Oneway ANOVA
Advanced Research Methods in
Psychology
- lecture -
Matthew Rockloff
1
When to use a Oneway ANOVA 1



Oneway ANOVA is a generalization of
the independent samples t-test.
Recall that the independent samples ttest is used to compare the mean
values of 2 different groups.
A Oneway ANOVA does the same thing,
but it has the advantage of allowing
comparisons between more than 2
groups.
2
When to use a Oneway ANOVA 2
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
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In psychology, for example, we often
want to contrast several conditions in
an experiment; such as a control, a
standard treatment, and a newer
“experimental” treatment.
Because Oneway ANOVA is simply a
generalization of the independent
samples t-test, we use this procedure
(to follow) to recalculate our previous
2 groups example.
Later, we will do an example with
more than 2 groups.
3
Example 7.1


Let’s return to our example
of the pizza vs. beer diet.
Our research question is:
“Is there any weight
gain difference between
a 1-week exclusive diet
of either pizza or beer?”
4
Example 7.1 (cont.)
X1
Xj
S2xj
1
2
2
2
3
=2
= 0.4
X2
3
4
4
4
5
4
0.4
5
Example 7.1 (cont.)
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
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An Oneway-ANOVA is a generalization
of the independent samples t-test in
which we can specify more than 2
conditions.
If we only specify 2 conditions,
however, the results will be exactly
the same as the t-test.
The calculations are somewhat
different, but the resulting “p-value”
will be the same, and therefore the
research conclusion will always be the
same.
6
Example 7.1 (cont.)


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ANOVA operates on the principle of
“partitioning the variance”.
There is a total amount of variance
in the set of data previous.
This total variance is found by
subtracting each value (e.g.,
1,2,2…) from the mean for all 10
people (  T  3), squaring the result,
summing the squares, and dividing
by the number of values (i.e., 10):
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Example 7.1 - Formula
S 
2
t
(



)

T
2
N
or 
S 
2
t
2
2
2
2
2
2
2
2
2
2
(1 - 3)  (2 - 3)  (2 - 3)  (2 - 3)  (3 - 3)  (3 - 3)  (4 - 3)  (4 - 3)  (4 - 3)  (5 - 3)
10
8
 1.4
Example 7.1 (cont.)

This total variance (S2t=1.4) can
be partitioned, or divided, into 2
parts:
• the variance within, and
• the variance between.
9
Example 7.1
– Variance

within
The variance within is calculated
by averaging the variances within
each condition.
 For the previous example

10
Example 7.1 – Variance within (cont)
S
2
within
S
S


2
within
J
2
xj
, where J = number of conditions
0.4  0.4

 0.4
2
11
Example 7.1 – Variance between
The variance between is
calculated by taking the variance
of the means of all conditions.
In our example, of course, we
only have 2 means:


S
2
between
S
2
between
 Variance ( 1 ,  2 ,... X J )
(


j
 T )
J
or
2
for a balanced study.
12
Example 7.1 – Variance between (cont.)
In our example:
(2  3)  (4  3)

1
2
2
S
2
between
2
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Example 7.1 (cont.)

Now we can write a formula for the
partition of the variance into its
components:
S2total = S2between+S2within
, or
1.4 = 1 + 0.4


The formula above will allow you to
check your hand calculations.
If you’ve done everything right, all
variances should “add up” to the total
variance.
14
Example 7.1 – ANOVA table

Next, we need to fill-in the so-called
ANOVA table:
Source
of
Variance
(SV)
Source
of
Squares
(SS)
Degrees
of
Freedom
(df)
Mean
Squares
(MS)
F-ratio
(F)
Critical
Value
(CV)
Reject
Decision
(Reject?)
Between
N-
J-1
SSb/dfb
MSb/MSw
See back
of table
of Stats
Text
Is F-ratio
> CV ?
SSw/dfw
S2between
Within
N-S2within
J(n-1)
Total
N-S2total
N-1
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Example 7.1 – ANOVA table (cont.)

