Regression with SAS
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

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 SAS 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 two ways that you can conduct analyses using these coding schemes: 1) using proc glm with estimate statements to define "contrast" coefficients that specify levels of the categorical variable that are to be compared, and 2) using proc reg. When using proc reg 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.  Method 1 uses a type of coding we will call "contrast coding" while method 2 uses a type of coding we will call "regression coding".  

The Example Data File

The examples in this page will use dataset called hsb2.sas7bdat 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.

proc means data = c:\sasreg\hsb2 mean n;
  class race;
  var write;
run;
The MEANS Procedure

  Analysis Variable : write writing score

                  N
        race    Obs            Mean      N
------------------------------------------
           1     24      46.4583333     24

           2     11      58.0000000     11

           3     20      48.2000000     20

           4    145      54.0551724    145
------------------------------------------

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: PROC GLM

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 proc glm.  As you see, a separate estimate statement is used for each contrast.

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 1 versus level 4' race 1 0 0 -1;
  estimate 'level 2 versus level 4' race 0 1 0 -1;
  estimate 'level 3 versus level 4' race 0 0 1 -1;
run;
quit;

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 t-value associated with this test is -3.82.  The results of the second contrast, comparing the mean of write for levels 2 and 4 is not statistically significant (t = 1.40, p = .1638), while the third contrast is statistically significant.  Please note that while we have included the full SAS output for this example, we will only show the relevant output in later examples to conserve space.

The GLM Procedure

Dependent Variable: write   writing score

                                        Sum of
Source                      DF         Squares     Mean Square    F Value    Pr > F

Model                        3      1914.15805       638.05268       7.83    <.0001

Error                      196     15964.71695        81.45264

Corrected Total            199     17878.87500

R-Square     Coeff Var      Root MSE    write Mean

0.107063      17.10111      9.025111      52.77500

Source                      DF       Type I SS     Mean Square    F Value    Pr > F

race                         3     1914.158046      638.052682       7.83    <.0001

Source                      DF     Type III SS     Mean Square    F Value    Pr > F

race                         3     1914.158046      638.052682       7.83    <.0001

                                              Standard
Parameter                     Estimate           Error    t Value    Pr > |t|

level 1 versus level 4     -7.59683908      1.98886958      -3.82      0.0002
level 2 versus level 4      3.94482759      2.82250377       1.40      0.1638
level 3 versus level 4     -5.85517241      2.15275967      -2.72      0.0071

Method 2: 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 proc reg.

data simple;
  set c:\sasreg\hsb2;
  if race = 1 then x1 = 3/4; else x1 = -1/4;
  if race = 2 then x2 = 3/4; else x2 = -1/4;
  if race = 3 then x3 = 3/4; else x3 = -1/4;
run;

proc reg data = simple;
  model write = x1 x2 x3;
run;
quit;

You will notice that the regression coefficients in the table below are the same as the contrast coefficients that we saw using proc glm.  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). You also can see that the t values and significance levels are also the same as those from the proc glm output.  Please note that while we have included the full SAS output for this example, we will only show the relevant output in later examples to conserve space.

The REG Procedure
Model: MODEL1
Dependent Variable: write writing score

                             Analysis of Variance

                                    Sum of           Mean
Source                   DF        Squares         Square    F Value    Pr > F

Model                     3     1914.15805      638.05268       7.83    <.0001
Error                   196          15965       81.45264
Corrected Total         199          17879

Root MSE              9.02511    R-Square     0.1071
Dependent Mean       52.77500    Adj R-Sq     0.0934
Coeff Var            17.10111

                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67838        0.98212      52.62      <.0001
x1                             1       -7.59684        1.98887      -3.82      0.0002
x2                             1        3.94483        2.82250       1.40      0.1638
x3                             1       -5.85517        2.15276      -2.72      0.0071

