# Between-Groups ANOVA ```Between-Groups
ANOVA
Chapter 12
&gt; When to use an F distribution
• Working with more than two samples
&gt; ANOVA
• Used with two or more nominal
independent variables and an interval
dependent variable
Why not use multiple t-tests?
&gt; The problem of too many t tests
• Fishing for a finding
• Problem of Type I error
The F Distribution
&gt; Analyzing variability to compare means
• F = variance between groups
variance within groups
&gt; That is, the difference among the
sample means divided by the average
of the sample variances
Types of Variance
&gt; Between groups: estimate of the
population variance based on
differences among group means
&gt; Within groups: estimate of population
variance based on differences within
(3 or more) sample distributions
&gt; If between-groups variance is 8 and
within-groups variance is 2, what would
F be?
Types of ANOVA
&gt; One-Way: hypothesis test including one
nominal variable with more than two
levels and a scale DV
&gt; Within-Groups: more than two samples,
with the same participants; also called
repeated-measures
&gt; Between-Groups: more than two
samples, with different participants in
each sample
Assumptions of ANOVAs
&gt; Random selection of samples
&gt; Normally distributed sample
&gt; Homoscedasticity: samples come from
populations with the same variance
One-Way Between-Groups ANOVA
&gt; Everything about ANOVA but the calculations
&gt; 1. Identify the populations, distribution, and
assumptions.
&gt; 2. State the null and research hypotheses.
&gt; 3. Determine the characteristics of the
comparison distribution.
&gt; 4. Determine the critical value, or cutoff.
&gt; 5. Calculate the test statistic.
&gt; 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?
&gt; If there are three levels of the independent
variable?
&gt; If there are a total of 20 participants in each of
the three levels?
&gt; 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
&gt; Quantifies overlap
&gt; Two ways to estimate population
variance
• Between-groups variability
• Within-groups variability
The Logic of
ANOVA
The Source Table
&gt; Presents important calculations and
final results in a consistent, easy-toread format
Bringing it All Together
&gt; What is the ANOVA telling us to do
&gt; 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
&gt; Step 1. Compare the variance (MS) by
diving the sum squares by the degrees
of freedom.
&gt; Step 2. Divide the between-groups MS
by the within-groups MS value.
&gt; 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
&gt; 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
&gt; When you have three groups, and F is
significant, how do you know where the
difference(s) are?
• Tukey HSD
• Bonferonni
&gt; A priori (planned) comparisons
Tukey HSD Test
&gt; Widely used post hoc test that uses
means and standard error
M1  M 2
HSD 
sM
MS within
sM 
N
The Bonferroni Test
&gt; A post-hoc test that provides a more strict
critical value for every comparison of
means.
&gt; 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.
&gt; Divide the p level by the number of
comparisons.
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