Two-way Anova Using SPSS

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Conducting Two-Way ANOVA using SPSS

Psychology 304b

This handout will review how to do 2-way ANOVA, comparisons of marginal means, interaction

contrasts, and simple effects analyses using SPSS. In addition, I will discuss how to perform

non-orthogonal ANOVAs using SPSS.

Two general approaches will be used depending on the analysis. One involves the pull-down

menus you see when you first enter SPSS. The second involves syntax files, in which the user

enters commands similar to those that we have used for SAS. The former approach is best for

relatively simple analyses but the latter offers the maximal number of options. Most of the

analyses will be done using the Generalized Linear Model program in SPSS with the univariate

option. However, at some points we will use the multivariate (MANOVA) option within the

GLM program.

The Data

I will assume that you know how to create a SPSS .sav file and to read it in using a GET

statement if youre using the syntax editor. The data file used for this example is on the web site

and named twoway.sav. It contains data from the biofeedback X drug group example used for

the SAS two-way ANOVA programs. I should note that the particular data set used here is the

one with 6 subjects per cell (36 in all) that was used for the SAS program demonstrating

interaction contrasts and simple effects analysis. Part of the data file looks like this:


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Two-way ANOVA

To do the omnibus two-way ANOVA, it is probably easiest to just use the pull-down menu

option. Click on Analyze >>> General Linear Model >> Univariate:

Then, indicate the dependent variable and the factors in the appropriate boxes.


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I would also use the Options to generate marginal and cell means:

Then hit Continue and then OK and you will initiate the analysis. As youll see from the output,

these steps also generate the following Syntax code (that you could use if you did this using the

Syntax editor):

UNIANOVA bp BY bio_gp drug_gp/METHOD=SSTYPE(3)/INTERCEPT=INCLUDE/EMMEANS=TABLES(bio_gp)/EMMEANS=TABLES(drug_gp)/EMMEANS=TABLES(bio_gp*drug_gp)/CRITERIA=ALPHA(.05)/DESIGN=bio_gp drug_gp bio_gp*drug_gp.

In fact, not all these statements are necessary if you wanted to reproduce the pull-down menu.

You could get away with just:

UNIANOVA bp BY bio_gp drug_gp/EMMEANS=TABLES(bio_gp)

/EMMEANS=TABLES(drug_gp)/EMMEANS=TABLES(bio_gp*drug_gp)

The output is as follows:


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Univariate Analysis of Variance

Tests of Between-Subjects Effects

Dependent Variable:bp

SourceType III Sum of

Squares Df Mean Square F Sig.

Corrected Model 6552.000a 5 1310.400 9.593 .000

Intercept 1340964.000 1 1340964.000 9816.720 .000

bio_gp 1296.000 1 1296.000 9.488 .004

drug_gp 4104.000 2 2052.000 15.022 .000

bio_gp * drug_gp 1152.000 2 576.000 4.217 .024

Error 4098.000 30 136.600

Total 1351614.000 36

Corrected Total 10650.000 35

a. R Squared = .615 (Adjusted R Squared = .551)

You will note that the ANOVA output and means are identical to the omnibus SAS output in the

program demonstrating simple effects analyses and interaction contrasts. You can disregard the

intercept source above (representing modeling of the effect of grand mean). The key effectshere are the two main effects for bio_gp and drug_gp and the bio_gp*drug_gp interaction term.

Estimated Marginal Means

1. bio_gp


bio_gp Mean Std. Error

95% Confidence Interval

Lower Bound Upper Bound

1 187.000 2.755 181.374 192.626

2 199.000 2.755 193.374 204.626


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2. drug_gp


drug_gp Mean Std. Error



1 178.000 3.374 171.110 184.890

2 202.000 3.374 195.110 208.890

3 199.000 3.374 192.110 205.890

3. bio_gp * drug_gp


bio_gp drug_gp Mean Std. Error



1 1 168.000 4.771 158.255 177.745

2 204.000 4.771 194.255 213.745

3 189.000 4.771 179.255 198.745

2 1 188.000 4.771 178.255 197.745

2 200.000 4.771 190.255 209.745

3 209.000 4.771 199.255 218.745

Main Effect Comparisons: Pairwise Comparisons

There are several different ways to do main effect comparisons in SPSS, some of which can be

surprisingly tedious and complex. Pairwise comparisons, though, are pretty simple.

