1. Advanced Research Methods in Psychology - lecture - Matthew Rockloff
Oneway ANOVA
Advanced Research Methods in Psychology
- lecture -
Matthew Rockloff

2. When to use a Oneway ANOVA 1
Oneway ANOVA is a generalization of the independent samples t-test.
Recall that the independent samples t-test 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.

3. When to use a Oneway ANOVA 2
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.

4. 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?”

5. Example 7.1 (cont.) X1 X2 1 3 2 4 5
Xj = 2
S2xj = 0.4
0.4

6. Example 7.1 (cont.)
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.

7. Example 7.1 (cont.)
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 ( ), squaring the result, summing the squares, and dividing by the number of values (i.e., 10):

8. Example Formula
or 

9. 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.

10. Example 7.1 – Variance within
The variance within is calculated by averaging the variances within each condition.
For the previous example 

11. Example 7.1 – Variance within (cont)
, where J = number of conditions

12. 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: or
for a balanced study.

13. Example 7.1 – Variance between (cont.)
In our example:

14. 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 =
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.

15. See back of table of Stats Text
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-S2between
J-1
SSb/dfb
MSb/MSw
See back of table of Stats Text
Is F-ratio > CV ?
Within
N-S2within
J(n-1)
SSw/dfw
Total
N-S2total
N-1

16. 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)

17. 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

18. Example 7.1 (cont.)
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).

19. 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.

20. 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).

21. Example 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.

22. Example 7.1 - Using SPSS (cont.)
In addition, we must code for the “diet” variable (per above): 

23. Example 7.1 - Using SPSS (cont.)

24. 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:

25. Example 7.1 - Using SPSS (cont.)
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:

26. Example 7.1 – SPSS output viewer
Running this syntax produces the following in the SPSS output viewer:

27. The warning can be safely ignored.
Example 7.1 – SPSS (cont.)
A warning is given which states that the sub-command “/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.

28. Example 7.1 – SPSS (cont.)
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.

29. 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 (two-tailed).

30. Example 7.1 – NB: APA style
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.

31. Advanced Research Methods in Psychology
Thus concludes 
Oneway ANOVA
Advanced Research Methods in Psychology
- Week 6 lecture -
Matthew Rockloff