Regression with SPSS
Chapter 5: Additional coding systems for categorical variables in regression analysis 

Chapter Outline
    5.1 Simple Coding
    5.2 Forward Difference Coding
    5.3 Backward Difference Coding
    5.4 Helmert Coding
    5.5 Reverse Helmert Coding
    5.6 Deviation Coding
    5.7 Orthogonal Polynomial Coding
    5.8 User-Defined Coding
   
5.9 Summary
    5.10 For more information

Introduction

Categorical variables require special attention in regression analysis because, unlike dichotomous or continuous variables, they cannot by entered into the regression equation just as they are.  For example, if you have a variable called race that is coded 1 = Hispanic, 2 = Asian 3 = Black 4 = White, then entering race in your regression will look at the linear effect of race, which is probably not what you intended. Instead, categorical variables like this need to be recoded into a series of variables which can then be entered into the regression model.  There are a variety of coding systems that can be used when coding categorical variables.  Ideally, you would choose a coding system that reflects the comparisons that you want to make.  In Chapter 3 of the Regression with SPSS Web Book we covered the use of categorical variables in regression analysis focusing on the use of dummy variables, but that is not the only coding scheme that you can use.  For example, you may want to compare each level to the next higher level, in which case you would want to use "forward difference" coding, or you might want to compare each level to the mean of the subsequent levels of the variable, in which case you would want to use "Helmert" coding.  By deliberately choosing a coding system, you can obtain comparisons that are most meaningful for testing your hypotheses.  Regardless of the coding system you choose, the test of the overall effect of the categorical variable (i.e., the overall effect of race) will remain the same.  Below is a table listing various types of contrasts and the comparison that they make.  
 
Name of contrast Comparison made
Simple Coding Compares each level of a variable to the reference level
Forward Difference Coding Adjacent levels of a variable (each level minus the next level)
Backward Difference Coding Adjacent levels of a variable (each level minus the prior level)
Helmert Coding Compare levels of a variable with the mean of the subsequent levels of the variable
Reverse Helmert Coding Compares levels of a variable with the mean of the previous levels of the variable
Deviation Coding Compares deviations from the grand mean
Orthogonal Polynomial Coding Orthogonal polynomial contrasts
User-Defined Coding User-defined contrast

There are a couple of notes to be made about the coding systems listed above.  The first is that they represent planned comparisons and not post hoc comparisons.  In other words, they are comparisons that you plan to do before you begin analyzing your data, not comparisons that you think of once you have seen the results of preliminary analyses.  Also, some forms of coding make more sense with ordinal categorical variables than with nominal categorical variables. Below we will show examples using race as a categorical variable, which is a nominal variable.  Because simple effect coding compares the mean of the dependent variable for each level of the categorical variable to the mean of the dependent variable at for the reference level, it makes sense with a nominal variable.  However, it may not make as much sense to use a coding scheme that tests the linear effect of race.  As we describe each type of coding system, we note those coding systems with which it does not make as much sense to use a nominal variable.  Also, you may notice that we follow several rules when creating the contrast coding schemes.  For more information about these rules, please see the section on User-Defined Coding.

This page will illustrate three ways that you can conduct analyses using these coding schemes: 1) using the glm command with /lmatrix to define "contrast" coefficients that specify levels of the categorical variable that are to be compared, 2) using the glm command with /contrast to specify one of the SPSS predefined coding schemes, or 3) using regression. When using regression to do contrasts, you first need to create k-1 new variables (where k is the number of levels of the categorical variable) and use these new variables as predictors in your regression model. While methods 1 and 3 both involve manually specifying "contrasts", method 1 uses a type of coding we will call "contrast coding";  method 3 uses a type of coding we will call "regression coding".  

There are benefits and drawbacks of each of these three methods.  For example, methods 1 and 3 allow you to manually code the contrasts and give you absolute control over the coding, but the drawback is that it is relatively easy to make an error in the coding.  By contrast, method 2 automates the process by letting SPSS do the coding for you, but you are limited to just the pre-defined coding schemes that SPSS has created.  Method 3 can be the most difficult, but it can be used with any kind of regression procedure.

The Example Data File

The examples in this page will use dataset called hsb2.sav and we will focus on the categorical variable race, which has four levels (1 = Hispanic, 2 = Asian, 3 = African American and 4 = white) and we will use write as our dependent variable.  Although our example uses a variable with four levels, these coding systems work with variables that have more or fewer categories. No matter which coding system you select, you will always have one fewer recoded variables than levels of the original variable.  In our example, our categorical variable has four levels so we will have three new variables (a variable corresponding to the final level of the categorical variables would be redundant and therefore unnecessary).

Before considering any analyses, let's look at the mean of the dependent variable, write, for each level of race.  This will help in interpreting the output from later analyses.

means tables = write by race.
Case Processing Summary

Cases
Included Excluded Total
N Percent N Percent N Percent
writing score * RACE 200 100.0% 0 .0% 200 100.0%
Report
writing score
RACE Mean N
hispanic 46.4583 24
asian 58.0000 11
african-amer 48.2000 20
white 54.0552 145
Total 52.7750 200

5.1 Simple Coding

The results of simple coding are very similar to dummy coding in that each level is compared to the reference level. In the example below, level 4 is the reference level and the first comparison compares level 1 to level 4, the second comparison compares level 2 to level 4, and the third comparison compares level 3 to level 4. 

Method 1: GLM with /LMATRIX

The table below shows the simple coding making the comparisons described above.  The first contrast compares level 1 to level 4, and level 1 is coded as 1 and level 4 is coded as -1.  Likewise, the second contrast compares level 2 to level 4 by coding level 2 as 1 and level 4 as -1.  As you can see with contrast coding, you can discern the meaning of the comparisons simply by inspecting the contrast coefficients.  For example, looking at the contrast coefficients for c3, you can see that it compares level 3 to level 4.

SIMPLE contrast coding
Level of race New variable 1 (c1) New variable 2 (c2) New variable 3 (c3)
1 (Hispanic) 1 0 0
2 (Asian) 0 1 0
3 (African American) 0 0 1
4 (white) -1 -1 -1

Below we illustrate how to form these comparisons using the glm command with /lmatrix.  As you see, a separate /lmatrix subcommand is used for each contrast.

glm write by race
 /lmatrix "level 1 versus level 4" race 1 0 0 -1
 /lmatrix "level 2 versus level 4" race 0 1 0 -1
 /lmatrix "level 3 versus level 4" race 0 0 1 -1.

Each of the above /lmatrix subcommands produced two tables shown below, "Contrast Results (K Matrix)" and "Test Results".  The contrast estimate for the first contrast compares the mean of the dependent variable, write, for levels 1 and 4 yielding -7.597 and is statistically significant (p<.000). The F-value associated with this test is given in the "Test Results" table and is 14.590. The p-value given in the "Contrast Results (K Matrix)" table and the p-value in the "Test Results" table are the same because they both refer to the same test of the contrast coefficient to zero.  The results of the second contrast, comparing the mean of write for levels 2 and 4 is not statistically significant (F = 1.953, p = .164), while the third contrast is statistically significant.

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -7.597
Hypothesized Value 0
Difference (Estimate - Hypothesized) -7.597
Std. Error 1.989
Sig. .000
95% Confidence Interval for Difference Lower Bound -11.519
Upper Bound -3.675
a Based on the user-specified contrast coefficients (L') matrix: group 1 versus group 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 1188.388 1 1188.388 14.590 .000
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 3.945
Hypothesized Value 0
Difference (Estimate - Hypothesized) 3.945
Std. Error 2.823
Sig. .164
95% Confidence Interval for Difference Lower Bound -1.622
Upper Bound 9.511
a Based on the user-specified contrast coefficients (L') matrix: group 2 versus group 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 159.108 1 159.108 1.953 .164
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -5.855
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.855
Std. Error 2.153
Sig. .007
95% Confidence Interval for Difference Lower Bound -10.101
Upper Bound -1.610
a Based on the user-specified contrast coefficients (L') matrix: group 3 versus group 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 602.550 1 602.550 7.398 .007
Error 15964.717 196 81.453

Method 2: GLM with /CONTRAST

Instead of using the /lmatrix subcommand, we can achieve the same results using the /contrast subcommand with the glm command. Instead of specifying the numbers to be used in the contrast as we did above, we can simply type in the name of the contrast that we wish to use, and SPSS will do the coding for us. We will use the /print = test(lmatrix) subcommand to have SPSS print out the coding scheme that it used to make the contrasts. You will notice that the table entitled "Contrast Coefficients (L' Matrix)" is the same as the table we used in method 1 above. 

glm write by race
 /contrast (race)=simple
 /print = test(lmatrix).