Individual Pairwise Comparisons (Unadjusted for Multiplicity)

Lets start with comparisons un-adjusted for multiplicity (e.g., if you were just doing one

contrast). One good way for you to go is to use the syntax editor. Specify the contrast as part ofthe EMMEANS option. For example, lets say we want to do all pairwise comparisons among

the 3 drug groups. Your code would look like this:

GLM bp by bio_gp drug_gp/EMMEANS TABLES (drug_gp) COMPARE(drug_gp).


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Youll see in the output the results of pairwise comparisons among the three drug groups:

Pairwise Comparisons


(I)drug_gp

(J)drug_gp

MeanDifference (I-

J) Std. Error Sig.a

95% Confidence Interval forDifference

a


1 2 -24.000 4.771 .000 -33.745 -14.255

3 -21.000 4.771 .000 -30.745 -11.255

2 1 24.000 4.771 .000 14.255 33.745

3 3.000 4.771 .534 -6.745 12.745

3 1 21.000 4.771 .000 11.255 30.745

2 -3.000 4.771 .534 -12.745 6.745

Based on estimated marginal means

*. The mean difference is significant at the .050 level.

a. Adjustment for multiple comparisons: Least Significant Difference (equivalent tono adjustments).

You can see that drug group 1 (i.e., drug X) differs significantly from group 2(Y) and 3 (Z) butthat groups 2 and 3 dont differ.

Theres another way to do this too. You could just use the pull-down menu, go to post-hoc

contrasts (I know this sounds weird), specify LSD contrasts but just ignore the results of the

overall ANOVA. The idea here is to just get out the results of individual contrasts with anindividual per comparison type 1 error rate of .05. The key strokes are Analyze >> General

Linear Models >> Univariate >> Enter DV and Fixed Factors if you havent done so >> Post

Hoc >> LSD >>Continue >> Continue

Multiple Pairwise ComparisonsPairwise comparisons corrected for multiplicity are easy. Just use the pull-down menu.

The key strokes are Analyze >> General Linear Models >> Univariate >> Enter DV and FixedFactors if you havent done so >> Post Hoc. Then you will see this menu:


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Click on the factor of interest (here drug_gp) and indicate the post-hoc tests you want (e.g.,Bonferroni, Tukey, Scheffe). Then hit Continue, and then OK and you will get the MC results in

addition to the results of the ANOVA. You should be able to interpret the output clearly. For

example, below is the output of the Tukey test for these data:

Multiple Comparisons

bp

Tukey HSD

(I)drug_gp

(J)drug_gp

Mean Difference(I-J) Std. Error Sig.



1 2 -24.0000 4.77144 .000 -35.7629 -12.2371

3 -21.0000 4.77144 .000 -32.7629 -9.2371

2 1 24.0000 4.77144 .000 12.2371 35.7629

3 3.0000 4.77144 .806 -8.7629 14.7629


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3 1 21.0000 4.77144 .000 9.2371 32.7629

2 -3.0000 4.77144 .806 -14.7629 8.7629

Based on observed means.

The error term is Mean Square(Error) = 136.600.

*. The mean difference is significant at the 0.05 level.

Homogeneous Subsets

bp

Tukey HSD

drug_gp N

Subset

1 2

1 12 178.0000

3 12 199.0000

2 12 202.0000

Sig. 1.000 .806

Means for groups in homogeneous subsets aredisplayed.

Based on observed means.

The error term is Mean Square(Error) = 136.600.

As you can see group 1 differs from groups 2 and 3 but the latter two dont differ from each

other.

