One-way Analysis of Variance
1-Factor ANOVA
Previously…
• We learned how to determine the probability
that one sample belongs to a certain
population.
• Then we learned how to determine the
probability that two samples belong to the
same population.
Now…
• We will learn how to determine the
probability that two or more samples were
taken from the same population.
• While the t tests use standard deviation units
(standard error of the mean and standard
error of the difference), this new analysis uses
variance.
One-Way Analysis of Variance
• One-way (or one factor) just means that we are
looking at the effect of one independent variable.
• With an ANOVA, we can partition the variance
into categories to determine how much of the
variance is due to our experimental procedure
and how much is due to individual differences or
experimental error.
• Instead of a t score, the ANOVA produces an F
score, which has its own table.
One-way ANOVA Hypothesis
• The null hypothesis for the ANOVA with three
groups is:
• H0: μ1 = μ2 = μ3
• The alternative hypothesis is:
• H1: μ1 ≠ μ2 ≠ μ3
One-way ANOVA Hypothesis
• Notice, at the end of the ANOVA, we cannot
say which group’s results are responsible for
rejecting the null.
• To do that, we have to conduct a post hoc
analysis, which we will get to later.
Terms
refers to a raw score from a group (g).
refers to the mean of a group.
refers to the mean of all scores in all groups
or the “grand mean.”
Key Deviations
• Remember we use deviation scores to find
sum of squares (SS) and SS to find variance.
The mean score for a group minus the
grand mean.
A raw score in a group minus the
group mean.
A raw score minus the grand mean.
Picturing the Variability
• If you have only a little within-groups variability,
but a lot of between groups variability, it is easy
to see that there is an effect.
• If you have a lot of within groups variability, but
only a little between groups variability, it is hard
to see that there is an effect.
• See p. 241.
Picturing the Variability
Between Groups
Variability
Total Variability
Within Groups
Variability
Three Sources of Variability
1. Individual Differences
Within Groups Variability
2. Experimental Error
3. Treatment Effect
Between Groups Variability
Sum of Squares
• To run the ANOVA, we are going to first find the
sum of squares for each variance of interest.
Total SS:
Within-groups SS:
Between-groups SS:
Between-groups SS
• This one might need a little explaining:
• All it means is that you subtract the grand mean
from the group mean and square the result, then
multiply it times the group N. Do this for each
group and add them up.
Step 1: Find SS
X
Group 1
Mean = 5
Group 2
Mean = 9
Grand Mean = 7
2
(2-7)2 = 25
(2-5)2 = 9
(5-7)2 = 4
4
(4-7)2 = 9
(4-5)2 = 1
(5-7)2 = 4
8
(8-7)2 = 1
(8-5)2 = 9
(5-7)2 = 4
6
(6-7)2 = 1
(6-5)2 = 1
(5-7)2 = 4
8
(8-7)2 = 1
(8-9)2 = 1
(9-7)2 = 4
9
(9-7)2 = 4
(9-9)2 = 0
(9-7)2 = 4
8
(8-7)2 = 1
(8-9)2 = 1
(9-7)2 = 4
11
(11-7)2 = 16 (11-9)2 = 4
(9-7)2 = 4
Step 2: ANOVA Summary Table
SOURCE
SS
df
MS
F
Between
SSb
dfb = K – 1 *
SSb/dfb = MSb
MSb/MSw = F
Within
SSw
dfw = N - K
SSw/dfw = MSw
Total
SStot
dftot = N - 1
K = # of groups
Step 2: ANOVA Summary Table
SOURCE
SS
df
MS
F
Between
SSb = 32
dfb = K – 1 = 1
SSb/dfb = MSb
32/1 = 32
MSb/MSw = F
32/4.33 = 7.38
Within
SSw = 26
dfw = N – K = 6
SSw/dfw = MSw
26/6 = 4.33
Total
SStot = 58
dftot = N – 1 = 7
Step 3: Find Fcrit
• Look at Table C in Appendix 4
• Find the column associated with dfb
• Find the row associated with dfw
• In our example, the critical value at the .05
level is 5.99, and 13.7 at the .01 level.
Step 4: Make a Decision
• If we set α = .05, we can reject the null
because Fobt (7.38) is bigger than Fcrit (5.99).
• If we set α = .01, we retain the null because
Fobt (7.38) is smaller than Fcrit (13.7).
Step 5: Interpret the Results
• Because the obtained F is larger than the critical
value at the .05 level, we reject the hypothesis
that the samples came from the same population
and conclude that the treatments varied in their
effectiveness.
• Because the obtained F is smaller than the critical
value at the .01 level, we retain the hypothesis
that the samples came from the same population
and conclude that the treatments did not vary in
their effectiveness.
What is F?
• F is the ratio of variability between groups to variability
within groups. (MSb/MSw)
• If the samples came from the same population, we
would expect both between and within group
variability to be about the same, so we would expect F
to be 1.
• If the samples do not come from the same population,
we would expect the between group variability to be
greater than the within group variability, which would
make F bigger (because between group variability is in
the numerator).
What is F?
• Also, just in case you were wondering, if you
ran an independent samples t test on the data
in our example (remember we had only two
groups), you would get the same exact results.
• This is because the math behind the 1-way
ANOVA and the 2-sample t test is the same.
Activity #1
Number of tasks completed:
• Group 1 (downers) = 4, 1, 5, 2
• Group 2 (placebo) = 5, 6, 8, 9
• Group 3 (uppers) = 8, 7, 9, 8
Now What?
• We rejected the null, so we are pretty sure
that μ1 ≠ μ2 ≠ μ3. In other words, we are pretty
sure that people in one of the groups are
somehow different from people in another
group, but we don’t know which groups.
• We need to do a post hoc analysis. There are
many of them, but we will talk about the
Fisher LSD.
Fisher LSD (Least Significant
Difference) Test
• The reason we don’t just run a bunch of t tests when
we have more than two groups to compare is that we
increase the probability of Type I Error (incorrectly
rejecting the null).
• Why is this?
• Because, the probability of finding a difference
between groups 1 and 2 OR groups 1 and 3 OR groups
2 and 3 is much greater than the probability of finding
a difference between any two of the groups
(remember the addition rule of probability).
Fisher LSD (Least Significant
Difference) Test
• The Fisher LSD increases the difference required
to find a significant result.
• Here is the formula:
• Get t from the table using df = N – K.
• N1 and N2 are the sample sizes for the first and
second samples we are comparing.
Fisher LSD (Least Significant
Difference) Test
• For three groups, you will have three
comparisons:
– Group 1 mean – Group 2 mean
– Group 1 mean – Group 3 mean
– Group 2 mean – Group 3 mean
Fisher LSD (Least Significant
Difference) Test
• If all of your sample sizes are the same, you
will only need to compute LSD once.
• Remember what LSD means, and it will make
sense. Least significant difference means the
smallest difference between the sample
means that can be significant.
Fisher LSD (Least Significant
Difference) Test
• If the difference between the group means
you are comparing is at least as large as your
obtained LSD, then the difference between
those groups is “significant” at the alpha level
you used.
Activity #2
• Using α = .01, conduct a LSD post hoc analysis
on our activity #1 data.
• Interpret the results.
Homework
• Study for Chapter 11 Quiz
• Read Chapter 12
• Do Chapter 11 HW