Factorial ANOVA in SPSS

In the dataset to be used for this example, there are two N-level variables (“treatment”

and “problem”) for each person—“treatment” has two levels (CBT [cognitive-behavioral

therapy] vs. IPT [interpersonal psychotherapy]), and “problem” has three levels (anxiety

disorder vs. depression vs. eating disorder). Each person (row in the table) has one type

of diagnosis, and receives one treatment. Then each person also has a “symptom” score,

indicating their level of symptom severity after treatment (so this is your outcome

variable, to see if the treatment was effective—lower scores on “symptom” mean the

treatment helped). The goal of this two-way ANOVA is to determine whether people who

receive different treatments have different outcomes, and whether the type of diagnosis

that the person is being treated for moderates the effect of treatment types on symptom

levels. (In other words, the question is whether there’s an interaction between treatment

type and problem type, in terms of these two variables’ effects on symptom levels).

To do this analysis, we start (as usual) from the “Analyze” menu. One way to get a basic

F-test is to use the “Compare Means” command, but there’s another way to do it: select

“General Linear Model,” and then select “Univariate.” (A “univariate” ANOVA has one

dependent variable—that’s true in this case, where the only dependent variable is

“symptom” level). For this example (as well as ANCOVA, MANOVA, and other

procedures) it’s necessary to use the “General Linear Model” instead of the “Compare

Means/ANOVA” command. This is because it’s a more generalized procedure.

The dialog window asks you to select your independent and dependent variables. There

are two N-level predictors as independent variables—they go in the “Fixed Factors” box

(this term refers to any N-level predictor in an ANOVA procedure). Then your criterion

variable—“symptom”—goes in the DV spot:

Hit the “Model” button to continue.

The “model” dialog box gives you a chance to customize the list of predictors that you

want to test out. The default model is a “full factorial” model, which means that SPSS

will test all of the possible main effects, and also all of the possible interactions among

the fixed factors you selected in the previous step. If you leave the “full factorial” setting

in place for this example, you’ll get exactly the same result as I do by customizing the

model. However, I’m going to do the customization, to show you exactly what we’re

testing for.

To set up a custom model, I select the predictors that I’m interested in from the list on the

left, select the type of test that I want to run from the drop-down menu in the middle of

the screen, and then use the arrow to move the predictors to the right-hand list. First, I

select “main effects” from the drop-down menu, and use the arrow to create a “main

effect” test for each individual predictor’s effect on the criterion variable. Next, I select

“interaction” from the drop-down menu. This time, I have to select both variables that I

want an interaction for, and use the arrow with both of these predictors selected. This

gives me the interaction term, which looks like this: “treatment*problem”.

When you’ve duplicated what I have shown above (or if you just left the “full factorial”

default setting in place), click “Continue” to go on.

Note: For now, leave this other drop down menu set to “Type III”

sums of squares. Next week we’ll change this setting, and look at

the difference between “Type I” and “Type III” sums of squares,

when we start to work with ANCOVA (analysis of covariance).

Back in the main dialog box for univariate ANOVA, click on the “Plots” button to see

this dialog box:

This dialog allows you to generate the type of line graph that we saw in the lecture notes.

One of the predictor variables goes on the horizontal axis of the graph, and the other is

used to define separate lines (refer to lecture notes for an example of the line graphs that

show main effects and interaction effects). Let’s put “treatment” on the horizontal axis,

and use “problem” as the variable to define three separate lines on the graph. The y-axis

of the graph will automatically be the dependent variable (“symptom”).

Click “Continue” to go on.

Once you have put the variables into the correct

boxes in the dialog, you have to hit the “Add”

button, and get the graph to show on this list, in

order for it to appear on your output.

Back in the main dialog window, click on “Options” to see this sub-dialog:

This window lets you get various types of descriptive statistics. First, select the variables

you want summary statistics on from the left-hand list, and use the arrow to move them to

the right-hand list. Then use the check boxes down below to select the types of summary

statistics you want to see. Let’s get “descriptive statistics” and also the “observed power”

for each statistical test.

Click “Continue” to go on. Then, back in the main dialog box, we’re ready to see the

results, so click “OK” in the main window for this analysis.

Here are some highlights of the output:

This table (“Tests of Between-Subjects Effects”) gives you the results of the F-tests for the two main effects and the interaction. Tests of Between-Subjects Effects Dependent Variable: Symptom Severity (higher = worse)

Source Type III Sum of

Squares

df Mean Square

F Sig. Noncent. Parameter

Observed Power

Corrected Model

238.938 5 47.788 23.894 .000 119.469 1.000

Intercept 9642.857 1 9642.857 4821.429 .000 4821.429 1.000TREATMNT 61.714 1 61.714 30.857 .000 30.857 .999

PROBLEM 77.169 2 38.585 19.292 .000 38.585 .998TREATMNT *

PROBLEM71.631 2 35.815 17.908 .000 35.815 .997

Error 20.000 10 2.000Total 10209.000 16

Corrected Total

258.938 15

a Computed using alpha = .05 b R Squared = .923 (Adjusted R Squared = .884)

p-value for the first main effect (effect of “treatment” type on symptom levels)

p-value for the second main effect (effect of “problem” type on symptom levels)

p-value for the interaction of the two predictors (effect of the 6 possible combinations of

“treatment” type and “problem” type on symptom levels)

As before, “Error” is the within-groups sum of squares (it’s the same for all three F-tests

reported in this table). “Corrected Total” is the total sum of squares (equal to the SS for

each of the three variables tested—the two main effects and the interaction—plus the

Error SS).

Finally, notice this R-squared value. This tells you the “total percentage of variability in

the criterion variable that can be accounted for by all three predictors together.” It’s a

“total” R2 because it’s measuring the effect of all of the predictors together (the two main

effects and the interaction) on the dependent variable.

These are the

power for each

test, shown

here because of

the “option”

we selected in

the analysis

The next three tables that you see give the different averages, standard deviations, etc. for

each sub-group. Here’s the example of averages by diagnosis:

2. Patient Diagnosis Dependent Variable: Symptom Severity (higher = worse)

Mean Std. Error 95% Confidence Interval

Patient Diagnosis

Lower Bound Upper Bound

Anx 23.000 .645 21.562 24.438Depr 24.000 .645 22.562 25.438

Eating 28.000 .577 26.714 29.286

Finally, here’s the graph that we selected using the “Plots” command:

Estimated Marginal Means of Symptom Severity (higher = worse)

Type of Therapy

IPTCBT

Estim

ate

d M

arg

inal M

eans

34

32

30

28

26

24

22

20

18

Patient Diagnosis

Anx

Depr

Eating

You can see the “X” pattern created by the lines—it means there’s a significant

interaction. Similarly, the difference in height between the blue and red lines shows a

significant main effect for “diagnosis,” and the upward slope of the lines shows a

significant main effect for “treatment.” Depression is the oddball diagnosis, where IPT

has a better effect than CBT. With the other dx groups, CBT has better results.

Patients with eating

disorders still had higher

levels of symptom severity

after treatment