Main Effect Comparisons: Complex Comparisons

Individual Complex Comparisons

Specifying complex comparisons with SPSS is harder than you might expect. There are

two main ways to do this and theyre both pretty tricky. Let me show you the method I think willbe easiest for you. Im not going to explain why this works this is simply a how-to, cookbook

approach (though I certainly DO assume you know comparisons at well more than a cookbook

level). Really understanding why this works probably requires a class in linear models and/ormultivariate analysis.

Lets say we wanted to do the complex contrast comparing the mean of drug groups 1 and 2 todrug group 3. Our contrast coefficients could be 1 1 -2. I would do this using the MANOVA

command in the syntax editor. The key thing is to use the CONTRAST option and to specify


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what SPSS calls a SPECIAL contrast because its not one of their standard defaults. The key

thing here is that you have to create a square matrix of coefficients with the numbers of rows andcolumns equal to the number of marginal means associated with the factor of interest. Here that

number = 3. So you need a 3 X 3 matrix. The first row of this matrix must be all 1s (NOT the

contrast youre interested in). Make the second row the coefficients of the contrast youre

interested in. Make the third row the coefficients of an additional contrast (often the best choicehere is a contrast thats orthogonal to the first but that need not be the case). Obviously if you

have a second contrast of interest then use it for row 3. If you have 4 groups you need to enter a

4 X 4 matrix with all rows being 1s, then put three contrasts in rows 2-4 and make sure one ofthem is the contrast youre interested in.

The overall syntax here is this:

MANOVA bp by bio_gp (1,2) drug_gp (1,3)

/ERROR = W/Contrast(drug_gp) = SPECIAL (1 1 1

1 1 -21 -2 1).

As you can see, were using the MANOVA command that denotes the dv and the two factors

(including information about the levels of each factor (i.e., from 1 to 2 and from 1 to 3). Put

ERROR=W to make sure the MSW is used as the error term. Then put in the Contrast statement.Note how row 2 is our contrast of interest (1 1 -2). I put in a second complex contrast in the third

row as well. If you run this you get the following output:

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - -

* * * * * * * * * * * * * * * * * A n a l y s i s o f V a r i a n c e * * * * * ** * * * * * * * * * *

36 cases accepted.0 cases rejected because of out-of-range factor values.0 cases rejected because of missing data.6 non-empty cells.

1 design will be processed.

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - - -

* * * * * * * * * * * * * * * * * A n a l y s i s o f V a r i a n c e -- Design1 * * * * * * * * * * * * * * * * *

Tests of Significance for bp using UNIQUE sums of squaresSource of Variation SS DF MS F Sig of F

WITHIN CELLS 4098.00 30 136.60bio_gp 1296.00 1 1296.00 9.49 .004drug_gp 4104.00 2 2052.00 15.02 .000bio_gp BY drug_gp 1152.00 2 576.00 4.22 .024


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(Model) 6552.00 5 1310.40 9.59 .000(Total) 10650.00 35 304.29

R-Squared = .615Adjusted R-Squared = .551

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

- - - - - - - - - - - - - - - - -Estimates for bp--- Individual univariate .9500 confidence intervals

bio_gp

Parameter Coeff. Std. Err. t-Value Sig. t Lower -95% CL- Upper

2 -6.0000000000 1.94793 -3.08019 .00440 -9.97821 -2.02179

drug_gp


3 -18.0000000000 8.26438 -2.17802 .03740 -34.87812 -1.12188

4 -27.0000000000 8.26438 -3.26703 .00272 -43.87812 -10.12188

bio_gp BY drug_gp


5 12.0000000000 8.26438 1.45201 .15688 -4.87812 28.878126 -24.0000000000 8.26438 -2.90403 .00685 -40.87812 -7.12188

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Here its kind of hard to find what you want. However, the key thing here are the parameters for

drug_gp. The first one shown (# 3 in bold) is the one for our contrast of interest. In fact, if youuse the contrast coefficients of 1, 1, and -2 and the marginal means, you will find that the value

of the contrast is -18. You will see the associated t statistic and p value.

Multiple Complex Comparisons.