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: PROC GLM

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

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 1 versus level 2' race 1 -1 0 0;
  estimate 'level 2 versus level 3' race 0 1 -1 0;
  estimate 'level 3 versus level 4' race 0 0 1 -1;
run;
quit;
                                              Standard
Parameter                     Estimate           Error    t Value    Pr > |t|

level 1 versus level 2     -11.5416667      3.28612920      -3.51      0.0006
level 2 versus level 3       9.8000000      3.38783369       2.89      0.0043
level 3 versus level 4      -5.8551724      2.15275967      -2.72      0.0071

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: 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

data forward;
  set c:\sasreg\hsb2;

  if race = 1 then x1 = 3/4; else x1 = -1/4;

  if race = 1 or race = 2 then x2 = 1/2;
  if race = 3 or race = 4 then x2 = -1/2;

  if race = 4 then x3 = -3/4; else x3 = 1/4;

run;

proc reg data = forward;
  model write = x1 x2 x3;
run;
quit;
                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67838        0.98212      52.62      <.0001
x1                             1      -11.54167        3.28613      -3.51      0.0006
x2                             1        9.80000        3.38783       2.89      0.0043
x3                             1       -5.85517        2.15276      -2.72      0.0071

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 (Hispanics minus Asians).  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: PROC GLM

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

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 1 versus level 2' race -1 1 0 0;
  estimate 'level 2 versus level 3' race 0 -1 1 0;
  estimate 'level 3 versus level 4' race 0 0 -1 1;
run;
quit;
                                              Standard
Parameter                     Estimate           Error    t Value    Pr > |t|

level 1 versus level 2      11.5416667      3.28612920       3.51      0.0006
level 2 versus level 3      -9.8000000      3.38783369      -2.89      0.0043
level 3 versus level 4       5.8551724      2.15275967       2.72      0.0071

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: 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 = 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

data backward;
  set c:\sasreg\hsb2;

  if race = 1 then x1 = -3/4; else x1 = 1/4;

  if race = 1 or race = 2 then x2 = -1/2;
  if race = 3 or race = 4 then x2 = 1/2;

  if race = 4 then x3 = 3/4; else x3 = -1/4;

run;

proc reg data = backward;
  model write = x1 x2 x3;
run;
quit;
                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67838        0.98212      52.62      <.0001
x1                             1       11.54167        3.28613       3.51      0.0006
x2                             1       -9.80000        3.38783      -2.89      0.0043
x3                             1        5.85517        2.15276       2.72      0.0071

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: PROC GLM

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 proc glm with estimate statements.  Note that on the first estimate statement we indicate -.33333 and not just -.33.  We need to use this many decimals so the sum of all of the contrast coefficients (i.e., 1 + -.333333 + -.333333 + -.333333) is sufficiently close to zero, otherwise SAS will say that the term cannot be estimated.

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 1 versus levels 2, 3 & 4' race 1 -.33333 -.33333 -.33333;
  estimate 'level 2 versus levels 3 & 4' race 0 1 -.5 -.5;
  estimate 'level 3 versus level 4' race 0 0 1 -1;
run;
quit;
                                                      Standard
Parameter                             Estimate           Error    t Value    Pr > |t|

level 1 versus levels 2, 3 & 4     -6.96006384      2.17520603      -3.20      0.0016
level 2 versus levels 3 & 4         6.87241379      2.92632513       2.35      0.0198
level 3 versus level 4             -5.85517241      2.15275967      -2.72      0.0071

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: 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 porc reg.

data helmert;
  set c:\sasreg\hsb2;
  if race = 1 then x1 = .75; else x1 = -.25;

  if race = 1 then x2 = 0;
  if race = 2 then x2 = 2/3;
  if race = 3 or race = 4 then x2 = -1/3;

  if race = 1 or race = 2 then x3 = 0;
  if race = 3 then x3 = 1/2;
  if race = 4 then x3 = -1/2;

run;

proc reg data = helmert;
  model write = x1 x2 x3;
run;
quit;

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.

