Let’s use an example dataset, crf24.sav, adapted from Kirk (1968, First Edition).
get file 'c:\temp\crf24.sav'.
These data are from a 2×4 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
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
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.