You can use the general approach noted above when you have multiple complex

comparisons or a combination of complex and pairwise comparisons. Note here that your typicalalternatives will be Bonferroni or Scheffe. If its Bonferroni, just use the general approach

described above (you might have to run more than one MANOVA to get all the contrasts in) and

note the critical per comparison t and alpha levels youre shooting for. If youre doing Scheffes,

take the t value shown in the SPSS output and square it to get your observed F. Then compare itto the Scheffe critical value = (df for the main effect of interest) X critical F for that effect for the

overall ANOVA at alpha = .05.

Simple Effects Analyses

Its fairly easy to run simple effects analyses using SPSS. Again use the syntax editor. Lets say

we want to estimate the simple effects of drug at biofeedback. That is, we want to compare thethree drug groups at biofeedback present and absent. To do this, use the following code:

GLM bp by bio_gp drug_gp

/EMMEANS Tables (bio_gp*drug_gp) Compare(drug_gp).


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At the bottom of the output, youll see the simple effects F tests:

Univariate Tests


bio_gp Sum of Squares Df Mean Square F Sig.

1 Contrast 3924.000 2 1962.000 14.363 .000

Error 4098.000 30 136.600

2 Contrast 1332.000 2 666.000 4.876 .015

Error 4098.000 30 136.600

Each F tests the simple effects of drug_gp within each level combination of the other effects shown.These tests are based on the linearly independent pairwise comparisons among the estimatedmarginal means.

This result is identical to what we computed by hand and via SAS.

Interaction Contrasts

To do interaction contrasts in SPSS, I would again use the syntax editor and use a specification

thats similar to what you would use with SAS. Below is the syntax for specifying the

interaction contrast between drugs x and y and biofeedback. The key thing here is to notesomething similar that I noted for SAS. Note how in the GLM statement bio_gp appears first.

This means that it moves more slowly. So that the six cells in order in the LMATRIX portion

are bio present/X, bio present/Y, bio-present Z, bio_absent/X, bio_absent Y, bio_absent Z.


/LMATRIX = 'x vs y by biofeedback interaction contrast' bio_gp*drug_gp 1 -1 0 -1 1 0.

The key thing here is to note something similar that I noted for SAS. Note how in the GLM

statement bio_gp appears first. This means that it moves more slowly. So that the six cells in

order in the LMATRIX portion are bio present/X, bio present/Y, bio-present Z, bio_absent/X,bio_absent Y, bio_absent Z. Another way to think of this that might be easier because it retains

the 2-dimensional row X column format is that biofeedback conditions constitute the rows

(because they come first in the by statement and drug_gp constitutes the columns. For example,

we could rewrite the LMATRIX statement above in the following manner:


/LMATRIX = 'x vs y by biofeedback interaction contrast' bio_gp*drug_gp 1 -1 0-1 1 0.

Either way will work. The output is as follows:


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Contrast Results (K Matrix)a

Contrast

DependentVariable

bp

L1 Contrast Estimate -24.000

Hypothesized Value 0

Difference (Estimate - Hypothesized) -24.000

Std. Error 9.543

Sig. .017

95% Confidence Interval for

Difference

Lower Bound -43.489

Upper Bound -4.511

a. Based on the user-specified contrast coefficients (L') matrix: x vs y by biofeedbackinteraction contrast

Test Results


SourceSum ofSquares df

MeanSquare F Sig.

Contrast 864.000 1 864.000 6.325 .017

Error 4098.000 30 136.600

The results for this contrast are identical to what we observed for SAS.

Non-orthogonal ANOVA

Remember that for non-orthogonal ANOVAs you will have the choice of Type 1 or Type 3 SS

approaches. The default in SPSS will be Type 3 SS. If you want to use Type 1, just use the pull-


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down menu. The key strokes are Analyze >> General Linear Models >> Univariate Enter DV

and Fixed Factors if you havent done so >> Model. You will see the following menu:

Just enter Type 1 in the SS box. Then hit Continue and OK. This would be equivalent to thefollowing statements in the syntax editor:

UNIANOVA bp BY bio_gp drug_gp

/METHOD=SSTYPE(1).

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Two-way Anova Using SPSS