t and F statistics compare?
13
ANOVA Summary Table The F Ratio
Source SS df MS F
Between SSB k - 1 MSB
MSB/MSW Within SSW N - k MSW Total
SST N - 1
Where k number of groups N total number of
observations (subjects) in all groups
Ho m1 m2 m3....H1 some ms unequal
14
ANOVA Summary Table The F Ratio
Source SS df MS F
Between SSB k - 1 MSB
MSB/MSW Within SSW N - k MSW Total
SST N - 1
Where k number of groups N total number of
observations (subjects) in all groups
Ho m1 m2 m3....H1 some ms unequal
15
ANOVA Summary Table The F Ratio
Degrees of Freedom
Source SS df MS F
Between SSB k - 1 MSB
MSB/MSW Within SSW N - k MSW Total
SST N - 1
Where k number of groups N total number of
observations (subjects) in all groups
Ho m1 m2 m3....H1 some ms unequal
16
An Example
As df WithinN-K we have 15 subjects, (5 in each
group)
As df Btwk-1 we have 3 grps
Source SS df MS Fobt
Between 203.3 2 101.7
22.59 Within 54.0 12 4.5 Total
257.3 14
F Table will show that Fcrit (2,12 a.05) 3.89
As df here is N-1 we have a sample size of 15
17
A portion of the F distribution table. Entries in
roman type are critical values for the .05 level
of significance, and bold type values are for the
.01 level of significance. The critical values
for df 2.12 have been highlighted
18
Effect Size and Anova

• Statistical significance means are mean
  differences due to chance?
• Significance does not tell us about the magnitude
  of the differences
• Effect size for ANOVA is r2 the percentage of
  variance accounted for.
• How much of the diffrences between scores is
  accounted for by the difference between
  treatments?

19
Summary of the ANOVA Technique
Between Group Variance Estimate VB
Between Group Sum of Squares SSB2
Total Variability
Fobt
Within-Groups variance estimate VW
Within Grp sum of squares SSw2
20
Sums of Squared Deviations from the Mean

• Instead of dealing with absolute deviations we
  deal with squared deviations about the mean
• SST SSw SSb

21
Simple Formulae for Sums of Squares

• Total SS
• Between
• Within

Note Ntotal sample size ngroup size kgroup
number
22
Logic of ANOVA(Summary)

• Every variance estimate made is an estimate of
  variance around a true population mean
• The best estimate of the true mean is the SAMPLE
  GRAND MEAN (by virtue of numbers)
• Every deviation from the grand mean has two
  components
• deviation of score from group mean
• deviation of group mean from grand mean
• If there is no experimental effect then the
  deviation from the group mean will be the same as
  the deviation from the grand mean

23
ANOVA LOGIC(SummaryII)

• When the groups come from the same underlying
  population F ratio is equal to 1.00
• When Fgt1.00 we use tables generated from random
  samples taken when there is no experimental
  effect
• Tables indicate how often we should expect F
  ratios that deviate from 1.00 as a function of
  sampling error and provide critical values which
  indicate how often we can expect to find F ratios
  of a certain size when H0 is true

24
Degrees of Freedom (df) and Mean Square (MS)

• Sum of squares (SS) depends very much on the
  number of measurements
• Sum of 10 squared deviations gt sum of 5
• Need to find average Sum of Squared Deviations
• Average SS of single sample Variance
• Average SS after partitioning Mean Squared
  deviation (Mean Square)
• Number that the average is based on df degrees
  of freedom