As you see in the output below, the table titled "Contrast Coefficients (L' Matrix)" shows the coding scheme that was used for each comparison.  The table entitled "Contrast Results (K Matrix)" shows the results of the various contrasts.  In our example, the difference between level 1 of race and level 4 of race is statistically significant.  You will notice that the contrast estimate is the difference between the mean for the dependent variable for the first level minus the mean of the dependent variable for the omitted level.  In other words, the mean for level 1 minus the mean for level 4 which is 46.4583 - 54.0552 = -7.597.  The row labeled "Sig." is .000, indicating that this difference is significant, and this is followed by a confidence interval for the difference. The next part of the table compares level 2 of race and level 4 of race and shows that this difference is not statistically significant and the next part of the table shows the difference between level 3 of race and level 4 of race is statistically significant. You might note that while the significance ("Sig.") is given for each of these tests, there is no "t" value, but you could obtain this by dividing the "Contrast Estimate" by the "Std. Error", i.e., -7.597 / 1.989. 

The table entitled "Test Results" indicates that the test of the overall effect race is statistically significant.  In other words, it is a test of all of the contrasts taken together. 

Contrast Coefficients (L' Matrix)

RACE Simple Contrast(a)
Parameter Level 1 vs. Level 4 Level 2 vs. Level 4 Level 3 vs. Level 4
Intercept 0 0 0
[RACE=1.00] 1 0 0
[RACE=2.00] 0 1 0
[RACE=3.00] 0 0 1
[RACE=4.00] -1 -1 -1
The default display of this matrix is the transpose of the corresponding L matrix.
a Reference category = 4
Contrast Results (K Matrix)

Dependent Variable
RACE Simple Contrast(a) writing score
Level 1 vs. Level 4 Contrast Estimate -7.597
Hypothesized Value 0
Difference (Estimate - Hypothesized) -7.597
Std. Error 1.989
Sig. .000
95% Confidence Interval for Difference Lower Bound -11.519
Upper Bound -3.675
Level 2 vs. Level 4 Contrast Estimate 3.945
Hypothesized Value 0
Difference (Estimate - Hypothesized) 3.945
Std. Error 2.823
Sig. .164
95% Confidence Interval for Difference Lower Bound -1.622
Upper Bound 9.511
Level 3 vs. Level 4 Contrast Estimate -5.855
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.855
Std. Error 2.153
Sig. .007
95% Confidence Interval for Difference Lower Bound -10.101
Upper Bound -1.610
a Reference category = 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 1914.158 3 638.053 7.833 .000
Error 15964.717 196 81.453

Method 3: Regression

The regression coding is a bit more complex than contrast coding.  In our example below, level 4 is the reference level and x1 compares level 1 to level 4, x2 compares level 2 to level 4, and x3 compares level 3 to level 4.  For x1 the coding is 3/4 for level 1, and -1/4 for all other levels.  Likewise, for x2 the coding is 3/4 for level 2, and -1/4 for all other levels, and for x3 the coding is 3/4 for level 3, and -1/4 for all other levels.  It is not intuitive that this regression coding scheme yields these comparisons; however, if you desire simple comparisons, you can follow this general rule to obtain these comparisons.

SIMPLE regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
1 (Hispanic) 3/4 -1/4 -1/4
2 (Asian) -1/4 3/4 -1/4
3 (African American) -1/4 -1/4 3/4
4 (white) -1/4 -1/4 -1/4

Below we show the more general rule for creating this kind of coding scheme using regression coding, where k is the number of levels of the categorical variable (in this instance, k=4).

SIMPLE regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
1 (Hispanic) (k-1) / k -1 / k -1 / k
2 (Asian) -1 / k (k-1) / k -1 / k
3 (African American) -1 / k -1 / k (k-1) / k
4 (white) -1 / k -1 / k -1 / k

Below we illustrate how to create x1, x2 and x3 and enter these new variables into the regression model using the regression command.

if race = 1 x1 = 3/4.
if any(race,2,3,4) x1 = -1/4.

if race = 2 x2 = 3/4.
if any(race,1,3,4) x2 = -1/4.

if race = 3 x3 = 3/4.
if any(race,1,2,4) x3 = -1/4.
execute.
regression
 /dependent = write
 /method = enter x1 x2 x3.

You will notice that the regression coefficients in the table below are the same as the contrast coefficients that we saw using the glm command.  Both the regression coefficient for x1 and the contrast estimate for c1 are the mean of write for level 1 of race (Hispanic) minus the mean of write for level 4 (white). Likewise, the regression coefficient for x2 and the contrast estimate for c2 are the mean of write for level 2 (Asian) minus the mean of write for level 4 (white). The F-value shown in the glm output is the square of the t-value shown in the regression table below. You also can see that the significance levels are also the same as those from the glm output.



Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.678 .982
52.619 .000
X1 -7.597 1.989 -.261 -3.820 .000
X2 3.945 2.823 .095 1.398 .164
X3 -5.855 2.153 -.186 -2.720 .007
a Dependent Variable: writing score

5.2 Forward Difference Coding

In this coding system, the mean of the dependent variable for one level of the categorical variable is compared to the mean of the dependent variable for the next (adjacent) level.  In our example below, the first comparison compares the mean of write for level 1 with the mean of write for level 2 of race (Hispanics minus Asians).  The second comparison compares the mean of write for level 2 minus level 3, and the third comparison compares the mean of write for level 3 minus level 4.  This type of coding may be useful with either a nominal or an ordinal variable.   

Method 1: GLM with /LMATRIX

FORWARD DIFFERENCE contrast coding
Level of race New variable 1 (c1) New variable 2 (c2) New variable 3 (c3)
  Level 1 v. Level 2 Level 2 v. Level 3 Level 3 v. Level 4
1 (Hispanic)  1 0 0
2 (Asian)  -1 1 0
3 (African American)  0 -1 1
4 (white)  0 0 -1

glm write by race
 /lmatrix "level 1 versus level 2" race 1 -1 0 0
 /lmatrix "level 2 versus level 3" race 0 1 -1 0
 /lmatrix "level 3 versus level 4" race 0 0 1 -1.
Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -11.542
Hypothesized Value 0
Difference (Estimate - Hypothesized) -11.542
Std. Error 3.286
Sig. .001
95% Confidence Interval for Difference Lower Bound -18.022
Upper Bound -5.061
a Based on the user-specified contrast coefficients (L') matrix: group 1 versus group 2
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 1004.785 1 1004.785 12.336 .001
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 9.800
Hypothesized Value 0
Difference (Estimate - Hypothesized) 9.800
Std. Error 3.388
Sig. .004
95% Confidence Interval for Difference Lower Bound 3.119
Upper Bound 16.481
a Based on the user-specified contrast coefficients (L') matrix: group 2 versus group 3
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 681.574 1 681.574 8.368 .004
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -5.855
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.855
Std. Error 2.153
Sig. .007
95% Confidence Interval for Difference Lower Bound -10.101
Upper Bound -1.610
a Based on the user-specified contrast coefficients (L') matrix: group 3 versus group 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 602.550 1 602.550 7.398 .007
Error 15964.717 196 81.453

With this coding system, adjacent levels of the categorical variable are compared.  Hence, the mean of the dependent variable at level 1 is compared to the mean of the dependent variable at level 2:  46.4583 - 58 = -11.542, which is statistically significant.  For the comparison between levels 2 and 3, the calculation of the contrast coefficient would be 58 - 48.2 = 9.8, which is also statistically significant.  Finally, comparing levels 3 and 4, 48.2 - 54.0552 = -5.855, a statistically significant difference.  One would conclude from this that each adjacent level of race is statistically significantly different.

