General Linear Model (GLM):
3-Way, Between-Subjects Designs

Reading: SPSS Base 9.0 User's Guide: Chapter 20, GLM Univariate Analysis
                Syntax - SPSS Advanced Models 9.0, pp. 392-416
Homework:
Download: glm_3way.sav        (Download Tips)

1. Overview

This set of notes describes the changes to the 2-way procedures that are need to run a 3-way factorial analysis of variance (ANOVA).  Using these notes you should be able to generalize to any size factorial design.

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2. The Data

The data comes from a test of terror management theory (Greenberg, Pyszczynski, & Solomon, 1986; Solomon, Greenberg, & Pyszczynski, 1991).  The study is called Death, Sex, Love, and Neurosis: Why is Sex such a Problem? by Goldenberg, Pyszczynski, McCoy, and Solomon (1998). It was designed to test the hypothesis that "physical sex is threatening to neurotics because they have not successfully legitimized sex via an appropriate worldview that imbues sex with acceptable cultural meaning."  The participants were 105 undergraduates who filled out a set of measures described as "personality assessments."

One of the scales was Eysenck's Personality Inventory.  This scale was used to divide the participants into two groups, those below the median neuroticism score (low neurotics) and those above the median (high neurotics). 

One manipulated independent variable was priming of thoughts about sex or romantic love.  This was done by having the participants rate each of 10 physical or 10 romantic items according to how appealing they were ( 0 = not at all appealing, 7 = extremely appealing).

The second manipulated independent variable was the presence or absence of  topic that legitimized thoughts of sex.  Participants were asked to "Briefly describe the emotions that the thought of being in love arouses in you" (the legitimizing topic) or "Briefly describe the emotions that the thought of eating a good meal arouses in you" (the neutral, control topic). 

Finally, the accessibility of death thoughts (the dependent variable) was measured by providing participants with 25 word fragments, 5 of which could be completed by either neutral or death related words.  For example, the word fragment COFF___ would be completed either as "coffee" or "coffin."   Death accessibility was the total number of death-related words completed by each person.   This score could range from 0 to 5.

It was hypothesized that "high neurotics would respond to reminders of physical sex, followed by a non-legitimizing topic (eating a good meal) with greater death accessibility than low neurotics and greater death accessibility than after reminders of romantic sex.  However, when the sex prime is followed with a task that explicitly asked the participants to think about being in love, we predicted that the increase in death thought accessibility that neurotics show in response to thoughts of physical sex would be reduced or eliminated."  In short, the authors predicted a significant 3-way interaction.

The variables in the glm_3way.sav file are shown in Table 1.

The effects and degrees of freedom in this 3-way ANOVA are shown in Table 1.

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3. Testing the Assumptions

Assumption 1 (independence). The data are independent, there are different participants in each cell of the design.

Assumption 2 (scale of measurement). The scale of measurement for the death accessibility score is ratio, there is a rational zero point for this scale.

Assumption 3 (normality). It is assumed that the distributions in each of the eight cells of the design are normal. The analysis of variance is robust if each of the distributions are symmetric, or if the any skewness is in the same direction for each cell.

In order to use explore to test the assumptions for each of the eight cells the 2 x 2 x 2 design you need to revise the VARIABLES subcommand as shown in Table 4. Each of the factors, neurotic, priming, and legitimz, need to be separated by the keyword BY.

The explore output is not reproduced here.  The Shapiro-Wilks tests were significant for seven of the eight cells.  But none of the eight cells indicated any normality problems when tested by the 95% confidence intervals for the skewness and kurtosis. It was decided that the normality assumption was not seriously violated.

Assumption 4 (homogeneity of variance). The homogeneity of variance assumption can be tested in GLM or by running the Examine procedure.  The Examine procedure will suggest a power transformation that you could use to reduce the homogeneity problem. GLM will not suggest a transformation.

In this case the data were homogeneous, Levene(7, 97) = 1.56, p = .156.

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4. Run the GLM General Factorial Procedure

Select the GLM General Factorial procedure by clicking on

Analyze
   General Linear Model
       Univariate ...

