### SPSS FAQ How can I do ANOVA contrasts in SPSS?

Let's use an example dataset, crf24, adapted from Kirk (1968, First Edition).

get file 'd:\crf24.sav'.

These data are from a 2x4 factorial design but the same data can also be used for one-way ANOVA examples.  The variable y is the dependent variable.  The variable a is an independent variable with two levels, while b is an independent variable with four levels.

#### Using the contrast command in a one-way ANOVA

glm y by b.
Between-Subjects Factors

N
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 194.500(a) 3 64.833 44.276 .000
Intercept 924.500 1 924.500 631.366 .000
B 194.500 3 64.833 44.276 .000
Error 41.000 28 1.464

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .826 (Adjusted R Squared = .807)
means tables = y by b
/ cells mean.
Case Processing Summary

Cases
Included Excluded Total
N Percent N Percent N Percent
Y * B 32 100.0% 0 .0% 32 100.0%
Report
Mean
B Y
1 2.75
2 3.50
3 6.25
4 9.00
Total 5.38

It is quite clear that there is a significant overall F for the independent variable b (F(3, 28) = 44.276, p = .000).  Now, let's devise some contrasts that we can test:
1) group 3 versus group 4
2) the average of groups 1 and 2 versus the average of groups 3 and 4
3) the average of groups 1, 2, and 3 versus group 4

glm y by b
/contrast(b)=special (0 0 1 -1).
Between-Subjects Factors

N
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 194.500(a) 3 64.833 44.276 .000
Intercept 924.500 1 924.500 631.366 .000
B 194.500 3 64.833 44.276 .000
Error 41.000 28 1.464

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .826 (Adjusted R Squared = .807)
Contrast Results (K Matrix)

Dependent Variable
B Special Contrast Y
L1 Contrast Estimate -2.750
Hypothesized Value 0
Difference (Estimate - Hypothesized) -2.750
Std. Error .605
Sig. .000
95% Confidence Interval for Difference Lower Bound -3.989
Upper Bound -1.511
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 30.250 1 30.250 20.659 .000
Error 41.000 28 1.464

This contrast is statistically significant (F(1, 28) = 20.659, p = .000).

glm y by b
/contrast(b)=special (1 1 -1 -1).
Between-Subjects Factors

N
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 194.500(a) 3 64.833 44.276 .000
Intercept 924.500 1 924.500 631.366 .000
B 194.500 3 64.833 44.276 .000
Error 41.000 28 1.464

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .826 (Adjusted R Squared = .807)
Contrast Results (K Matrix)

Dependent Variable
B Special Contrast Y
L1 Contrast Estimate -9.000
Hypothesized Value 0
Difference (Estimate - Hypothesized) -9.000
Std. Error .856
Sig. .000
95% Confidence Interval for Difference Lower Bound -10.753
Upper Bound -7.247
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 162.000 1 162.000 110.634 .000
Error 41.000 28 1.464

This contrast is also statistically significant (F(1, 28) = 110.634, p = .000).

glm y by b
/contrast(b)=special (1 1 1 -3).
Between-Subjects Factors

N
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 194.500(a) 3 64.833 44.276 .000
Intercept 924.500 1 924.500 631.366 .000
B 194.500 3 64.833 44.276 .000
Error 41.000 28 1.464

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .826 (Adjusted R Squared = .807)
Contrast Results (K Matrix)

Dependent Variable
B Special Contrast Y
L1 Contrast Estimate -14.500
Hypothesized Value 0
Difference (Estimate - Hypothesized) -14.500
Std. Error 1.482
Sig. .000
95% Confidence Interval for Difference Lower Bound -17.536
Upper Bound -11.464
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 140.167 1 140.167 95.724 .000
Error 41.000 28 1.464

This contrast is also statistically significant (F(1, 28) = 95.724, p = .000).

Note that you can enter multiple contrasts in a single subcommand, as shown below.  Each contrast must be separated by a comma.  While you get the significance tests for each individual test, you do not get the t-value.  To obtain the t-value, you will have to divide the contrast estimate by the std. error in the Contrast Results (K Matrix) table.

glm y by b
/contrast(b)=special (0 0 1 -1, 1 1 -1 -1,  1 1 1 -3).
Between-Subjects Factors

N
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 194.500(a) 3 64.833 44.276 .000
Intercept 924.500 1 924.500 631.366 .000
B 194.500 3 64.833 44.276 .000
Error 41.000 28 1.464

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .826 (Adjusted R Squared = .807)
Contrast Results (K Matrix)

Dependent Variable
B Special Contrast Y
L1 Contrast Estimate -2.750
Hypothesized Value 0
Difference (Estimate - Hypothesized) -2.750
Std. Error .605
Sig. .000
95% Confidence Interval for Difference Lower Bound -3.989
Upper Bound -1.511
L2 Contrast Estimate -9.000
Hypothesized Value 0
Difference (Estimate - Hypothesized) -9.000
Std. Error .856
Sig. .000
95% Confidence Interval for Difference Lower Bound -10.753
Upper Bound -7.247
L3 Contrast Estimate -14.500
Hypothesized Value 0
Difference (Estimate - Hypothesized) -14.500
Std. Error 1.482
Sig. .000
95% Confidence Interval for Difference Lower Bound -17.536
Upper Bound -11.464
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 192.250 2 96.125 65.646 .000
Error 41.000 28 1.464

#### Using the contrast command in a two-way ANOVA

Now let's try the same contrasts on b but in a two-way ANOVA.
glm y by a b.
Between-Subjects Factors

N
A 1 16
2 16
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 217.000(a) 7 31.000 40.216 .000
Intercept 924.500 1 924.500 1199.351 .000
A 3.125 1 3.125 4.054 .055
B 194.500 3 64.833 84.108 .000
A * B 19.375 3 6.458 8.378 .001
Error 18.500 24 .771

