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
X X
n2
2
1
variance
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
MSSS
df
FMS
MSb
w
b
w
var
var
FMS
MS
SSdfSSdf
b
w
b
b
w
w
SS SS SStot b w
SS X
X
ntot 2
2
SS
X
n
X
n
X
n
X
nb
k
k
1
2
1
2
2
2
2 2
...
Where K represents the number of groups
SS X
X
nX
X
nX
X
nw k
k
k
1
2 1
2
122 2
2
2
2
2
...
SS SS SS
SS SS SS
tot b w
tot b w
MSSS
df
df n
df k
df n k
tot
b
w
1
1