Move the variable deathnum to the Dependent Variable: window.

Neurotic, priming, and legitimz are all fixed factors, move them to the Fixed Factor(s): window. In this study there are no random factors nor covariates.

At this point we don't know which, if any, of the 3 main effects or 4 interaction will be significant.  So, just run the analysis and then determine which means to print out and how to proceed with comparisons of means for any significant effects.

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5. The Basic GLM Output

The ANOVA table is reproduced in Table 5. 

Dependent Variable: number of death words

There are two significant effects, a neuroticism main effect, F(1, 97) = 7.52, p = .007, and a three-way interaction, F(1, 97) = 8.62, p = .004.

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6. Interpreting Significant Effects: Displaying the Means

Main Effect and Interaction Means

Main effect and interaction means can be displayed by going to the Estimated Marginal Means section of the Options... dialog box and moving the effect(s) of interest from the Factors(s) and Factor Interactions: box to the Display Means for: box.

In this case move the significant neurotic main effect and the three-way interaction to the Display Means for: box.

Select Descriptive Statistics in the Display section of the  Options... dialog box to display descriptive statistics (mean, standard deviation, and n) for each cell in the design.

The estimated marginal means for the neuroticism main effect, and the 3-way interaction are shown in Tables 6 and 7.  The descriptive statistics are shown in Table 8.

Dependent Variable: Number of death words

Dependent Variable: Number of death words

Profile Plots

Profile plots for two-way interactions are obtained by entering one of the factors as the Horizontal Axis: and the other factor as the Separate Lines: factor.

Profile plots for three-way interactions can be thought of as a series of 2-way interactions plotted separately at each level of a third factor. Suppose that you wish to plot the three-way interaction A x B x C as the A x B interaction within each of the levels of C. The enter A (or B) as the Horizontal Axis: variable, B (or A) as the Separate Lines: variable and C as the Separate Plots: variable. Two plots will be created. An A x B plot within C₁ and an A x B plot within C₂.

In this example it was hypothesized (a) that there would be an interaction between neuroticism and priming, and (b) that this interaction would be greater for participants in the legitimizing thought not present condition (having a good mean) than in the legitimizing thought present condition (being in love).  Thus, the interaction between neuroticism and priming should be displayed within each level of the legitimizing thought condition.   The interaction between neuroticism and priming states that priming should have a greater effect for high neurotics than for low neurotics.  Because the focus of the interaction is on the effects of priming within each level of priming you can focus the readers attention on this difference by plotting separate lines for the two neuroticism conditions. 

Set up the plot by going to the Plots... dialog box in GLM.  To set up the basic 2-way interaction between priming and neuroticism move  priming to the Horizontal Axis: and neurotic to Separate Lines:. To plot this interaction separately for the two legitimizing topics conditions move legitmz to Separate Plots:. Then click on Add to move the priming by neurotic by legitimz interaction plot to the Plots: window.  The resulting plots are shown in Figure 1. The neuroticism by priming interaction for being in love legitimizing topic is shown in the left panel, that interaction for the having a good meal legitimizing topic is shown in the right panel. 

Figure 1.  Unedited interaction plots

It was predicted that the 2-way interaction would be larger in the good meal condition than in the being in love condition.  A glance the interactions in Figure 1 suggests that the magnitudes of the interactions are about the same, although the effect of the priming task seems to be in a different direction for the two legitimizing topic conditions.   However, these raw plots are a bit misleading.  The range of scores for the dependent variable, plotted on the y-axis, is not the same for the two legitimizing topic conditions. The y-axis is scaled according to the range of means found in each of   the legitimizing conditions. The y-axis for the left panel has a low score of 0.6 and a high score of 1.8  while the y-axis on the right panel has a low score of 0.8 and a high score of 2.2. This difference in scale makes it difficult to make a visual comparison between the two plots.  The plots should be edited so that the y-axis scales are the same for both plots.  The convention is to use the possible range of scores for the y-axis scale.  In this study the possible range scores for the number of death responses variable is from 0 to 5.  The low value on the y-axis for each plot should be 0, the high value should be 5.