                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67836        0.98212      52.62      <.0001
x1                             1       -6.96003        2.17521      -3.20      0.0016
x2                             1        6.87241        2.92633       2.35      0.0198
x3                             1       -5.85517        2.15276      -2.72      0.0071

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: PROC GLM

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 proc glm with estimate statements.  Note that on the third estimate statement we indicate -.33333 and not just -.33.  We need to use this many decimals so the sum of all of the contrast coefficients (i.e., -.333333 + - .333333 + - .333333 + 1) is sufficiently close to zero, otherwise SAS will say that the term cannot be estimated.

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 2 versus level1' race -1 1 0 0;
  estimate 'level 3 versus levels 1 & 2' race -.5 -.5 1 0;
  estimate 'level 4 versus levels 1, 2 & 4' race -.33333 -.33333 -.33333 1;
run;
quit;

An alternate way, which solves the problem of the repeating decimals, is shown below.  Only one output is shown because the two outputs are identical.

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 2 versus level 1' race -1 1 0 0;
  estimate 'level 3 versus levels 1 & 2' race -.5 -.5 1 0;
  estimate 'level 4 versus levels 1, 2 & 4' race -1 -1 -1 3 / divisor=3;
run;
quit;
                                                      Standard
Parameter                             Estimate           Error    t Value    Pr > |t|

level 2 versus level1               11.5416667      3.28612920       3.51      0.0006
level 3 versus levels 1 & 2         -4.0291667      2.60236299      -1.55      0.1232
level 4 versus levels 1, 2 & 4       3.1690296      1.48797250       2.13      0.0344

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: 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 proc reg.

data diff;
  set c:\sasreg\hsb2;
  if race = 1 then x1 = -1/2;
  if race = 2 then x1 = 1/2;
  if race = 3 or race = 4 then x1 = 0;

  if race = 1 or race = 2 then x2 = -1/3;
  if race = 3 then x2 = 2/3;
  if race = 4 then x2 = 0;

  if race = 4 then x3 = 3/4; else x3 = -1/4;

run;

proc reg data = diff;
  model write = x1 x2 x3;
run;
quit;
                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67839        0.98212      52.62      <.0001
x1                             1       11.54167        3.28613       3.51      0.0006
x2                             1       -4.02917        2.60236      -1.55      0.1232
x3                             1        3.16905        1.48799       2.13      0.0344

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: PROC GLM

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 proc glm.

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 1 versus levels 2, 3 & 4' race .75 -.25 -.25 -.25;
  estimate 'level 2 versus levels 1, 3 & 4' race -.25 .75 -.25 -.25;
  estimate 'level 3 versus levels 1, 2 & 4' race -.25 -.25 .75 -.25;
run;
quit;
                                                      Standard
Parameter                             Estimate           Error    t Value    Pr > |t|

level 1 versus levels 2, 3 & 4     -5.22004310      1.63140849      -3.20      0.0016
level 2 versus levels 1, 3 & 4      6.32162356      2.16031394       2.93      0.0038
level 3 versus levels 1, 2 & 4     -3.47837644      1.73230472      -2.01      0.0460

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 = .0016), and the t-value for this test of -3.20.  The results for the next two contrasts were computed in a similar manner.

Method 2: 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 proc reg.

data deviation;
  set c:\sasreg\hsb2;
  if race = 1 then x1 = 1;
  if race = 2 or race = 3 then x1 = 0;
  if race = 4 then x1 = -1;

  if race = 2 then x2 = 1;
  if race = 1 or race = 3 then x2 = 0;
  if race = 4 then x2 = -1;

  if race = 3 then x3 = 1;
  if race = 1 or race = 2 then x3 = 0;
  if race = 4 then x3 = -1; 
run;

proc reg data = deviation;
  model write = x1 x2 x3;
run;
quit;

In this example, both the regression coefficient for x1 is the mean of write for level 1 (Hispanic) minus the grand mean of write.  Likewise, the regression coefficient for x2 is the mean write for level 2 (Asian) minus the grand mean of write, and so on. As we saw in the previous analyses, all three contrasts are statistically significant.