Method 2: GLM with /CONTRAST

As with the previous examples, we will conduct the above analysis again, this time using the /contrast(race)=repeated subcommand to request forward difference contrasts.  You can compare the results below to those above to verify that the results are identical to those obtained using the /lmatrix subcommand.

glm write by race
 /contrast (race)=repeated
 /print = test(lmatrix).
Contrast Coefficients (L' Matrix)

RACE Repeated Contrast
Parameter Level 1 vs. Level 2 Level 2 vs. Level 3 Level 3 vs. Level 4
Intercept 0 0 0
[RACE=1.00] 1 0 0
[RACE=2.00] -1 1 0
[RACE=3.00] 0 -1 1
[RACE=4.00] 0 0 -1
The default display of this matrix is the transpose of the corresponding L matrix.
Contrast Results (K Matrix)

Dependent Variable
RACE Repeated Contrast writing score
Level 1 vs. Level 2 Contrast Estimate -11.542
Hypothesized Value 0
Difference (Estimate - Hypothesized) -11.542
Std. Error 3.286
Sig. .001
95% Confidence Interval for Difference Lower Bound -18.022
Upper Bound -5.061
Level 2 vs. Level 3 Contrast Estimate 9.800
Hypothesized Value 0
Difference (Estimate - Hypothesized) 9.800
Std. Error 3.388
Sig. .004
95% Confidence Interval for Difference Lower Bound 3.119
Upper Bound 16.481
Level 3 vs. Level 4 Contrast Estimate -5.855
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.855
Std. Error 2.153
Sig. .007
95% Confidence Interval for Difference Lower Bound -10.101
Upper Bound -1.610

Again, we see that the results are the same as those obtained using the /lmatrix subcommand: all three comparisons are statistically significant.

Method 3: Regression

For the first comparison, where the first and second levels are compared, x1 is coded 3/4 for level 1 and the other levels are coded -1/4.  For the second comparison where level 2 is compared with level 3, x2 is coded 1/2 1/2 -1/2 -1/2, and for the third comparison where level 3 is compared with level 4, x3 is coded 1/4 1/4 1/4 -3/4.  

FORWARD DIFFERENCE regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
  Level 1 v. Level 2 Level 2 v. Level 3 Level 3 v. Level 4
1 (Hispanic)  3/4 1/2 1/4
2 (Asian)  -1/4 1/2 1/4
3 (African American)  -1/4 -1/2 1/4
4 (white)  -1/4 -1/2 -3/4

The general rule for this regression coding scheme is shown below, where k is the number of levels of the categorical variable (in this case k=4).

FORWARD DIFFERENCE regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
  Level 1 v. Level 2 Level 2 v. Level 3 Level 3 v. Level 4
1 (Hispanic)  (k-1)/k (k-2)/k (k-3)/k
2 (Asian)  -1/k (k-2)/k (k-3)/k
3 (African American)  -1/k -2/k (k-3)/k
4 (white)  -1/k -2/k -3/k

if race = 1 x1 = 3/4.
if any(race,2,3,4) x1 = -1/4.

if any(race,1,2) x2 = 1/2.
if any(race,3,4) x2 = -1/2.

if any(race,1,2,3) x3 = 1/4.
if race = 4 x3 = -3/4.
execute.

regression
 /dep write
 /method = enter x1 x2 x3.


Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.678 .982
52.619 .000
X1 -11.542 3.286 -.397 -3.512 .001
X2 9.800 3.388 .394 2.893 .004
X3 -5.855 2.153 -.277 -2.720 .007
a Dependent Variable: writing score

You can see the regression coefficient for x1 is the mean of write for level 1 (Hispanic) minus the mean of write for level 2 (Asian).  Likewise, the regression coefficient for x2 is the mean of write for level 2 (Asian) minus the mean of write for level 3 (African American), and the regression coefficient for x3 is the mean of write for level 3 (African American) minus the mean of write for level 4 (white).

5.3 Backward Difference Coding

In this coding system, the mean of the dependent variable for one level of the categorical variable is compared to the mean of the dependent variable for the prior adjacent level.  In our example below, the first comparison compares the mean of write for level 2 with the mean of write for level 1 of race (Asians minus Hispanics).  The second comparison compares the mean of write for level 3 minus level 2, and the third comparison compares the mean of write for level 4 minus level 3.  This type of coding may be useful with either a nominal or an ordinal variable.   

Method 1: GLM with /LMATRIX

BACKWARD DIFFERENCE contrast coding
Level of race New variable 1 (c1) New variable 2 (c2) New variable 3 (c3)
  Level 1 v. Level 2 Level 2 v. Level 3 Level 3 v. Level 4
1 (Hispanic)  -1 0 0
2 (Asian)  1 -1 0
3 (African American)  0 1 -1
4 (white)  0 0 1

glm write by race
 /lmatrix "level 1 versus level 2" race -1 1 0 0
 /lmatrix "level 2 versus level 3" race 0 -1 1 0
 /lmatrix "level 3 versus level 4" race 0 0 -1 1.
Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 11.542
Hypothesized Value 0
Difference (Estimate - Hypothesized) 11.542
Std. Error 3.286
Sig. .001
95% Confidence Interval for Difference Lower Bound 5.061
Upper Bound 18.022
a Based on the user-specified contrast coefficients (L') matrix: level 1 versus level 2
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 1004.785 1 1004.785 12.336 .001
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -9.800
Hypothesized Value 0
Difference (Estimate - Hypothesized) -9.800
Std. Error 3.388
Sig. .004
95% Confidence Interval for Difference Lower Bound - 16.481
Upper Bound - 3.119
a Based on the user-specified contrast coefficients (L') matrix: level 2 versus level 3
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 681.574 1 681.574 8.368 .004
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 5.855
Hypothesized Value 0
Difference (Estimate - Hypothesized) 5.855
Std. Error 2.153
Sig. .007
95% Confidence Interval for Difference Lower Bound 1.610
Upper Bound 10.101
a Based on the user-specified contrast coefficients (L') matrix: level 3 versus level 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 602.550 1 602.550 7.398 .007
Error 15964.717 196 81.453

With this coding system, adjacent levels of the categorical variable are compared, with each level compared to the prior level.  Hence, the mean of the dependent variable at level 2 is compared to the mean of the dependent variable at level 1:  58-46.4583 = 11.542, which is statistically significant.  For the comparison between levels 3 and 2, the calculation of the contrast coefficient is 48.2 - 58 = -9.8, which is also statistically significant.  Finally, comparing levels 4 and 3, 54.0552 - 48.2 = 5.855, a statistically significant difference.  One would conclude from this that each adjacent level of race is statistically significantly different.

Method 2: GLM with /CONTRAST

While SPSS has the /contrast (race)=repeated subcommand for comparing each level to the next level, it does not have an equivalent command for comparing each level to a prior level.  This would need to be done via the /lmatrix subcommand, or by reversing the coding of the categorical variable.

Method 3: Regression

For the first comparison, where the first and second levels are compared, x1 is coded 3/4 for level 1 while the other levels are coded -1/4.  For the second comparison where level 2 is compared with level 3, x2 is coded 1/2 1/2 -1/2 -1/2, and for the third comparison where level 3 is compared with level 4, x3 is coded 1/4 1/4 1/4 -3/4. 

BACKWARD DIFFERENCE regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
  Level 2 v. Level 1 Level 3 v. Level 2 Level 4 v. Level 3
1 (Hispanic) - 3/4 -1/2 -1/4
2 (Asian)  1/4 -1/2 -1/4
3 (African American)  1/4 1/2 -1/4
4 (white)  1/4 1/2 3/4

The general rule for this regression coding scheme is shown below, where k is the number of levels of the categorical variable (in this case, k is 4).

BACKWARD DIFFERENCE regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
  Level 1 v. Level 2 Level 2 v. Level 3 Level 3 v. Level 4
1 (Hispanic)  -(k-1)/k -(k-2)/k -(k-3)/k
2 (Asian) 1/k -(k-2)/k -(k-3)/k
3 (African American)  1/k 2/k -(k-3)/k
4 (white)  1/k 2/k 3/k

if race = 1 x1 = -3/4.
if any(race,2,3,4) x1 = 1/4.

if any(race,1,2) x2 = -1/2.
if any(race,3,4) x2 = 1/2.

if any(race,1,2,3) x3 = -1/4.
if race = 4 x3 = 3/4.
execute.

regression
 /dep write
 /method = enter x1 x2 x3.


Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.678 .982
52.619 .000
X1 11.542 3.286 .397 3.512 .001
X2 -9.800 3.388 -.394 -2.893 .004
X3 5.855 2.153 .277 2.720 .007
a Dependent Variable: writing score

In the above example, the regression coefficient for x1 is the mean of write for level 2 minus the mean of write for level 1 (58- 46.4583 = 11.542).  Likewise, the regression coefficient for x2 is the mean of write for level 3 minus the mean of write for level 2, and the regression coefficient for x3 is the mean of write for level 4 minus the mean of write for level 3.

5.4 Helmert Coding

Helmert coding compares each level of a categorical variable to the mean of the subsequent levels.  Hence, the first contrast compares the mean of the dependent variable for level 1 of race with the mean of all of the subsequent levels of race (levels 2, 3, and 4), the second contrast compares the mean of the dependent variable for level 2 of race with the mean of all of the subsequent levels of race (levels 3 and 4), and the third contrast compares the mean of the dependent variable for level 3 of race with the mean of all of the subsequent levels of race (level 4). While this type of coding system does not make much sense with a nominal variable like race, it is useful in situations where the levels of the categorical variable are ordered say, from lowest to highest, or smallest to largest, etc.

For Helmert coding, we see that the first comparison comparing level 1 with levels 2, 3 and 4 is coded 1, -1/3, -1/3 and -1/3, reflecting the comparison of level 1 with all other levels.  The second comparison is coded 0, 1, -1/2 and -1/2, reflecting that it compares level 2 with levels 3 and 4.  The third comparison is coded 0, 0, 1 and -1, reflecting that level 3 is compared to level 4.

Method 1: GLM with /LMATRIX

HELMERT contrast coding
Level of race New variable 1 (c1) New variable 2 (c2) New variable 3 (c3)
  Level 1 v. Later Level 2 v. Later Level 3 v. Later
1 (Hispanic)  1 0 0
2 (Asian)  -1/3 1 0
3 (African American)  -1/3 -1/2 1
4 (white)  -1/3 -1/2 -1

Below we illustrate how to form these comparisons using the glm command with /lmatrix.  Note the use of fractions on the first  /lmatrix subcommand.  You cannot use .333 instead of 1/3:  SPSS will give an error message and fail to calculate the contrast coefficient.  The problem is that .333 + .333 + .333 - 1 is not sufficiently close to zero. (If you wanted to use decimals, you would need to use something like .333333 so that  .333333 + .333333 + .333333 - 1 would be sufficiently close to zero.)

glm write by race
 /lmatrix "level 1 versus levels 2 3 and 4" race 1 -1/3 -1/3 -1/3.
 /lmatrix "level 2 versus levels 3 and 4" race   0  1    -1/2  -1/2.
 /lmatrix "level 3 versus level 4" race          0  0      1   -1.

The contrast estimate for the comparison between level 1 and the remaining levels is calculated by taking the mean of the dependent variable for level 1 and subtracting the mean of the dependent variable for levels 2, 3 and 4: 46.4583 - [(58 + 48.2 + 54.0552) / 3] = -6.960, which is statistically significant.  This means that the mean of write for level 1 of race is statistically significantly different from the mean of write for levels 2 through 4.  As noted above, this comparison probably is not meaningful because the variable race is nominal.  This type of comparison would be more meaningful if the categorical variable was ordinal.  

To calculate the contrast coefficient for the comparison between level 2 and the later levels, you subtract the mean of the dependent variable for levels 3 and 4 from the mean of the dependent variable for level 2:  58 - [(48.2 + 54.0552) / 2] = 6.872, which is statistically significant.  The contrast estimate for the comparison between level 3 and level 4 is the difference between the mean of the dependent variable for the two levels:  48.2 - 54.0552 = -5.855, which is also statistically significant.

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -6.960
Hypothesized Value 0
Difference (Estimate - Hypothesized) -6.960
Std. Error 2.175
Sig. .002
95% Confidence Interval for Difference Lower Bound -11.250
Upper Bound -2.670
a Based on the user-specified contrast coefficients (L') matrix: group 1 versus groups 2 3 and 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 833.927 1 833.927 10.238 .002
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 6.872
Hypothesized Value 0
Difference (Estimate - Hypothesized) 6.872
Std. Error 2.926
Sig. .020
95% Confidence Interval for Difference Lower Bound 1.101
Upper Bound 12.644
a Based on the user-specified contrast coefficients (L') matrix: group 2 versus groups 3 and 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 449.240 1 449.240 5.515 .020
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -5.855
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.855
Std. Error 2.153
Sig. .007
95% Confidence Interval for Difference Lower Bound -10.101
Upper Bound -1.610
a Based on the user-specified contrast coefficients (L') matrix: group 3 versus group 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 602.550 1 602.550 7.398 .007
Error 15964.717 196 81.453

Method 2: GLM with /CONTRAST

As with the previous examples, we will conduct the analysis above again, this time using the /contrast subcommand.  

glm write by race
 /contrast (race)=helmert
 /print = test(lmatrix).

This output shows the three comparisons: the mean of write for level 1 of race to the mean of write for the other three levels (called "later" in this output), the mean of write for level 2 of race to the mean of write for the other two levels, etc.  Again, all three comparisons are statistically significant.

Contrast Coefficients (L' Matrix)

RACE Helmert Contrast
Parameter Level 1 vs. Later Level 2 vs. Later Level 3 vs. Level 4
Intercept .000 .000 .000
[RACE=1.00] 1.000 .000 .000
[RACE=2.00] -.333 1.000 .000
[RACE=3.00] -.333 -.500 1.000
[RACE=4.00] -.333 -.500 -1.000
The default display of this matrix is the transpose of the corresponding L matrix.
Contrast Results (K Matrix)

Dependent Variable
RACE Helmert Contrast writing score
Level 1 vs. Later Contrast Estimate -6.960
Hypothesized Value 0
Difference (Estimate - Hypothesized) -6.960
Std. Error 2.175
Sig. .002
95% Confidence Interval for Difference Lower Bound -11.250
Upper Bound -2.670
Level 2 vs. Later Contrast Estimate 6.872
Hypothesized Value 0
Difference (Estimate - Hypothesized) 6.872
Std. Error 2.926
Sig. .020
95% Confidence Interval for Difference Lower Bound 1.101
Upper Bound 12.644
Level 3 vs. Level 4 Contrast Estimate -5.855
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.855
Std. Error 2.153
Sig. .007
95% Confidence Interval for Difference Lower Bound -10.101
Upper Bound -1.610

Method 3: Regression

Below we see an example of Helmert regression coding.  For the first comparison (comparing level 1 with levels 2, 3 and 4) the codes are 3/4 and -1/4 -1/4 -1/4.  The second comparison compares level 2 with levels 3 and 4 and is coded 0 2/3 -1/3 -1/3.  The third comparison compares level 3 to level 4 and is coded 0 0 1/2 -1/2. 

HELMERT regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
  Level 1 v. Later Level 2 v. Later Level 3 v. Later
1 (Hispanic)  3/4 0 0
2 (Asian)  -1/4 2/3 0
3 (African American)  -1/4 -1/3 1/2
4 (white)  -1/4 -1/3 -1/2

Below we illustrate how to create x1, x2 and x3 and enter these new variables into the regression model using the regression command.

if race = 1 x1 = 3/4.
if any(race,2,3,4) x1 = -1/4.

if race = 1 x2 = 0.
if race = 2 x2 = .667.
if any(race,3,4) x2 = -1/3.

if any(race,1,2) x3 = 0.
if race = 3 x3 = 1/2.
if race = 4 x3 = -1/2.
execute.

regression
 /dep write
 /method = enter x1 x2 x3.

As you see below, the regression coefficient for x1 is the mean of write for level 1 (Hispanic) versus all subsequent levels (levels 2, 3 and 4).  Likewise, the regression coefficient for x2 is the mean of write for level 2 minus the mean of write for levels 3 and 4.  Finally, the regression coefficient for x3 is the mean of write for level 3 minus the mean of write for level 4.



Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.677 .982
52.635 .000
X1 -6.958 2.175 -.239 -3.199 .002
X2 6.872 2.926 .177 2.348 .020
X3 -5.855 2.153 -.204 -2.720 .007
a Dependent Variable: writing score

5.5 Reverse Helmert Coding

Reverse Helmert coding (also know as difference coding) is just the opposite of Helmert coding: instead of comparing each level of categorical variable to the mean of the subsequent level(s), each is compared to the mean of the previous level(s).  In our example, the first contrast codes the comparison of the mean of the dependent variable for level 2 of race to the mean of the dependent variable for level 1 of race.  The second comparison compares the mean of the dependent variable level 3 of race with both levels 1 and  2 of race, and the third comparison compares the mean of the dependent variable for level 4 of race with levels 1, 2 and 3. Clearly, this coding system does not make much sense with our example of race because it is a nominal variable.  However, this system is useful when the levels of the categorical variable are ordered in a meaningful way.  For example, if we had a categorical variable in which work-related stress was coded as low, medium or high, then comparing the means of the previous levels of the variable would make more sense. 

For reverse Helmert coding, we see that the first comparison comparing levels 1 and 2 are coded -1 and 1 to compare these levels, and 0 otherwise.  The second comparison comparing levels 1, 2 with level 3 are coded -1/2, -1/2,  1 and 0, and the last comparison comparing levels 1, 2 and 3 with level 4 are coded -1/3, -1/3, -1/3 and 1.

Method 1: GLM with /LMATRIX

REVERSE HELMERT contrast coding
  New variable 1 (c1) New variable 2 (c2) New variable 3 (c3)
  Level 2 v. Level 1 Level 3 v. Previous Level 4 v. Previous
1 (Hispanic) -1 -1/2 -1/3
2 (Asian)  1 -1/2 -1/3
3 (African American)  0 1 -1/3
4 (white)  0 0 1

Below we illustrate how to form these comparisons using the glm command with /lmatrix.  Note the use of fractions on the /lmatrix subcommand.  As mentioned above, you need to use numbers that sum to zero, such as 1/3 + 1/3 + 1/3 - 1.  You cannot use .333 instead of 1/3:  SPSS will give an error message and fail to calculate the contrast coefficient.  The problem is that .333 + .333 + .333 - 1 is not sufficiently close to zero.  (If you wanted to use decimals, you would need to use something like .333333 so that  .333333 + .333333 + .333333 - 1 would be sufficiently close to zero.) 

glm write by race
 /lmatrix "level 2 versus level 1"          race   -1    1    0 0
 /lmatrix "level 3 versus levels 1 and 2"   race  -1/2  -1/2    1 0
 /lmatrix "level 4 versus levels 1 2 and 3" race -1/3 -1/3 -1/3 1.

The contrast estimate for the first comparison shown in this output was calculated by subtracting the mean of the dependent variable for level 2 of the categorical variable from the mean of the dependent variable for level 1:  58 - 46.4583 = 11.542.  This result is statistically significant.  The contrast estimate for the second comparison (between level 3 and the previous levels) was calculated by subtracting the mean of the dependent variable for levels 1 and 2 from that of level 3:  48.2 - [(46.4583 + 58) / 2] = -4.029.  This result is not statistically significant, meaning that there is not a reliable difference between the mean of write for level 3 of race compared to the mean of write for levels 1 and 2 (Hispanics and Asians).  As noted above, this type of coding system does not make much sense for a nominal variable such as race.  For the comparison of level 4 and the previous levels, you take the mean of the dependent variable for the those levels and subtract it from the mean of the dependent variable for level 4:  54.0552 - [(46.4583 + 58 + 48.2) / 3] = 3.169.  This result is statistically significant.  

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 11.542
Hypothesized Value 0
Difference (Estimate - Hypothesized) 11.542
Std. Error 3.286
Sig. .001
95% Confidence Interval for Difference Lower Bound 5.061
Upper Bound 18.022
a Based on the user-specified contrast coefficients (L') matrix: group 2 versus group 1
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 1004.785 1 1004.785 12.336 .001
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -4.029
Hypothesized Value 0
Difference (Estimate - Hypothesized) -4.029
Std. Error 2.602
Sig. .123
95% Confidence Interval for Difference Lower Bound -9.161
Upper Bound 1.103
a Based on the user-specified contrast coefficients (L') matrix: group 3 versus groups 1 and 2
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 195.254 1 195.254 2.397 .123
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 3.169
Hypothesized Value 0
Difference (Estimate - Hypothesized) 3.169
Std. Error 1.488
Sig. .034
95% Confidence Interval for Difference Lower Bound .235
Upper Bound 6.104
a Based on the user-specified contrast coefficients (L') matrix: group 4 versus groups 1 2 and 3
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 369.460 1 369.460 4.536 .034
Error 15964.717 196 81.453

Method 2: GLM with /CONTRAST

As with the previous examples, we will conduct the analysis above again, this time using the /contrast(race)=difference subcommand to request reverse Helmert contrasts.

glm write by race
 /contrast(race)=difference
 /print = test(lmatrix).

These contrasts are interpreted in the same way as the contrasts obtained using method 1.  Again, we see that the first and third contrasts are statistically significant, while the second one is not.

Contrast Coefficients (L' Matrix)

RACE Difference Contrast
Parameter Level 2 vs. Level 1 Level 3 vs. Previous Level 4 vs. Previous
Intercept .000 .000 .000
[RACE=1.00] -1.000 -.500 -.333
[RACE=2.00] 1.000 -.500 -.333
[RACE=3.00] .000 1.000 -.333
[RACE=4.00] .000 .000 1.000
The default display of this matrix is the transpose of the corresponding L matrix.
Contrast Results (K Matrix)

Dependent Variable
RACE Difference Contrast writing score
Level 2 vs. Level 1 Contrast Estimate 11.542
Hypothesized Value 0
Difference (Estimate - Hypothesized) 11.542
Std. Error 3.286
Sig. .001
95% Confidence Interval for Difference Lower Bound 5.061
Upper Bound 18.022
Level 3 vs. Previous Contrast Estimate -4.029
Hypothesized Value 0
Difference (Estimate - Hypothesized) -4.029
Std. Error 2.602
Sig. .123
95% Confidence Interval for Difference Lower Bound -9.161
Upper Bound 1.103
Level 4 vs. Previous Contrast Estimate 3.169
Hypothesized Value 0
Difference (Estimate - Hypothesized) 3.169
Std. Error 1.488
Sig. .034
95% Confidence Interval for Difference Lower Bound .235
Upper Bound 6.104

Method 3: Regression

The regression coding for reverse Helmert coding is shown below.  For the first comparison, where the first and second level are compared, x1 is coded -1/2 and 1/2 and 0 otherwise.  For the second comparison, the values of x2 are coded -1/3 -1/3  2/3 and 0.  Finally, for the third comparison, the values of x3 are coded -1/4 -1/4 -/14 and 3/4.   

REVERSE HELMERT regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
1 (Hispanic) -1/2 -1/3 -1/4
2 (Asian) 1/2 -1/3 -1/4
3 (African American) 0 2/3 -1/4
4 (white) 0 0 3/4

Below we illustrate how to create x1, x2 and x3 and enter these new variables into the regression model using the regression command.

if race = 1      x1 = -1/2.
if race = 2      x1 =  1/2.
if any(race,3,4) x1 = 0.

if any(race,1,2) x2 = -1/3.
if race = 3      x2 =  2/3.
if race = 4      x2 = 0.

if any(race,1,2,3) x3 = -1/4.
if race = 4        x3 =  3/4.
execute.

regression
 /dep write
 /method = enter x1 x2 x3.


Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.679 .982
52.616 .000
X1 11.542 3.286 .252 3.512 .001
X2 -4.029 2.602 -.108 -1.548 .123
X3 3.168 1.488 .150 2.129 .035
a Dependent Variable: writing score

In the above examples, both the regression coefficient for x1 and the contrast estimate for c1 would be the mean of write for level 1 (Hispanic) minus the mean of write for level 2 (Asian).  Likewise, the regression coefficient for x2 and the contrast estimate for c2 would be the mean of write for levels 1 and 2 combined minus the mean of write for level 3.  Finally, the regression coefficient for x3 and the contrast estimate for c3 would be the mean of write for levels 1, 2 and 3 combined minus the mean of write for level 4.

5.6 Deviation Coding

This coding system compares the mean of the dependent variable for a given level to the overall mean of the dependent variable.  In our example below, the first comparison compares level 1 (Hispanics) to all levels of race, the second comparison compares level 2 (Asians) to all levels of race, and the third comparison compares level 3 (African Americans) to all levels of race.

