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

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.

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.

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

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.

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.

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

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

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.

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

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.

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).

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

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

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.

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

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.

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

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

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.

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

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.

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.

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

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.

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

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.

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.

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

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.

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

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.

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.  

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

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.

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

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.

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).

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.

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.

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.

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:spsshsb2.sav".
sort cases by race.
save outfile = "c:temprace.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:temprace.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.

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.

• SPSS Textbook Examples from Design and Analysis: Chapter 6
• SPSS Textbook Examples from Design and Analysis: Chapter 7
• SPSS Textbook Examples: Applied Regression Analysis, Chapter 8
• An Overview of SPSS GLM
  (courtesy of SPSS)
• MANOVA and GLM (courtesy of SPSS)
• Understanding and Interpreting
  Parameter Estimates in Regression and ANOVA (courtesy of SPSS)
• Data Setup Comparing Means in SPSS  (courtesy
  of SPSS)
• One-Way
  ANOVA Contrast Code Problems From Charles Judd and Gary McClelland 
• Two-way contrast code solutions