Lecture 9: One Way ANOVA Between Subjects

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Lecture 9One Way ANOVABetween Subjects

• Laura McAvinue
• School of Psychology
• Trinity College Dublin

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Analysis of Variance

• A statistical technique for testing for
  differences between the means of several groups
• One of the most widely used statistical tests
• T-Test
• Compare the means of two groups
• Independent samples
• Paired samples
• ANOVA
• No restriction on the number of groups

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T-test
Group 1 ?? ?? ?? ?? ?? ??
Group 2 ?? ?? ?? ?? ?? ??
Mean
Mean
Is the mean of one group significantly different
to the mean of the other group?

• t-test H0 - ?1 ?2 H1 ?1? ?2

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F-test
Group 2 ?? ?? ?? ?? ?? ??
Group 3 ?? ?? ?? ?? ?? ??
Group 1 ?? ?? ?? ?? ?? ??
Mean
Mean
Mean
Is the mean of one group significantly different
to the means of the other groups?
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Analysis of Variance
One way ANOVA
Factorial ANOVA
More than One Independent Variable
One Independent Variable
Between subjects
Repeated measures / Within subjects
Two way
Three way
Four way
Different participants
Same participants
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A few examples

• Between subjects one way ANOVA
• The effect of one independent variable with three
  or more levels on a dependent variable
• What are the independent dependent variables in
  each of the following studies?
• The effect of three drugs on reaction time
• The effect of five styles of teaching on exam
  results
• The effect of age (old, middle, young) on recall
• The effect of gender (male, female) on hostility

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Rationale

• Lets say you have three groups and you want to
  see if they are significantly different
• Recall inferential statistics
• Sample Population
• Your question
• Are these 3 groups representative of the same
  population or of different populations?

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Population
Draw 3 samples
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2
Did the manipulation alter the samples to such an
extent that they now represent different
populations?
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Drug 1
Drug 2
Drug 3
Manipulate the samples
DV
µ1
µ2
µ3
measure effect of manipulation on a DV
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Recall sampling error the sampling distribution
of the mean

• The means of samples drawn from the same
  population will differ a little due to random
  sampling error
• When comparing the means of a number of groups,
  your task
• Difference due to a true difference between the
  samples (representative of different
  populations)?
• Difference due to random sampling error
  (representative of the same population)?
• If a true difference exists, this is due to your
  manipulation, the independent variable

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Steps of NHST

• Specify the alternative / research hypothesis
• At least one mean is significantly different
  from the others
• At least one group is representative of a
  separate population
• Set up the null hypothesis
• The hypothesis that all population means are
  equal
• All groups are representative of the same
  population
• Omnibus Ho µ1 µ2 µ3

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Steps of NHST

• Collect your data
• Run the appropriate statistical test
• Between subjects one way ANOVA
• Obtain the test statistic associated p-value
• F statistic
• Compare the F statistic you obtained with the
  distribution of F when Ho is true
• Determine the probability of obtaining such an F
  value when Ho is true

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Steps of NHST

• Decide whether to reject or fail to reject Ho on
  the basis of the p value
• If the p value is very small (lt.5), reject Ho
• Conclude that at least one sample mean is
  significantly different to the other means
• Not all groups are representative of the same
  population

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How is ANOVA done?

• Assume Ho is true
• Assume that all three groups are representative
  of the same population
• Make two estimates of the variance of this
  population
• If Ho is true, then these two estimates should be
  about the same
• If Ho is false, these two estimates should be
  different

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Two estimates of population variance

• Within group variance
• Pooled variability among participants in each
  treatment group
• Between group variance
• Variability among group means

If Ho is true Between Groups Variance Within
Groups Variance 1
If Ho is false Between Groups Variance Within
Groups Variance gt 1
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Calculations

• Step
• 1 Sum of squares
• 2 Degrees of freedom
• 3 Mean square
• 4 F ratio
• 5 p value

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Total Variance In data SStotal




Within groups Variance SSwithin
Between groups variance SSbetween



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SStotal

• ? (xij - Grand Mean )2
• Based on the difference between each score and
  the grand mean
• The sum of squared deviations of all
  observations, regardless of group membership,
  from the grand mean

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SSbetween

• n? (Group meanj - Grand Mean )2
• Based on the differences between groups
• Related to the variance of the group means
• The sum of squared deviations of the group means
  from the grand mean, multiplied by the number of
  observations in each group

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SSwithin

• ? (xij - Group Meanj )2
• Based on the variability within each group
• Calculate SS within each group add
• The sum of squared deviations within each group
  or
• SStotal - SSbetween

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Degrees of Freedom

• Total variance
• N 1
• Total no. of observations - 1
• Between groups variance
• K 1
• No. of groups 1
• Within groups variance
• k (n 1)
• No. of groups (no. in each sample 1)
• Whats left over!