As you can see, the logic of the contrast coding is fairly straightforward.  The first comparison compares level 1 to levels 2, 3 and 4.  A value of 3/4 is assigned to level 1 and a value of -1/4 is assigned to levels 2, 3 and 4.  Likewise, the second comparison compares level 2 to levels 1, 3 and 4. A value of 3/4 is assigned to level 2 and a value of -1/4 is assigned to levels 1, 3 and 4. A similar pattern is followed for assigning values for the third comparison.  Note that you could substitute 3 for 3/4 and 1 for 1/4 and you would get the same test of significance, but the contrast coefficient would be different.

Method 1: GLM with /LMATRIX

DEVIATION contrast coding
Level of race New variable 1 (c1) New variable 2 (c2) New variable 3 (c3)
  Level 1 v. Mean Level 2 v. Mean Level 3 v. Mean
1 (Hispanic) 3/4 -1/4 -1/4
2 (Asian) -1/4 3/4 -1/4
3 (African American) -1/4 -1/4 3/4
4 (white) -1/4 -1/4 -1/4

Below we illustrate how to form these comparisons using the glm command with /lmatrix.  As you see, a separate /lmatrix subcommand is used for each contrast.

glm write by race
 /lmatrix "level 1 versus levels 2 3 and 4" race  3/4 -1/4 -1/4 -1/4
 /lmatrix "level 2 versus levels 1 3 and 4" race -1/4  3/4 -1/4 -1/4
 /lmatrix "level 3 versus levels 1 2 and 4" race -1/4 -1/4  3/4 -1/4.

In the first "Contrast Results (K Matrix)" table, the contrast estimate is the mean for level 1 minus the grand mean.  However, this grand mean is not the mean of the dependent variable that is listed in the output of the means command above.  Rather it is the mean of means of the dependent variable at each level of the categorical variable:  (46.4583 + 58 + 48.2 + 54.0552) / 4 = 51.678375.  This contrast estimate is then 46.4583 - 51.678375 = -5.220.  The difference between this value and zero (the null hypothesis that the contrast coefficient is zero) is statistically significant (p = .002), and the "Test Results" table below that shows the F-value for this test of 10.328.  The results for the next two contrasts were computed in a similar manner.

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -5.220
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.220
Std. Error 1.631
Sig. .002
95% Confidence Interval for Difference Lower Bound -8.437
Upper Bound -2.003
a Based on the user-specified contrast coefficients (L') matrix: group 1 versus groups 1 2 and 3
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 833.927 1 833.927 10.238 .002
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 6.322
Hypothesized Value 0
Difference (Estimate - Hypothesized) 6.322
Std. Error 2.160
Sig. .004
95% Confidence Interval for Difference Lower Bound 2.061
Upper Bound 10.582
a Based on the user-specified contrast coefficients (L') matrix: group 2 versus groups 1 3 and 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 697.475 1 697.475 8.563 .004
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -3.478
Hypothesized Value 0
Difference (Estimate - Hypothesized) -3.478
Std. Error 1.732
Sig. .046
95% Confidence Interval for Difference Lower Bound -6.895
Upper Bound -6.203E-02
a Based on the user-specified contrast coefficients (L') matrix: group 3 versus groups 1 2 and 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 328.405 1 328.405 4.032 .046
Error 15964.717 196 81.453

Method 2: GLM with /CONTRAST

Now let's conduct the same analysis using the /contrast subcommand instead of the /lmatrix subcommand.  Instead of providing the values for the contrasts that we want to perform, we can have SPSS provide those for us by indicating the type of coding that we wish to use, in this case, deviation coding via the /contrast (race)=deviation subcommand.

glm write by race
 /contrast (race)=deviation
 /print = test(lmatrix).

The contrasts estimates in the table entitled "Contrast Results (K Matrix)" are the mean of the particular level minus the grand (unweighted) mean.  This grand mean is not the mean of the dependent variable that is listed in the output of the means command above.  Rather it is the mean of means of the dependent variable at each level of the categorical variable:  (46.4583 + 58 + 48.2 + 54.0552) / 4 = 51.678375.  The contrast estimate for level 1 versus mean is then 46.4583 - 51.678375 = -5.220.  The difference between this value and zero (the null hypothesis that the contrast coefficient is zero) is statistically significant (p = .002).  The contrast coefficients for the other comparisons are calculated in the same manner.  

Contrast Coefficients (L' Matrix)

RACE Deviation Contrast(a)
Parameter Level 1 vs. Mean Level 2 vs. Mean Level 3 vs. Mean
Intercept .000 .000 .000
[RACE=1.00] .750 -.250 -.250
[RACE=2.00] -.250 .750 -.250
[RACE=3.00] -.250 -.250 .750
[RACE=4.00] -.250 -.250 -.250
The default display of this matrix is the transpose of the corresponding L matrix.
a Omitted category = 4
Contrast Results (K Matrix)

Dependent Variable
RACE Deviation Contrast(a) writing score
Level 1 vs. Mean Contrast Estimate -5.220
Hypothesized Value 0
Difference (Estimate - Hypothesized) -5.220
Std. Error 1.631
Sig. .002
95% Confidence Interval for Difference Lower Bound -8.437
Upper Bound -2.003
Level 2 vs. Mean Contrast Estimate 6.322
Hypothesized Value 0
Difference (Estimate - Hypothesized) 6.322
Std. Error 2.160
Sig. .004
95% Confidence Interval for Difference Lower Bound 2.061
Upper Bound 10.582
Level 3 vs. Mean Contrast Estimate -3.478
Hypothesized Value 0
Difference (Estimate - Hypothesized) -3.478
Std. Error 1.732
Sig. .046
95% Confidence Interval for Difference Lower Bound -6.895
Upper Bound -6.203E-02
a Omitted category = 4

Method 3: Regression

As you see in the example below, the regression coding is accomplished by assigning 1 to level 1 for the first comparison (because level 1 is the level to be compared to all others), a 1 to level 2 for the second comparison (because level 2 is to be compared to all others), and 1 to level 3 for the third comparison (because level 3 is to be compared to all others).  Note that a  -1 is assigned to level 4 for all three comparisons (because it is the level that is never compared to the other levels) and all other values are assigned a 0.  This regression coding scheme yields the comparisons described above.

DEVIATION regression coding
Level of race New variable 1 (x1) New variable 2 (x2) New variable 3 (x3)
  Level 1 v. Mean Level 2 v. Mean Level 3 v. Mean
1 (Hispanic) 1 0 0
2 (Asian) 0 1 0
3 (African American) 0 0 1
4 (white) -1 -1 -1

Below we illustrate how to create x1, x2 and x3 and enter these new variables into the regression model using the regression command.

if race = 1 x1 = 1.
if any(race,2,3) x1 = 0.
if race = 4 x1 = -1.

if race = 2 x2 = 1.
if any(race,1,3) x2 = 0.
if race = 4 x2 = -1.

if race = 3 x3 = 1.
if any(race,1,2) x3 = 0.
if race = 4 x3 = -1.
execute.

regression
 /dep write
 /method = enter x1 x2 x3.

In this example, both the regression coefficient for x1 and the contrast estimate for c1 would be the mean of write for level 1 (Hispanic)  minus the mean of write for levels 2, 3 and 4 combined.  Likewise, the regression coefficient for x2 and the contrast estimate for c2 would be the mean of write for level 2 (Asian) minus the mean of write for levels 1, 3, and 4 combined.  As we saw in the previous analyses, all three contrasts are statistically significant.



Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.678 .982
52.619 .000
X1 -5.220 1.631 -.382 -3.200 .002
X2 6.322 2.160 .385 2.926 .004
X3 -3.478 1.732 -.242 -2.008 .046
a Dependent Variable: writing score

5.7 Orthogonal Polynomial Coding

Orthogonal polynomial coding is a form of trend analysis in that it is looking for the linear, quadratic and cubic trends in the categorical variable.  This type of coding system should be used only with an ordinal variable in which the levels are equally spaced.  Examples of such a variable might be income or education.  The table below shows the contrast coefficients for the linear, quadratic and cubic trends for the four levels.  These could be obtained from most statistics books on linear models.