To edit a plot move the cursor to a plot and double left click on it. After the chart has opened in the Chart Editor, move the cursor to the y-axis and double click on one of the scale numbers. This should open a Scale Axis dialog box. Change the Minimum value to 0 and the Maximum value to 5. Then change the major and minor divisions to 1.  This will scale the y-axis in whole numbers rather than in decimals. While you are there you can change the label of the y-axis.  It currently reads "Estimated Marginal Means."  Change it to "Mean Number of Death Words."  I like to center the label by changing Title Justification to "center."  The title is too long to be displayed,  I have changed the Title 1 to "Neuroticism by Priming Interaction" and the Title 2 to "Legitimizing topic = being in love." 

When you are finish making changes to the chart click File, Close. 

Go to the other chart and make the same changes to the y-axis and Title 1.  Change Title 2 to "Legitimizing topic = having a good meal."

The edited charts are shown in Figure 2.

Figure 2. Edited interaction plots

These plots (produced by GLM)  have no error bars, so you cannot visually inspect the plots to see which means might be significantly different from each other. Figure 3 was created from Interactive Charts - Error Bars. The variable deathnum was moved to the y-axis; priming was moved to the x-axis; neurotic was moved to the Color: box of the Legend Variables section; and legitimz was moved to the Panel Variables box. It was edited to increase the scale of the y-axis, the size of the text elements, and the positioning of the legends.

Figure 3.  Interaction plots with error bars: Created from Interactive Charts

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7. Interpreting Significant Effects: Post-Hoc Pairwise Comparisons

Main Effects

When the main effect has only 1 df (2 cells) there is no need to run any post hoc test, the two means are different from each other if the main effect is significant. The significant neurotic mean effect in this analysis might be reported as follows:

When reporting results the emphasis should be on the description of the effects rather than on the statistic itself.  Means and standard deviations should be reported for each level of the effect.  The F statistic is normally at the end of the description.  Notice that the elements of the F statistic are not included within parentheses (except for the number of degrees of freedom).

The tables that report the estimated marginal means display standard errors (SE) rather than standard deviations (SD). You can report the standard errors along with the means when the report the results.

Post hoc tests for significant main effects with more than two levels can be performed in two ways in GLM: 

(a)  Go to the Post Hoc... dialog box.  Select the desired main effect(s), then select the type(s) of   test(s)  you wish to perform, and then press continue. The post hoc tests are the same as those described in the oneway procedure.

(b) Go to the Estimated Marginal Means section of the Options... dialog box.  Select the desire main effect(s) from the Factor and Factor Interactions window and move them over to the Display Means for: window. Check the Compare main effects option, and press Continue. Using this option there are three possible ways to make adjustments to the confidence interval:  (a) none,  in which case the comparisons are made by an uncorrected t test,  this is also called the LSD test; (b) Bonferroni, in which case the alpha level is found as -

Bonferroni alpha = alpha/C

where C is the number of possible paired comparisons for that main effect test; and (c) Sidak, in which case the alpha level is found as -

Sidak alpha = 1 - (1 - alpha)^1/C

where C is the number of possible paired comparisons for that main effect test.

Interaction Effects:

There is no direct way to run post-hoc, paired-comparison tests on interaction effects using GLM.  Three alternatives are: running the tests by hand, creating a oneway effect from the interaction, and using a simple main effects analysis.

Interaction Effects: Running Tukey's HSD Test by Hand

For equal n designs the critical difference, y(HSD), is

where, MSerror is the MSerror from the analysis of variance; n is the number of cases in each cell; and q_a,p,v is obtained from the Percentage Points for the Studentized Range Statistic table at a given significance level, a, with p means, and v degrees of freedom for the MS error term.