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Mean Square

• SS / df
• The average variance between or within groups
• An estimate of the population variance
• MSbetween
• SSgroup / dfgroup
• MSwithin
• SSwithin / dfwithin

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F Ratio
MSbetween MSwithin
If Ho is true, F 1
If Ho is false, F gt 1
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MSbetween MSwithin
Treatment effect Differences due to
chance Differences due to chance
F
If treatment has no effect
0 Differences due to chance Differences due to
chance
F
1
If treatment has effect
EFFECT gt 0 Differences due to
chance Differences due to chance
gt 1
F
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MSBG
MSBG
MSBG
MSWG
MSWG
MSWG
Variance within groupsgt variance between
groups Flt1 Fail to reject Ho If there is more
variance within the groups, then any difference
observed is due to chance
Variance within groups Variance between
groups F 1 Fail to reject Ho If both sources of
variance are the same, then any difference
observed is due to chance
Variance within groups lt variance between
groups F gt1 Reject Ho The more the group means
differ relative to each other the more likely it
is that the differences are not due to chance.
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Size of F

• How much greater than 1 does F have to be to
  reject Ho?
• Compare the obtained F statistic to the
  distribution of F when Ho is true
• Calculate the probability of obtaining this F
  value when Ho is true
• p value
• If p lt .05, reject Ho
• Conclude that at least one of your groups is
  significantly different from the others

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ANOVA table
Source of variation SS df MS F p
Between groups n? (Group meanj - Grand Mean )2 K - 1 SSBG / dfBG MSBetween MSWithin Prob. of observing F-value when Ho is true
Within groups ? (xij - Group Meanj )2 K(n 1) SSWG / dfWG
Total ? (xij - Grand Mean )2 N - 1
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A few assumptions

• Data in each group should be
• Interval scale
• Normally distributed
• Histograms, box plots
• Homogeneity of variance
• Variance of groups should be roughly equal
• Independence of observations
• Each person should be in only one group
• Participants should be randomly assigned to
  groups

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Multiple Comparison Procedures

• Obtain a significant F statistic
• Reject Ho conclude that at least one sample
  mean is significantly different from the others
• But which one?
• H1 µ1 ? µ2 ? µ3
• H2 µ1 µ2 ? µ3
• H3 µ1 ? µ2 µ3
• Necessary to run a series of multiple comparisons
  to compare groups and see where the significant
  differences lie

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Problem with Multiple Comparisons

• Making multiple comparisons leads to a higher
  probability of making a Type I error
• The more comparisons you make, the higher the
  probability of making a Type I error
• Familywise error rate
• The probability that a family of comparisons
  contains at least one Type I error

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Problem with Multiple Comparisons

• ?familywise 1 - (1 - ?)c
• c number of comparisons
• Four comparisons run at ? .05
• ?familywise 1 - (1 - .05)4
• 1 - .8145
• .19
• You think you are working at ? .05, but youre
  actually working at ? .19

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Post hoc tests

• Bonferroni Procedure
• ? / c
• Divide your significance level by the number of
  comparisons you plan on making and use this more
  conservative value as your level of significance
• Four comparisons at ? .05
• .05 / 4 .0125
• Reject Ho if p lt .0125

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Post hoc tests

• Note Restrict the number of comparisons to the
  ones you are most interested in
• Tukey
• Compares each mean with each other mean in a way
  that keeps the maximum familywise error rate to
  .05
• Computes a single value that represents the
  minimum difference between group means that is
  necessary for significance

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Effect Size

• A statistically significant difference might not
  mean anything in the real world

Eta squared
Percentage of variability among observations that
can be attributed to the differences between the
groups
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A little less biased
Omega squared
How big is big? Similar to correlation
coefficient
Cohens d When comparing two groups
Meantreat Meancontrol SDcontrol