POLYNOMIAL
Level of race Linear (x1) Quadratic (x2) Cubic (x3)
1 (Hispanic)  -.671 .5 -.224
2 (Asian)  -.224 -.5 .671
3 (African American)  .224 -.5 -.671
4 (white)  .671 .5 .224

Method 1: GLM with /LMATRIX

glm write by race
 /lmatrix "linear" race -.671 -.224 .224 .671
 /lmatrix "quadratic" race .5 -.5 -.5 .5
 /lmatrix "cubic" race -.224 .671 -.671 .224.
Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 2.902
Hypothesized Value 0
Difference (Estimate - Hypothesized) 2.902
Std. Error 1.535
Sig. .060
95% Confidence Interval for Difference Lower Bound -.125
Upper Bound 5.930
a Based on the user-specified contrast coefficients (L') matrix: linear
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 291.104 1 291.104 3.574 .060
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -2.843
Hypothesized Value 0
Difference (Estimate - Hypothesized) -2.843
Std. Error 1.964
Sig. .149
95% Confidence Interval for Difference Lower Bound -6.717
Upper Bound 1.031
a Based on the user-specified contrast coefficients (L') matrix: quadratic
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 170.665 1 170.665 2.095 .149
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 8.277
Hypothesized Value 0
Difference (Estimate - Hypothesized) 8.277
Std. Error 2.316
Sig. .000
95% Confidence Interval for Difference Lower Bound 3.709
Upper Bound 12.846
a Based on the user-specified contrast coefficients (L') matrix: cubic
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 1040.029 1 1040.029 12.769 .000
Error 15964.717 196 81.453

To calculate the contrast estimates for these comparisons, you need to multiply the code used in the new variable by the mean for the dependent variable for each level of the categorical variable, and then sum the values.  For example, the code used in x1 for level 1 of race is -.671 and the mean of write for level 1 is 46.4583.  Hence, you would multiply -.671 and 46.4583 and add that to the product of the code for level 2 of x1 and its mean, and so on.  To obtain the contrast estimate for the linear contrast, you would do the following:  -.671*46.4583 + -.224*58 + .224*48.2 + .671*54.0552 = 2.905 (with rounding error).  This result is not statistically significant at the .05 alpha level, but it is close.  The quadratic component is also not statistically significant, but the cubic one is.  This suggests that, if the mean of the dependent variable was plotted against race, the line would tend to have two bends.  As noted earlier, this type of coding system does not make much sense with a nominal variable such as race.

Method 2: GLM with /CONTRAST

As with the previous examples, we will conduct the above analysis again, this time using the /contrast(race)=polynomial subcommand to request orthogonal polynomial contrasts.

glm write by race
 /contrast (race)=polynomial
 /print = test(lmatrix).
Contrast Coefficients (L' Matrix)

RACE Polynomial Contrast(a)
Parameter Linear Quadratic Cubic
Intercept .000 .000 .000
[RACE=1.00] -.671 .500 -.224
[RACE=2.00] -.224 -.500 .671
[RACE=3.00] .224 -.500 -.671
[RACE=4.00] .671 .500 .224
The default display of this matrix is the transpose of the corresponding L matrix.
a Metric = 1.000, 2.000, 3.000, 4.000
Contrast Results (K Matrix)

Dependent Variable
RACE Polynomial Contrast(a) writing score
Linear Contrast Estimate 2.905
Hypothesized Value 0
Difference (Estimate - Hypothesized) 2.905
Std. Error 1.534
Sig. .060
95% Confidence Interval for Difference Lower Bound -.121
Upper Bound 5.931
Quadratic Contrast Estimate -2.843
Hypothesized Value 0
Difference (Estimate - Hypothesized) -2.843
Std. Error 1.964
Sig. .149
95% Confidence Interval for Difference Lower Bound -6.717
Upper Bound 1.031
Cubic Contrast Estimate 8.273
Hypothesized Value 0
Difference (Estimate - Hypothesized) 8.273
Std. Error 2.316
Sig. .000
95% Confidence Interval for Difference Lower Bound 3.706
Upper Bound 12.840
a Metric = 1.000, 2.000, 3.000, 4.000

Again, we see that only the cubic effect is statistically significant.  In other words, if the mean of write was plotted against race, the line would tend to have two bends. 

Method 3: Regression

The regression coding for orthogonal polynomial coding is the same as the contrast coding.  Below you can see the SPSS code for creating x1, x2 and x3 that correspond to the linear, quadratic and cubic trends for race.

if race = 1 x1 = -.671.
if race = 2 x1 = -.224.
if race = 3 x1 = .224.
if race = 4 x1 = .671.

if race = 1 x2 = .5.
if race = 2 x2 = -.5.
if race = 3 x2 = -.5.
if race = 4 x2 = .5.

if race = 1 x3 = -.224.
if race = 2 x3 = .671.
if race = 3 x3 = -.671.
if race = 4 x3 = .224.
execute.

regression
 /dep write
 /method = enter x1 x2 x3.


Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.678 .982
52.619 .000
X1 2.900 1.534 .142 1.890 .060
X2 -2.843 1.964 -.109 -1.448 .149
X3 8.271 2.315 .278 3.573 .000
a Dependent Variable: writing score

The regression coefficients obtained from this analysis are the same as the contrast coefficients obtained using the glm command with either the /lmatrix or the /contrast subcommands.  

5.8 User Defined Coding

SPSS allows users to define their own coding systems.  For our example, we will make the following three comparisons: 

1) level 1 to level 3  
2) level 2 to levels 1 and 4 
3) levels 1 and 2 to levels 3 and 4.

In order to compare level 1 to level 3, we use the contrast coefficients 1 0 -1 0. To compare level 2 to levels 1 and 4 we use the contrast coefficients -1/2 1 0 -1/2 .  Finally, to compare levels 1 and 2 with levels 3 and 4 we use the coefficients 1/2 1/2 -1/2 -1/2.  Before proceeding to the SPSS code necessary to conduct these analyses, let's take a moment to more fully explain the logic behind the selection of these contrast coefficients.  

For the first contrast, we are comparing level 1 to level 3, and the contrast coefficients are 1 0 -1 0.  This means that the levels associated with the contrast coefficients with opposite signs are being compared.  In fact, the mean of the dependent variable is multiplied by the contrast coefficient.  Hence, levels 2 and 4 are not involved in the comparison:  they are multiplied by zero and "dropped out."  You will also notice that the contrast coefficients sum to zero.  This is necessary.  If the contrast coefficients do not sum to zero, the contrast is not estimable and SPSS will issue an error message. Which level of the categorical variable is assigned a positive or negative value is not terribly important:  1 0 -1 0 is the same as -1 0 1 0 in that both of these codings compare the first and the third levels of the variable.  However, the sign of the regression coefficient would change.  

Now let's look at the contrast coefficients for the second and third comparisons.  You will notice that in both cases we use fractions that sum to one (or minus one).  They do not have to sum to one (or minus one).  You may wonder why we would use fractions like -1/2 1 0 -1/2 instead of whole numbers such as -1 2 0 -1.  While -1/2 1 0 -1/2 and -1 2 0 -1 both compare level 2 with levels 1 and 4 and both will give you the same t-value and p-value for the regression coefficient, the contrast estimates/regression coefficients themselves would be different, as would their interpretation.  The coefficient for the -1/2 1 0 -1/2 contrast is the mean of level 2 minus the mean of the means for levels 1 and 4:  58 - (46.4583 + 54.0552)/2 = 7.74325.  (Alternatively, you can multiply the contrasts by the mean of the dependent variable for each level of the categorical variable: -1/2*46.4583 + 1*58.00 + 0*48.20 + -1/2*54.0552 = 7.74325.  Clearly these are equivalent ways of thinking about how the contrast coefficient is calculated.)  By comparison, the coefficient for the -1 2 0 -1 contrast is two times the mean for level 2 minus the means of the dependent variable for levels 1 and 4:  2*58 - (46.4583 + 54.0552) = 15.4865, which is the same as -1*46.4583 + 2*58 + 0*48.20 - 1*54.0552 = 15.4865. Note that the regression coefficient using the contrast coefficients -1 2 0 -1 is twice the regression coefficient obtained when -1/2 1 0 -1/2 is used.