If there are unequal ns in the cells being compared, then the harmonic n should be used.  The formula for the harmonic n (Nh) is --

Nh = p / (1/n(1) + 1/n(2) + ... + 1/n(p)) 

where p is the number of cells being compared and n is the number if cases in the ith cell of the comparison. For this data --

Nh = 8 / (1/14 + 1/13 + 1/16 + 1/10 + 1/12 + 1/14 + 1/12 + 1/14)
     8 / (0.07143 + 0.07692 + 0.06250 + 0.10000 + 0.83333 + 0.07143 + 0.08333 + 0.07143)
     8 / 0.62037
     12.896

For this analysis the MSerror = 0.777 and the df for the MSerror term is 97. The value of q_a,p,v with a = .05, 8 means, and 97 degrees of freedom is 4.39.  Substituting in the formula for the HSD critical difference --

HSD = 4.39* SQRT(0.777/12.896)
         = 4.39*0.2455
         = 1.078

Any pair of means will be significant different from each other at the .05 level of significance using Tukey's HSD if the difference between the means is greater than  1.078.

Using Tukey's HSD test with alpha = .05, the mean of the having a good meal-physical sex prime-high neurotic condition (M = 2.00) was higher than the mean for the being in love-romantic sex prime-low neurotic condition (M = 0.81) and the mean for the having a good meal-physical sex prime-low neurotic condition (M = 0.92).   There were no other significant paired comparisons.

Although this does describe the mean differences, the interpretation of these paired comparisons is awkward and cumbersome. 

Interaction Effects: Creating a Oneway Effect from the 3-Way Interaction

In this option we use the IF data transformation to create a factor that includes each of the cells in the 3-way interaction.  The syntax for creating a new factor, called AxBxC is shown in Table 13. The IF syntax can be entered by hand, or created using the COMPUTE transformation dialog box.  If you use the COMPUTE transformation dialog box you will need to individually paste each IF to the syntax window.  It takes more time to use the dialog box than to just enter the syntax commands directly.

IF (legitimz = 1 and priming = 1 and neurotic = 1) AxBxC = 1 .
IF (legitimz = 1 and priming = 1 and neurotic = 2) AxBxC = 2 .
IF (legitimz = 1 and priming = 2 and neurotic = 1) AxBxC = 3 .
IF (legitimz = 1 and priming = 2 and neurotic = 2) AxBxC = 4 .
IF (legitimz = 2 and priming = 1 and neurotic = 1) AxBxC = 5 .
IF (legitimz = 2 and priming = 1 and neurotic = 2) AxBxC = 6 .
IF (legitimz = 2 and priming = 2 and neurotic = 1) AxBxC = 7 .
IF (legitimz = 2 and priming = 2 and neurotic = 2) AxBxC = 8 .
VARIABLE LABEL  AxBxC 'factor for post hoc test of the interaction'.
VALUE LABELS  AxBxC  1 'legitimz = 1, priming = 1, neurotic = 1'
                     2 'legitimz = 1, priming = 1, neurotic = 2' 
                     3 'legitimz = 1, priming = 2, neurotic = 1'
                     4 'legitimz = 1, priming = 2, neurotic = 2'
                     5 'legitimz = 2, priming = 1, neurotic = 1'
                     6 'legitimz = 2, priming = 1, neurotic = 2' 
                     7 'legitimz = 2, priming = 2, neurotic = 1'
                     8 'legitimz = 2, priming = 2, neurotic = 2' .
Formats AxBxC(F1).
EXECUTE .

You should include the variable label and value label commands in the syntax or else go into the newly created variable, AxBxC, and manually add the labels. After you run the syntax commands you can go into GLM, enter the dependent variable and 'AxBxC' as the factor.  Then you can test the AxBxC "main effect" in either of the ways described above under Main Effects. 

Table 14 shows the IF transformations used to create a oneway factor for a significant B x C interaction in a 3-way, A x B x C design with equal ns.

