PSY 203: Analysis of Variance ANOVA

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PSY 203 Analysis of Variance (ANOVA)

• David Morrison

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Overview of Lectures

• Lecture 15 (Optimistically Today) Principles of
  ANOVA
• No Lecture 16 ANZAC Day
• Lecture 17 Repeated Measures ANOVA and post hoc
  tests
• Lecture 18 Multifactorial ANOVA
• Lecture 19 The interpretation of interaction in
  complex experimental design.

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ANOVALearning Objectives

• Underlying methodological principles of ANOVA
• Statistical principles Partitioning of
  variability
• Summary table for one-way ANOVA

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t-Test vs ANOVAThe Case for Multiple Groups

• t-tests can be used to compare mean differences
  for two groups
• Between Subject Designs
• Within Subject Designs
• Paired Samples Designs
• The test allows us to make a judgement concerning
  whether or not the observed differences are
  likely to have occurred by chance.
• Although helpful the t-test has its problems

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Multiple Groups

• Two groups are often insufficient
• No difference between Therapy X and Control Are
  other therapies effective?
• .05mmg/l Alcohol does not decrease memory ability
  relative to a control What about other doses?
• What if one control group is not enough?

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Multiple Groups 2

• With multiple groups we could make multiple
  comparisons using t-tests
• Problem We would expect some differences in
  means to be significant by chance alone
• How would we know which ones to trust?

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Comparing Multiple GroupsThe One-Way Design
Alcohol Level (Dose in mg/kg)
0 (Control) 10 20
.......

• Independent variable (factor)
• Dependent variable (measurement)
• Analysis Variation between and within conditions

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Analysis of Variance (F test)Advantages

• Provides an omnibus test avoids multiple
  t-tests and spurious significant results
• it is more stable since it relies on all of the
  data ...
• recall from your work on Std Error and T-tests
• the smaller the sample the less stable are the
  population parameter estimates.

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What does ANOVA do?

• Provides an F ratio that has an underlying
  distribution which we use to determine
  statistical significance between groups (just
  like a t-test or a z-test)
• e.g., Take an experiment in which subjects are
  randomly allocated to 3 groups
• The means and std deviations will all be
  different from each other
• We expect this because that is the nature of
  sampling (as you know!)

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The question is

• are the groups more different than we would
  expect by chance?

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How does ANOVA work?

• Instead of dealing with means as data points we
  deal with variation
• There is variation (variance) within groups
  (data)
• There is variance between group means (Exptl
  Effect)
• If groups are equivalent then the variance
  between and within groups will be equal.
• Expected variation is used to calculate
  statistical significance in the same way that
  expected differences in means are used in t-tests
  or z-tests

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What do we mean by variance?

• Population Variance
• The sum of squared deviations of each data point
  from the mean divided by the number of
  observations gives us the average sums of squared
  deviations known as the variance

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Sample Variance

• The sum of squared deviations divided by the
  number of observations -1 gives us the population
  variance estimate from sample data

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Why divide by n-1?

• Samples give us estimates of population
  parameters (population mean and variance)
• Dividing by n underestimates the population
  variance and this is easily demonstrated.

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The set of all the possible samples for n 2
selected from the population 003399 (Actual
mean4 Actual variance 14) The mean is computed
for each possible sample, and the variance is
computed two different ways (1) dividing by n,
which is incorrect and produces a biased
statistic and (2) dividing by n 1, which is
correct and produces an unbiased statistic.
Biased and unbiased estimates of the variance
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Another feature about n-1

• In many statistical tests we sum variances from
  groups and we lose a data point
• this is sometimes referred to as a degree of
  freedom (df).
• the sum of the deviation scores around a mean
  must add up to zero.
• for each sample estimate we therefore lose a
  degree of freedom all numbers on which the
  estimate is based are free to vary except one.
• it also happens to give us the denominator which
  provides us with the correct sample estimate of
  the population parameter

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Sample Variance
Imagine in the data were people who had been
subjected to a job redesign program
Average squared deviation from sample mean
The Total Variance in this study can be
partitioned into 2 parts

• Variance due to the effect of job design

• Variance as a function group membership

Job Satisfaction
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Partitioning of Variability
Within Group Variance for the Black Group
There is a separate variance estimate for the
White group calculated in the same way
The mean is that for the group
Job Satisfaction
Notice how the data for the groups fall in
different parts of the sample as a whole
This could be due to chance or the experimental
effect. Which is it?
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Partitioning of Variability
Gives us the between groups sums of squared
deviations from the mean
2. Squared Deviation of Black group mean from
from grand mean
12/no. of grps-1
1. Squared Deviation of White group mean from
grand mean
Job Satisfaction
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Between and Within Group Variance
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Put it another way

• Ultimately we want to know
• What is the variance within groups?
• Based on this information
• Is the variation of the group averages larger
  than expected by chance based on within group
  information?

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Do we have an Experimental Effect?
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Calculating Variance Components
STEP 1 Sums of squared deviations
Group means are estimates of the population mean
If this is large the groups are very different
As negative numbers are awkward to deal with we
square the deviations
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From the Sums of Squared Deviations about a mean
comes the variance
k number of groups n number of observations in
each group
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The F Ratio

• Between Groups Variance
• Within Groups Variance

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ANOVA Summary Table The F Ratio
Source SS df MS F
Between SSB k - 1 MSB
MSB/MSW Within SSW k(n-1) MSW Total
SST N - 1
Where k number of groups n
number of observations within each group N
total number of observations (subjects) in all
groups
Ho m1 m2 m3....H1 some ms unequal
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The Test Statistic for ANOVA is F but it is
similar to t
Which gives
If there is no experimental effect then
and F 1.0