                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67838        0.98212      52.62      <.0001
x1                             1       -5.22004        1.63141      -3.20      0.0016
x2                             1        6.32162        2.16031       2.93      0.0038
x3                             1       -3.47838        1.73230      -2.01      0.0460

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: PROC GLM

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'linear' race -.671 -.224 .224 .671;
  estimate 'quadratic' race .5 -.5 -.5 .5;
  estimate 'cubic' race -.224 .671 -.671 .224;
run;
quit;
                                            Standard
Parameter                   Estimate           Error    t Value    Pr > |t|

linear                    2.90227902      1.53520851       1.89      0.0602
quadratic                -2.84324713      1.96424409      -1.45      0.1494
cubic                     8.27749195      2.31648010       3.57      0.0004

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: Regression

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

data poly;
  set c:\sasreg\hsb2;
  if race = 1 then x1 = -.671;
  if race = 2 then x1 = -.224;
  if race = 3 then x1 = .224;
  if race = 4 then x1 = .671;

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

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

run;

proc reg data = poly;
  model write = x1 x2 x3;
run;
quit;
                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67838        0.98212      52.62      <.0001
x1                             1        2.89986        1.53393       1.89      0.0602
x2                             1       -2.84325        1.96424      -1.45      0.1494
x3                             1        8.27059        2.31455       3.57      0.0004

The regression coefficients obtained from this analysis are the same as the contrast coefficients obtained using proc glm.  

5.8 User Defined Coding

You can use SAS for any general kind of coding scheme.  For our example, we would like to 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 SAS 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 SAS 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: PROC GLM

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 estimate statements below.

proc glm data = c:\sasreg\hsb2;
  class race;
  model write = race;
  estimate 'level 1 versus level 3' race 1 0 -1 0;
  estimate 'level 2 versus levels 1 & 4' race -.5 1 0 -.5;
  estimate 'levels 1 & 2 versus levels 3 & 4' race .5 .5 -.5 -.5;
run;
quit;
                                                        Standard
Parameter                               Estimate           Error    t Value    Pr > |t|

level 1 versus level 3               -1.74166667      2.73248820      -0.64      0.5246
level 2 versus levels 1 & 4           7.74324713      2.89718584       2.67      0.0082
levels 1 & 2 versus levels 3 & 4      1.10158046      1.96424409       0.56      0.5756

The contrast estimate for the first 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 estimate is 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: Regression

As in the prior example, 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 SAS 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. 

proc iml;
  c = {  1 -.5  .5, 
         0   1  .5,
        -1   0 -.5,
         0 -.5 -.5 };
  x = c*inv( c`*c );
  print x;
run;
quit;

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

            X

     -0.5        -1       1.5
      0.5         1      -0.5
     -1.5        -1       1.5
      1.5         1      -2.5

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

data special;
  set c:\sasreg\hsb2;
  if race = 1 then x1 = -0.5;
  if race = 2 then x1 =   .5;
  if race = 3 then x1 = -1.5;
  if race = 4 then x1 =  1.5;

  if race = 1 or race = 3 then x2 = -1;
  if race = 2 or race = 4 then x2 =  1;

  if race = 1 or race = 3 then x3 = 1.5;
  if race = 2 then x3 = -.5;
  if race = 4 then x3 =-2.5;

run;

proc reg data = special;
  model write = x1 x2 x3;
run;
quit;

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.

                                 Parameter Estimates

                                      Parameter       Standard
Variable     Label            DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept         1       51.67838        0.98212      52.62      <.0001
x1                             1       -1.74167        2.73249      -0.64      0.5246
x2                             1        7.74325        2.89719       2.67      0.0082
x3                             1        1.10158        1.96424       0.56      0.5756

5.9 Summary

This page has described a number of different coding systems that you could use for categorical data, and two 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.  In general we would recommend using the easiest method that accomplishes your goals.

5.10 Additional Information

Here are some additional resources.

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