IF (A = 1 and B = 1 and C = 1) BxC = 1 .
IF (A = 1 and B = 1 and C = 2) BxC = 2 .
IF (A = 1 and B = 2 and C = 1) BxC = 3 .
IF (A = 1 and B = 2 and C = 2) BxC = 4 .
IF (A = 2 and B = 1 and C = 1) BxC = 1.
IF (A = 2 and B = 1 and C = 2) BxC = 2 .
IF (A = 2 and B = 2 and C = 1) BxC = 3 .
IF (A = 2 and B = 2 and C = 2) BxC = 4 .
VARIABLE LABEL  BxC 'factor for post hoc test of the interaction'.
VALUE LABELS  BxC  1 'B = 1, C = 1'
                   2 'B = 1, C = 2' 
                   3 'B = 2, C = 1'
                   4 'B = 2, C = 2'.
Formats BxC(F1).
EXECUTE .

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8. Interpreting Significant Effects: Simple Main Effects Analysis

Simple Main Effects for 2-way Interactions

Simple main effects analyses for significant 2-way interactions can be run using the syntax structure described in GLM: Simple Main Effects   notes.

Simple Interaction Effects for the 3-way Interaction

Review of a 2 x 2 interaction. The definition of a 2-way interaction is that the effects of one variable (e.g. A) are not the same across the levels of the other variable (e.g. B).  For a 2 x 2 ANOVA the null hypothesis of no interaction is expressed by the following equation -

int_2x2 = (A1B1 - A2B1) - (A1B2 - A2B2) = 0

This version of the interaction formula states that the change from A1 to A2 within B1 is the same as the change from A1 to A2 within B2. If the difference of the change scores is not zero, then there is an interaction effect. Visually we look to see if the slopes of the lines involved in the interaction are parallel.  A significant interaction implies that the slopes of the lines are different, that is, the change from A1 to A2 within B1 is not the same as the change from A1 to A2 within B2.

A simple main effects analysis tests for significant differences between  A1 and A2 within each level of B or between B1 and B2 within each level of A.

Extension to a 2 x 2 x 2 interaction. The definition of a 3-way interaction is that the interaction effects of two variables are not the same under the different levels of a third variable.  For example, if the A by B interaction within C1 is not the same as the A by B interaction within C2, then there is a 3-way interaction.   For a 2 x 2 x 2 ANOVA,  the null hypothesis for a 3-way interaction is expressed by the following formula

int_2x2x2 = [ (A1B1C1 - A2B1C1) - (A1B2C1 - A2B2C1)] - [(A1B1C2 - A2B1C2) - (A1B2C2 - A2B2C2)]

Visually you look at the plots of the effects of the 2-way interaction within each level of the third factor.  If the interactions are parallel within each level of the third factor, then there is no 3-way interaction.

When the 3-way interaction is significant, one way of interpreting the effects is to do a simple interaction analysis.  In a simple interaction analysis the 3-way interaction is broken down into a series of 2-way interactions, one 2-way within each level of the third variable.  Unfortunately you cannot use the "compare" syntax to run tests of simple interaction effects in SPSS 7.5, 8.0, or 9.0. 

As a sidebar I have linked some comments about this issue from David Nichols, an excellent statistical advisor at SPSS, Inc.

Simple, Simple Main Effects

One alternative to running simple interaction effects is to run simple, simple main effects.  Simple, simple main effects are the main effects of a factor within each of the combined levels of the remaining factors.  For example, given a 3-way ANOVA with factors A, B, and C, the simple, simple main effects of A are the A main effects within each cell of the B x C interaction.  That is, for a 2 x 2 x 2 ANOVA, the simple, main effects for A are defined as the difference of A1 - A2 within B1C1, B1C2, B2C1, and B2C2.  

In our example the authors are interested in how the effects of the priming factor are influenced by the neuroticism and legitimizing topic factors.  So it might prove fruitful to look at the simple, simple main effects of priming within each cell of the neuroticism by legitimizing factors interaction.  The syntax for creating simple, simple main effects for priming is shown in Table 15.  The EMMEANS table is created for the 3-way interaction, TABLES(legitimz*neurotic*priming).   The main effect of priming is the variable to be compared, COMPARE(priming).   The confidence intervals are adjusted using the Sidak adjustment, ADJ(SIDAK).

