Analysis of Variance: ANOVA
Group 1: control group/ no ind. Var.
Group 2: low level of the ind. Var.
Group 3: high level of the ind var.
If you evaluated the differences using a
t test:
group 1 vs. group 2
group 2 vs. group 3
group 1 vs. group 3
What is the probability of a type one error
somewhere in this analysis?
Type One error: rejecting the null when there
really isn’t any effect of the independent
variable.
Reject the null
The probability of a type one error in
a single comparison is whatever the
probability of a random event is.
P < .05
or
5% of the time
Probability over three comparisons:
type one error in comparison one
or
comparison two
or
comparison three
Probability of type one error:
.05 + .05 + .05 = .15
or
15% of the time
This inflation of the type one error rate is
known as:
Experimentwise error rate
To hold down the experimentwise
error rate we perform an analysis of
variance (ANOVA) instead of
multiple t tests.
variance between groups
F = --------------------------------------variance within groups
Var. between : how much each
group mean differs from all
other group means.
Var. within: how much
individuals within the groups
vary from each other.
What causes variability between:
treatment
+
random events
independent variable
confounding factors
What causes variability within:
Random events (individual differences)
treatment + random events
F = ------------------------------------------------
random events
When the only thing that has caused
the groups to differ is random events,
F should be a value very close to 1
S
2
variance
X  X


n 1
2
Sum of squares
(SS)
Degrees of
freedom (df)
In ANOVA:
Variance is known as the mean square
(MS)
Sum of Squares
Variance
=
------------------------------degrees of freedom
SS
MS 
df
varb
MSb
F

varw MS w
SSb
MSb
df b
F

MS w SS w
df w
SStot  SSb  SS w
SStot   X
2
X



n
2
X   X 




2
SSb
1
n1
2
n2
2
X   X 


...

2
k
nk
Where K represents the number of groups
n
2
SSw   X
2
1
X



1
n1
2
 X
2
2
X 



2
n2
2
... X k
2
X 



k
nk
2
SStot  SSb  SS w
SStot  SSb  SS w
SS
MS 
df
df tot  n  1
df b  k  1
df w  n  k