1. Lecture 9: One Way ANOVA Between Subjects
Laura McAvinue
School of Psychology
Trinity College Dublin

2. 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

3. Is the mean of one group significantly different to the
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

4. 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?

5. 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

6. 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

7. Rationale
Let’s 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?

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

9. 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

10. 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

11. Steps of NHST F statistic 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

12. 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 (<.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

13. 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

14. 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
> 1

15. Calculations Step… 1: Sum of squares 2: Degrees of freedom
3: Mean square
4: F ratio
5: p value

16. Total Variance In data
SStotal
Within groups
Variance
SSwithin
Between groups
variance
SSbetween

17. 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

18. 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

19. 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

20. Degrees of Freedom Total variance Between groups variance
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)
What’s left over!

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

22. F Ratio MSbetween MSwithin If Ho is true, F = 1
If Ho is false, F > 1

23. F F 1 > 1 F 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 > 0 + Differences due to chance
Differences due to chance
> 1 F

24. MSBG MSBG MSBG MSWG MSWG MSWG
Variance within groups>
variance between groups
F<1
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 <
variance between groups
F >1
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.

25. 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 < .05, reject Ho
Conclude that at least one of your groups is significantly different from the others

26. 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

27. 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

28. 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

29. 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

30. Problem with Multiple Comparisons
familywise = 1 - (1 - )c
c = number of comparisons
Four comparisons run at  = .05
familywise = 1 - ( )4 =
= .19
You think you are working at  = .05, but you’re actually working at  = .19

31. Post hoc tests Bonferroni Procedure Four comparisons at  = .05  / 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 < .0125

32. 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

33. 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

34. When comparing two groups Meantreat – Meancontrol SDcontrol
A little less biased…
Omega squared
How big is big? Similar to correlation coefficient
Cohen’s d
When comparing two groups
Meantreat – Meancontrol
SDcontrol