This results in two tables: (a) a set of univariate tests of the simple, simple effects of priming (Table 16), and (b) a set of pairwise comparisons of the simple, simple effects of priming (Table 17).

Dependent Variable: Number of death words

Each F tests the simple effects of Thought priming task within each level combination of the other effects shown. These tests are based on the linearly independent pairwise comparisons among the estimated marginal means.

The F test in Table 16 show that the only significant main effect for priming is in the having a good meal, high neurotic condition.  Go back to one of the graphic displays of the 3-way interaction and see how these simple, simple main effects fit the data.

Dependent Variable: Number of death words

a Adjustment for multiple comparisons: Sidak.

The authors of the study used MANOVA to test the simple interaction effects. (MANOVA is an SPSS procedure that preceded GLM.  It it still available in syntax form within SPSS for Windows 7.5, 8.0, and 9.0).  They found that the interaction between neuroticism and priming was significant in the having a good mean condition, but that there was no interaction between neuroticism and priming in the being in love condition.  Pairwise comparisons within the having a good mean condition showed a significant effect of priming for high neurotics, but not for low neurotics.  The simple, simple main effects analysis comes to the same conclusion.

This simple, simple main effects solution is generalizable to 4 way and larger interactions.

Pairwise comparisons for an interaction within levels of a third variable

The simple, simple main effects analysis looks at the main effects of one factor within the combined levels of two (or more) other factors.  Another way to conceptualize this is to look at pairwise comparisons for a 2-way interaction within each level of a third factor.  For example, suppose we wish to look at the neurotic*priming interaction within each level of the legitimizing topic (being in love and having a good meal).  The first thing we would need to do is to create oneway factor for the neurotic by priming interaction.  The  "IF" syntax shown in Table 18 will construct a oneway factor called PxN (Priming by Neurotic) with four levels.

IF (priming = 1 and neurotic = 1) PxN = 1 .
IF (priming = 1 and neurotic = 2) PxN = 2 .
IF (priming = 2 and neurotic = 1) PxN = 3 .
IF (priming = 2 and neurotic = 2) PxN = 4 .
VARIABLE LABEL PxN 'factor for test of the simple interaction'.
VALUE LABELS  PxN  1 'priming = 1, neurotic = 1'
                   2 'priming = 1, neurotic = 2' 
                   3 'priming = 2, neurotic = 1'
                   4 'priming = 2, neurotic = 2' .
                                      
Formats PxN (F1).
EXECUTE .

I have run that set of syntax commands and stored the results for you in the glm_3way.sav datafile.

The To run the analysis using the PxN factor enter PxN and legitimz as the factors and deathnum as the dependent variable.  In the Options... dialog box move legitimz, PxN, and the legitimz*PxN interaction from the Factor(s) and Factor Interactions: window to the Display Means for: window. Check the Compare main effects box and choose a confidence interval adjustment (e.g., Sidak). Next past the commands to the syntax editor. Edit the commands to test the PxN effect within each level of legitimz, as shown in Table 19 and run the syntax commands.

The pairwise comparisons table is shown in Table 20.

Dependent Variable: Number of death words

a Adjustment for multiple comparisons: Sidak.

Using the pairwise comparisons in Table 20, go back to one of the graphical displays and see which means are significantly different from each other.

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9. References

Goldenberg, J. L., Pyszczynski, T., McCoy, S., Greenberg, J., & Solomon, S. (1998). Death, sex, love and neurosis: Why is sex such a problem? Unpublished manuscript (submitted).

Greenberg, J., Pyszczynski, T., & Solomon, J. (1996). The causes and consequences of the need for self-esteem: A terror management theory. In R. F. Baumeister (Ed.). Public self and private self (pp. 189-212). New York: Springer-Verlag.

Solomon, J., Greenberg, J., & Pyszczynski, T. (1991). A terror management theory of social behavior: The psychological functions of self-esteem and cultural worldviews. In M. P. Zanna (Ed.). Advances in experimental social psychology (pp. 91-159). San Diego: Academic Press, Inc.

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©Lee A. Becker, 1997-1999  -revised 11/18/99