Method 1: GLM with /LMATRIX

In order to compare level 1 to level 3, we use the contrast coefficients 1 0 -1 0. To compare level 2 to levels 1 and 4 we use the contrast coefficients -1/2 1 0 -1/2 .  Finally, to compare levels 1 and 2 with levels 3 and 4, we use the coefficients 1/2 1/2 -1/2 -1/2.  These coefficients are used in the /lmatrix subcommands below.

glm write by race
 /lmatrix "compare level 1 to level 3" race 1 0 -1 0
 /lmatrix "compare level 2 to levels 1 and 4" race -1/2 1 0 -1/2
 /lmatrix "compare levels 1 and 2 to levels 3 and 4" race 1/2 1/2 -1/2 -1/2.

The first "Contrast Results" table shows the results of comparing level 1 to level 3.  The contrast estimate for this comparison is the mean of level 1 minus the mean for level 3, and the significance of this is .525, i.e., not significant.  The second "Contrast Results" output shows the contrast estimate to be 7.743, which is the mean of level 2 minus the mean of level 1 and level 4, and this difference is significant, p = 0.008.  The final contrast estimate is 1.1 which is the mean of levels 1 and 2 minus the mean of levels 3 and 4, and this contrast is not statistically significant, p = .576.

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate -1.742
Hypothesized Value 0
Difference (Estimate - Hypothesized) -1.742
Std. Error 2.732
Sig. .525
95% Confidence Interval for Difference Lower Bound -7.131
Upper Bound 3.647
a Based on the user-specified contrast coefficients (L') matrix: compare group 1 to group 3
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 33.092 1 33.092 .406 .525
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 7.743
Hypothesized Value 0
Difference (Estimate - Hypothesized) 7.743
Std. Error 2.897
Sig. .008
95% Confidence Interval for Difference Lower Bound 2.030
Upper Bound 13.457
a Based on the user-specified contrast coefficients (L') matrix: compare group 2 to groups 1 and 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 581.833 1 581.833 7.143 .008
Error 15964.717 196 81.453

Contrast Results (K Matrix)(a)

Dependent Variable
Contrast writing score
L1 Contrast Estimate 1.102
Hypothesized Value 0
Difference (Estimate - Hypothesized) 1.102
Std. Error 1.964
Sig. .576
95% Confidence Interval for Difference Lower Bound -2.772
Upper Bound 4.975
a Based on the user-specified contrast coefficients (L') matrix: compare groups 1 and 2 to groups 3 and 4
Test Results
Dependent Variable: writing score
Source Sum of Squares df Mean Square F Sig.
Contrast 25.618 1 25.618 .315 .576
Error 15964.717 196 81.453

Method 2: GLM with /CONTRAST

When using glm with the /contrast subcommand, you can specify your own contrast coefficients with the special keyword, followed by the contrasts you would like to test.  To compare level 1 to level 3, we use the contrast coefficients 1 0 -1 0 and then to compare level 2 to levels 1 and 4 we use the contrast coefficients -.5 1 0 -.5 and finally, to compare levels 1 and 2 with levels 3 and 4 we use the coefficients .5 .5 -.5 -.5 (note that when using the keyword special these values cannot be expressed as fractions, i.e., 1/2).  

glm write by race
 /contrast (race)=special(1 0 -1 0, -.5 1 0 -.5, .5 .5 -.5 -.5)
 /print = test(lmatrix).

As you can see, the glm results below correspond to the glm results above using method 1.

Contrast Coefficients (L' Matrix)

RACE Special Contrast
Parameter L1 L2 L3
Intercept .000 .000 .000
[RACE=1.00] 1.000 -.500 .500
[RACE=2.00] .000 1.000 .500
[RACE=3.00] -1.000 .000 -.500
[RACE=4.00] .000 -.500 -.500
The default display of this matrix is the transpose of the corresponding L matrix.
Contrast Results (K Matrix)

Dependent Variable
RACE Special Contrast writing score
L1 Contrast Estimate -1.742
Hypothesized Value 0
Difference (Estimate - Hypothesized) -1.742
Std. Error 2.732
Sig. .525
95% Confidence Interval for Difference Lower Bound -7.131
Upper Bound 3.647
L2 Contrast Estimate 7.743
Hypothesized Value 0
Difference (Estimate - Hypothesized) 7.743
Std. Error 2.897
Sig. .008
95% Confidence Interval for Difference Lower Bound 2.030
Upper Bound 13.457
L3 Contrast Estimate 1.102
Hypothesized Value 0
Difference (Estimate - Hypothesized) 1.102
Std. Error 1.964
Sig. .576
95% Confidence Interval for Difference Lower Bound -2.772
Upper Bound 4.975

Method 3: Regression

As in the prior examples, we will make the following three comparisons: 

1) level 1 to level 3,  
2) level 2 to levels 1 and 4 and   
3) levels 1 and 2 to levels 3 and 4.

For methods 1 and 2 it was quite easy to translate the comparisons we wanted to make into contrast codings, but it is not as easy to translate the comparisons we want into a regression coding scheme.  If we know the contrast coding system, then we can convert that into a regression coding system using the SPSS program shown below. As you can see, we place the three contrast codings we want into the matrix c and then perform a set of matrix operations on c, yielding the matrix x. We then display x using the print command. 

matrix.

compute c =
{ 1, -.5, .5  ;
  0,   1, .5  ;
 -1,   0, -.5 ;
  0, -.5, -.5 }.

compute x = c*inv( t(c)*c ).

print x.

end matrix.

 Below we see the output from this program showing the regression coding scheme we would use.

X
   -.500000000  -1.000000000   1.500000000
    .500000000   1.000000000   -.500000000
  -1.500000000  -1.000000000   1.500000000
   1.500000000   1.000000000  -2.500000000

This converted the contrast coding into the regression coding that we need for running this analysis with the regression command.  Below, we use if command to create x1, x2 and x3 according to the coding shown above and then enter them into the regression analysis.

if race = 1 x1 = -.5.
if race = 2 x1 =  .5.
if race = 3 x1 = -1.5.
if race = 4 x1 =  1.5.
if race = 1 x2 = -1.
if race = 2 x2 =  1.
if race = 3 x2 = -1.
if race = 4 x2 =  1.

if race = 1 x3 = 1.5.
if race = 2 x3 = -.5.
if race = 3 x3 = 1.5.
if race = 4 x3 =-2.5.
execute.
regression
 /dep write
 /method = enter x1 x2 x3.

Here is a shortcut to save typing all of the compute commands.  This assumes that race is coded 1 2 3 4.

get file = "d:\spss\hsb2.sav".
sort cases by race.
save outfile = "c:\temp\race.sav".

matrix.
compute c = { 1, -.5, .5  ;
              0,   1, .5  ;
             -1,   0, -.5 ;
              0, -.5, -.5 }.
compute x = c*inv( t(c)*c ).
save  x /outfile=* /var=x1 x2 x3
end matrix.

compute race = $CASENUM.
execute.

match files /table=* /file="c:\temp\race.sav" /by race.
execute.
regression
 /dep write
 /method = enter x1 x2 x3.

The first comparison of the mean of the dependent variable for level 1 to level 3 of the categorical variable was not statistically significant, while the comparison of the mean of the dependent variable for level 2 to that of levels 1 and 4 was.  The comparison of the mean of the dependent variable for levels 1 and 2 to that of levels 3 and 4 also was not statistically significant.



Coefficients(a)

Unstandardized Coefficients Standardized Coefficients t Sig.
Model B Std. Error Beta
1 (Constant) 51.678 .982
52.619 .000
X1 -1.742 2.732 -.192 -.637 .525
X2 7.743 2.897 .679 2.673 .008
X3 1.102 1.964 .194 .561 .576
a Dependent Variable: writing score

5.9 Summary

This page has described a number of different coding systems that you could use for categorical data, and three different strategies you could use for performing the analyses.  You can choose a coding system that yields comparisons that make the most sense for testing your hypotheses.  Among the three strategies (glm with /lmatrix, glm with /contrast and regression), each has its strengths and weaknesses.  You may have noticed that there is a balance between ease of use and flexibility/power.  The method that is easiest to use (glm with /contrast) is not as powerful as the other two methods. The most general method is the use of regression, but it can be the trickiest.  In general we would recommend using the easiest method that accomplishes your goals.

5.10 For more information

Here are some additional resources.

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