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .921 (Adjusted R Squared = .899)
glm y by a b
/contrast(b)=special (0 0 1 -1).
Between-Subjects Factors

N
A 1 16
2 16
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 217.000(a) 7 31.000 40.216 .000
Intercept 924.500 1 924.500 1199.351 .000
A 3.125 1 3.125 4.054 .055
B 194.500 3 64.833 84.108 .000
A * B 19.375 3 6.458 8.378 .001
Error 18.500 24 .771

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .921 (Adjusted R Squared = .899)
Contrast Results (K Matrix)

Dependent Variable
B Special Contrast Y
L1 Contrast Estimate -2.750
Hypothesized Value 0
Difference (Estimate - Hypothesized) -2.750
Std. Error .439
Sig. .000
95% Confidence Interval for Difference Lower Bound -3.656
Upper Bound -1.844
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 30.250 1 30.250 39.243 .000
Error 18.500 24 .771

glm y by a b
/contrast(b)=special (1 1 -1 -1).
Between-Subjects Factors

N
A 1 16
2 16
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 217.000(a) 7 31.000 40.216 .000
Intercept 924.500 1 924.500 1199.351 .000
A 3.125 1 3.125 4.054 .055
B 194.500 3 64.833 84.108 .000
A * B 19.375 3 6.458 8.378 .001
Error 18.500 24 .771

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .921 (Adjusted R Squared = .899)
Contrast Results (K Matrix)

Dependent Variable
B Special Contrast Y
L1 Contrast Estimate -9.000
Hypothesized Value 0
Difference (Estimate - Hypothesized) -9.000
Std. Error .621
Sig. .000
95% Confidence Interval for Difference Lower Bound -10.281
Upper Bound -7.719
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 162.000 1 162.000 210.162 .000
Error 18.500 24 .771

glm y by a b
/contrast(b)=special (1 1 1 -3).
Between-Subjects Factors

N
A 1 16
2 16
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 217.000(a) 7 31.000 40.216 .000
Intercept 924.500 1 924.500 1199.351 .000
A 3.125 1 3.125 4.054 .055
B 194.500 3 64.833 84.108 .000
A * B 19.375 3 6.458 8.378 .001
Error 18.500 24 .771

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .921 (Adjusted R Squared = .899)
Contrast Results (K Matrix)

Dependent Variable
B Special Contrast Y
L1 Contrast Estimate -14.500
Hypothesized Value 0
Difference (Estimate - Hypothesized) -14.500
Std. Error 1.075
Sig. .000
95% Confidence Interval for Difference Lower Bound -16.719
Upper Bound -12.281
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 140.167 1 140.167 181.838 .000
Error 18.500 24 .771

Note that the F-ratios in these contrasts are larger than the F-ratios in the one-way ANOVA example.  This is because the two-way ANOVA has a smaller mean square residual than the one-way ANOVA.

SPSS has a number of built-in contrasts that you can use, of which special (used in the above examples) is only one.  Below is a table listing those contrasts with an explanation of the contrasts that they make and an example of how the syntax works.  The repeated contrast compares group 1 with 2, 2 with 3, and 3 with 4 as shown in the Contrast Results (K Matrix) table in the results.

 Name of contrast Comparison made Simple Compares each level of a variable to the last level (or whichever level is specified) Deviation Compares deviations from the grand mean Difference Compares levels of a variable with the mean of the previous levels of the variable Helmert Compare levels of a variable with the mean of the subsequent levels of the variable Polynomial Orthogonal polynomial contrasts Repeated Adjacent levels of a variable Special User-defined contrast
glm y by a b
/contrast(b)=repeated.
Between-Subjects Factors

N
A 1 16
2 16
B 1 8
2 8
3 8
4 8
Tests of Between-Subjects Effects
Dependent Variable: Y
Source Type III Sum of Squares df Mean Square F Sig.
Corrected Model 217.000(a) 7 31.000 40.216 .000
Intercept 924.500 1 924.500 1199.351 .000
A 3.125 1 3.125 4.054 .055
B 194.500 3 64.833 84.108 .000
A * B 19.375 3 6.458 8.378 .001
Error 18.500 24 .771

Total 1160.000 32

Corrected Total 235.500 31

a R Squared = .921 (Adjusted R Squared = .899)
Contrast Results (K Matrix)

Dependent Variable
B Repeated Contrast Y
Level 1 vs. Level 2 Contrast Estimate -.750
Hypothesized Value 0
Difference (Estimate - Hypothesized) -.750
Std. Error .439
Sig. .100
95% Confidence Interval for Difference Lower Bound -1.656
Upper Bound .156
Level 2 vs. Level 3 Contrast Estimate -2.750
Hypothesized Value 0
Difference (Estimate - Hypothesized) -2.750
Std. Error .439
Sig. .000
95% Confidence Interval for Difference Lower Bound -3.656
Upper Bound -1.844
Level 3 vs. Level 4 Contrast Estimate -2.750
Hypothesized Value 0
Difference (Estimate - Hypothesized) -2.750
Std. Error .439
Sig. .000
95% Confidence Interval for Difference Lower Bound -3.656
Upper Bound -1.844
Test Results
Dependent Variable: Y
Source Sum of Squares df Mean Square F Sig.
Contrast 194.500 3 64.833 84.108 .000
Error 18.500 24 .771

For more information on coding contrasts, please see How can I use the lmatrix subcommand to understand a three-way interaction in ANOVA? .

#### References

Kirk, Roger E. (1968) Experimental Design: Procedures for the Behavioral Sciences. Monterey, California: Brooks/Cole Publishing.

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