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A SIMPLE FACTORIAL, 2X2, ANALYSIS OF VARIANCE
TREATMENT BY LENGTH (# CIGARETTES)
Suppose a researcher is working with smokers to help them cut down on their consumption of cigarettes. From experience, he
believes that those who have been smoking for 20 years are not as successful as those who have been smoking for 10 years.
Additionally, he believes that counseling with a money incentive is superior to counseling alone. He designs an experiment to test
these hypotheses.
He has 6 clients who have been smoking 20 years and 6 who have been smoking 10 years. All are smoking the same number of
cigarettes before any treatment. He randomly assigns half of each group to either (a) counseling, or (b) counseling plus money.
After treatment he counts the number of cigarettes consumed daily by each client.
Here is the data:
TREATMENT
20 years
LENGTH
10 years
Counseling
30
35
40
28
25
22
Counseling Plus $
28
30
32
16
20
24
He decides to do a factorial, 2X2 ANOVA, Treatment by Length.
The first task is to decide how to enter the data into SPSS for Windows. Since a 2X2 Factorial ANOVA always has two
independent variables and one dependent variable, you know you must define at least three variables, not counting ID numbers.
For this example, we will not use ID numbers.
Here are the three variables we must define:
1. numcigs - (dependent variable)
2. treatmen - (independent variable) [1 = counseling; 2 = money]
3. length - (independent variable) [1 = 10 years; 2 = 20 years]
How do we enter them?
_____ 1. Well, first you should click the variable view tab at the bottom left of the SPSS screen. There, you can define the variables
and the VALUE LABLES:
_____ 2. To designate the Value Labels, click in the Values column, the treatmen row until you see a small gray box with three
dots.
_____ 3. Click on the gray box and define the value labels by typing a 1 in the Value field, and counseling in the Value Label field,
then click Add. Now, type a 2 in the Value field and money in the Value Label field and click Add. Then, click OK:
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_____ 4. Now do the same thing to define Value Labels for the length variable:
_____ 5. Now, return to the data screen by clicking the Data View tab in the lower left corner of the SPSS screen:
Here is what the first few rows of your data screen will look like:
Now it is time to enter the data. Here is the raw data table, again:
Now, with a pencil, fill out the following SPSS data screen. In the table of scores above, work upper left quadrant to upper right
quadrant, to lower left quadrant, to lower right quadrant.
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Now, turn to the next page to see what the data should look like:
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You can make the job of checking these scores after you enter them a little less error prone by making SPSS display the VALUE
LABELS instead of the numbers for the treatmen and length variables. To do that:
Running the Two-Way ANOVA
_____ 1. Click Analyze, General Linear Model, and Univariate. The following box will open:
_____ 2. numcigs is the dependent variable, and treatmen and length are both independent variables (FACTORS). Since they do
not represent repeated measures (such are pretest/posttest), they are both Fixed Factors. Move the variable names into the appropriate
boxes as follows:
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There are six buttons along the right side of the above box. You will need to click on each and enter the information needed:
_____ 1. Click on Model and make sure the choice is for Full factorial and click CONTINUE.
_____ 2. Click on Contrasts and make sure that both treatmen and length are in the Factors: field. If both are not there, you made an
error on one of the previous steps.
_____ 3. Now click the Plots button. You can order graphs of any interaction found. To do so, highlight treatmen and click the right
arrow button to move it into the Horizontal Axis: field. Then highlight length and click the right arrow button to move it into the
Separate Lines: field. Then click the ADD button next to PLOTS:
Now, repeat this exactly, except this time, place length in the Horizontal Axis: field and treatmen in the
Separate Lines: field.
This will order two different graphs, reversing which of the two independent variables will appear on the X axis, and which will be
defined by separate lines in the body of the graph. When you are finished, the box should look like this (notice the large field at the
bottom of the box):
_____ 4. Now, click the CONTINUE button. Now, click the Post Hoc button. This will allow you to order post hoc analyses in case
of significant findings. This is necessary whenever there are more than two means in a main effect. In this analysis, there are only
two means in each of the independent variables. THEREFORE, NO POST HOC TESTS WILL BE NECESSARY. If the main
effect for treatmen is significant, there will be only two means. Therefore, we know that is the pair that differs (since it is the ONLY
pair) and we can simply look at those means and see which is highest. The same is true for length, if it proves to be significant.
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_____ 5. Click the Continue button.
_____ 6. Click the Options button. Move all the variables listed on the left into the field marked Display Means For: At the bottom
of the box, put check marks next to Descriptive statistics, Estimates of effect size, Observed power and Homogeneity tests. The box
should look like this:
_____ 7. Click the Continue button.
_____ 8. Click the OK button to run the analysis. PRINT OUT THE OUTPUT TO STUDY. It should be just like the handout I
gave you.
INTERPRETING THE RESULT OF THE FACTORIAL, 2X2, ANOVA
One of the first things you will need to help you understand the results of your analysis is a table of means with
cell means, row means, and column means filled in. Use the printout you just ran or the one I gave you to
locate these means and fill in the following table. Then, read on to see where you find these means on the
printout, and to see how this table should be filled out:
TREATMENT
Counseling
Counseling Plus $
20 years
Row Mean =
LENGTH
Row Mean =
10 years
Column Mean =
Column Mean =
GRAND MEAN =
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The means you need for the table can be found on page 1 of the output. It is in the second box on the page labeled Descriptive
Statistics. Check your box to be sure it looks like the following:
TREATMENT
Counseling
Counseling Plus $
20 years
LENGTH
10 years
35
30
Row Mean = 32.5
25
20
Row Mean = 22.5
Column Mean = 30
Column Mean =25
GRAND MEAN =
27.5
Just looking at these means do you predict that there is a significant main effect for:
A. TREATMENT?
B. LENGTH?
Do you predict a significant interaction effect for
C. TREATMENT BY LENGTH?
The next thing we need is a traditional two-way ANOVA source table. See if you can use the printout to fill in the empty spaces in the
following table. Then, go on to the next page where you will find a completed table and more information.
Here is the source table from the factorial ANOVA:
SOURCE
Rows (length)
Columns (treatmen)
Interaction
Within (Error)
Total
SS
df
MS
F
p
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Here is how the source table from the factorial 2X2 ANOVA should look:
SOURCE
Rows (length)
Columns (treatmen)
Interaction
Within (Error)
Total
SS
300.00
75.00
.00
108.00
483.00
df
1
1
1
8
11
MS
300.00
75.00
.00
13.50
F
22.22
5.56
.00
p
.002
.046
-
TREATMENT
Counseling
Counseling Plus $
20 years
LENGTH
10 years
35
30
Row Mean = 32.5
25
20
Row Mean = 22.5
Column Mean = 30
Column Mean =25
GRAND MEAN =
27.5
Here is a copy of interaction graphs:
The other graph on page 4 reverses the independent variables. In this one, the lower line represents those who received counseling
plus money, while the upper line represents those who received counseling alone. It makes very little difference which one we use. In
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a factorial ANOVA, we generally choose the one that will result in the fewest separate lines. However, if there are the same number
of levels in each independent variable as there are here (because it is 2 X 2), that is not a consideration.
We would then write up our analysis, in which we would suggest that counseling plus money is the best alternative for both types of
smokers. Further, we would suggest that 10-year smokers achieve more success in cutting down on smoking than do 20-year
smokers.
END