Between-Groups
ANOVA
Chapter 12
> When to use an F distribution
• Working with more than two samples
> ANOVA
• Used with two or more nominal
independent variables and an interval
dependent variable
Why not use multiple t-tests?
> The problem of too many t tests
• Fishing for a finding
• Problem of Type I error
The F Distribution
> Analyzing variability to compare means
• F = variance between groups
variance within groups
> That is, the difference among the
sample means divided by the average
of the sample variances
Types of Variance
> Between groups: estimate of the
population variance based on
differences among group means
> Within groups: estimate of population
variance based on differences within
(3 or more) sample distributions
Check Your Learning
> If between-groups variance is 8 and
within-groups variance is 2, what would
F be?
Types of ANOVA
> One-Way: hypothesis test including one
nominal variable with more than two
levels and a scale DV
> Within-Groups: more than two samples,
with the same participants; also called
repeated-measures
> Between-Groups: more than two
samples, with different participants in
each sample
Assumptions of ANOVAs
> Random selection of samples
> Normally distributed sample
> Homoscedasticity: samples come from
populations with the same variance
One-Way Between-Groups ANOVA
> Everything about ANOVA but the calculations
> 1. Identify the populations, distribution, and
assumptions.
> 2. State the null and research hypotheses.
> 3. Determine the characteristics of the
comparison distribution.
> 4. Determine the critical value, or cutoff.
> 5. Calculate the test statistic.
> 6. Make a decision.
Step 3. Characteristics
df between  N groups  1
df within  df1  df 2  df 3  ...df last
df1  n1  1
• What are the degrees of freedom?
> If there are three levels of the independent
variable?
> If there are a total of 20 participants in each of
the three levels?
> Step 4:
Critical
Values
Determine Cutoffs for an F Distribution
(Step 4)
Formulae
SStotal   ( X  GM )2
SSwithin   ( X  M )2
SSbetween   ( X  GM ) 2
SStotal  SS within  SSbetween
MS within 
SS within
df within
MS between 
MS between
F
MS within
SS between
df between
Logic behind the F Statistic
> Quantifies overlap
> Two ways to estimate population
variance
• Between-groups variability
• Within-groups variability
The Logic of
ANOVA
The Source Table
> Presents important calculations and
final results in a consistent, easy-toread format
Bringing it All Together
> What is the ANOVA telling us to do
about the null hypothesis?
> Do we reject or accept the null
hypothesis?
An F Distribution
Here the F statistic is 8.27 while the cutoff is 3.86. Do we
reject the null hypothesis?
Making a Decision
> Step 1. Compare the variance (MS) by
diving the sum squares by the degrees
of freedom.
> Step 2. Divide the between-groups MS
by the within-groups MS value.
> Step 3. Compare the calculated F to the
critical F (in Appendix B).
• If calculated is bigger than critical, we have
a significant difference between means
Calculating Effect Size
> R2 is a common measure of effect size
for ANOVAs.
SS between
R 
SS total
2
Post-Hoc Tests to Determine
Which Groups Are Different
> When you have three groups, and F is
significant, how do you know where the
difference(s) are?
• Tukey HSD
• Bonferonni
> A priori (planned) comparisons
Tukey HSD Test
> Widely used post hoc test that uses
means and standard error
M1  M 2
HSD 
sM
MS within
sM 
N
The Bonferroni Test
> A post-hoc test that provides a more strict
critical value for every comparison of
means.
> We use a smaller critical region to make it
more difficult to reject the null hypothesis.
• Determine the number of comparisons we
plan to make.
> Divide the p level by the number of
comparisons.