Here’s what we know so far:
• S2between = 1
• S2within
= 0.4
• S2total
=1.4
• J=2 (because there are 2 conditions)
• n=5 (because there are 5 people in each condition)
• N=10 (because there are 10 subjects in total)
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Example 7.1 – ANOVA table (cont.)

Now we can fill-in the table:
Source
of
Variance
(SV)
Source
of
Squares
(SS)
Degrees
of
Freedom
(df)
Mean
Squares
(MS)
F-ratio
(F)
Critical
Value
(CV)
Reject
Decision
(Reject?)
Between
10(1)= 10
2-1= 1
10/1= 10
10/0.5=
20
5.32
Is F-ratio
> CV ?
YES
Within
10(0.4)=
4
2(5-1)= 8
4/8= 0.5
Total
10(1.4)=
14
10-1= 9
17
Example 7.1 (cont.)
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This is a 2-tailed test because we had no
notion of which diet should have greater
weight gain.
In the back of a Statistic text we find the
critical value of this “F” is 5.32, by looking for
a 2-tailed F with 1 and 8 degrees of freedom.
The first, or numerator, degrees of freedom
are the degrees of freedom associated with
the Mean Squared Between (df=1).
The second, or denominator, degrees of
freedom are associated with the Means
Squared Within (df=8).
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Example 7.1 – Conclusion …

Our calculated F = 20 is higher
than the critical value, therefore
we reject the null hypothesis and
conclude that:
there is a significant
difference in weight
gain between the 2 diets.
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Example 7.1 – Conclusion (cont.)

More specifically, we can look at the
mean weight gain in each condition
(Mpizza = 2 and Mbeer = 4), and conclude
that:
The beer diet (M = 4.00)
has significantly higher
weight gain than the
pizza diet (M = 2.00),
F(1,8) = 20.00,
p < .05 (two-tailed).
20
Example 7.1 - Using SPSS

First, we need to add 2 variables to the
SPSS variable view:
• IndependentVariable = diet (coded as
1=Pizza and 2=Beer)
• DependentVariable = wtgain (or “weight
gain”)

As before, personid is added a
convenient – although not critical additional variable.
21
Example 7.1 - Using SPSS (cont.)

In addition, we must code for the
“diet” variable (per above):

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Example 7.1 - Using SPSS (cont.)
23
Example 7.1 - Using SPSS (cont.)

In the same
manner as the
independent
samples t-test,
we enter the
data in the
SPSS data
view:
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Example 7.1 - Using SPSS (cont.)
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The only “change” in performing the ANOVA
procedure is the new syntax:
Oneway DependentVariable by IndependentVariable
/ranges = scheffe.

In our example, the following syntax is
entered:
25
Example 7.1 – SPSS output viewer
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Running this syntax produces the
following in the SPSS output viewer:
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Example 7.1 – SPSS (cont.)
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A warning is given which states that the subcommand “/ranges = scheffe” was not
executed.
This procedure is only necessary when there
are more than 2 groups, because it helps to
test all possible pairs of means between
groups.
In our example, we can simply interpret the
ANOVA table to determine significant
difference between our 2 means for Pizza and
Beer.

The warning can be safely ignored.
27
Example 7.1 – SPSS (cont.)
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
The ANOVA table is simply a
reproduction of the table that was
computed by hand.
Unlike the hand calculated
results, SPSS provides an exact
probability value associated with
the F-value.
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Example 7.1 – Conclusion

The conclusion can therefore be
modified as follows:
The beer diet (M = 4.00) has
significantly higher weight gain
than the pizza diet (M = 2.00),
F(1,8) = 20.00, p < .01 (twotailed).
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Example 7.1 – NB: APA style
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Notice that the probability given by
SPSS was p = .002.
Per APA style, rounded to 2 significant
digits the probability becomes p=.00.
Probabilities, however, are never zero,
so we must modify this result to the
smallest p-value normally expressed in
APA style, p < .01.
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Thus concludes 
Oneway ANOVA
Advanced Research Methods in
Psychology
- Week 6 lecture -
Matthew Rockloff
31