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.

Name of contrast	Comparison made
Simple Coding	Compares each level of a variable to the reference level
Forward Difference Coding	Adjacent levels of a variable (each level minus the next level)
Backward Difference Coding	Adjacent levels of a variable (each level minus the prior level)
Helmert Coding	Compare levels of a variable with the mean of the subsequent levels of the variable
Reverse Helmert Coding	Compares levels of a variable with the mean of the previous levels of the variable
Deviation Coding	Compares deviations from the grand mean
Orthogonal Polynomial Coding	Orthogonal polynomial contrasts
User-Defined Coding	User-defined contrast

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

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

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

The Example Data File

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

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

means tables = write by race.

 

Case Processing Summary

	Cases
	Included		Excluded		Total
	N	Percent	N	Percent	N	Percent
writing score * RACE	200	100.0%	0	.0%	200	100.0%

 

Reportwriting score

RACE	Mean	N
hispanic	46.4583	24
asian	58.0000	11
african-amer	48.2000	20
white	54.0552	145
Total	52.7750	200

5.1 Simple Coding

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

Method 1: GLM with /LMATRIX

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

SIMPLE contrast coding

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

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

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

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

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-7.597
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-7.597
	Std. Error		1.989
	Sig.		.000
	95% Confidence Interval for Difference	Lower Bound	-11.519
		Upper Bound	-3.675
a Based on the user-specified contrast coefficients (L’) matrix: group 1 versus group 4

 

Test ResultsDependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	1188.388	1	1188.388	14.590	.000
Error	15964.717	196	81.453

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		3.945
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		3.945
	Std. Error		2.823
	Sig.		.164
	95% Confidence Interval for Difference	Lower Bound	-1.622
		Upper Bound	9.511
a Based on the user-specified contrast coefficients (L’) matrix: group 2 versus group 4

 

Test ResultsDependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	159.108	1	159.108	1.953	.164
Error	15964.717	196	81.453

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-5.855
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-5.855
	Std. Error		2.153
	Sig.		.007
	95% Confidence Interval for Difference	Lower Bound	-10.101
		Upper Bound	-1.610
a Based on the user-specified contrast coefficients (L’) matrix: group 3 versus group 4

 

Test ResultsDependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	602.550	1	602.550	7.398	.007
Error	15964.717	196	81.453

Method 2: GLM with /CONTRAST

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

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

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

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

Contrast Coefficients (L’ Matrix)

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

Contrast Results (K Matrix)

			Dependent Variable
RACE Simple Contrast(a)			writing score
Level 1 vs. Level 4	Contrast Estimate		-7.597
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-7.597
	Std. Error		1.989
	Sig.		.000
	95% Confidence Interval for Difference	Lower Bound	-11.519
		Upper Bound	-3.675
Level 2 vs. Level 4	Contrast Estimate		3.945
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		3.945
	Std. Error		2.823
	Sig.		.164
	95% Confidence Interval for Difference	Lower Bound	-1.622
		Upper Bound	9.511
Level 3 vs. Level 4	Contrast Estimate		-5.855
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-5.855
	Std. Error		2.153
	Sig.		.007
	95% Confidence Interval for Difference	Lower Bound	-10.101
		Upper Bound	-1.610
a Reference category = 4

 

Test ResultsDependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	1914.158	3	638.053	7.833	.000
Error	15964.717	196	81.453

Method 3: Regression

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

SIMPLE regression coding

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

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

SIMPLE regression coding

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

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

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

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

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

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

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

 

 

Coefficients(a)

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

5.2 Forward Difference Coding

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

Method 1: GLM with /LMATRIX

FORWARD DIFFERENCE contrast coding

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

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

 

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-11.542
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-11.542
	Std. Error		3.286
	Sig.		.001
	95% Confidence Interval for Difference	Lower Bound	-18.022
		Upper Bound	-5.061
a Based on the user-specified contrast coefficients (L’) matrix: group 1 versus group 2

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	1004.785	1	1004.785	12.336	.001
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		9.800
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		9.800
	Std. Error		3.388
	Sig.		.004
	95% Confidence Interval for Difference	Lower Bound	3.119
		Upper Bound	16.481
a Based on the user-specified contrast coefficients (L’) matrix: group 2 versus group 3

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	681.574	1	681.574	8.368	.004
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-5.855
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-5.855
	Std. Error		2.153
	Sig.		.007
	95% Confidence Interval for Difference	Lower Bound	-10.101
		Upper Bound	-1.610
a Based on the user-specified contrast coefficients (L’) matrix: group 3 versus group 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	602.550	1	602.550	7.398	.007
Error	15964.717	196	81.453

 

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

 

Method 2: GLM with /CONTRAST

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

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

 

 

Contrast Coefficients (L’ Matrix)

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

 

Contrast Results (K Matrix)

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

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

 

Method 3: Regression

 

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

FORWARD DIFFERENCE regression coding

 

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

 

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

FORWARD DIFFERENCE regression coding

 

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

 

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

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

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

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

 

 

 

Coefficients(a)

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

 

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

 

5.3 Backward Difference Coding

 

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

 

Method 1: GLM with /LMATRIX

 

BACKWARD DIFFERENCE contrast coding

 

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

 

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

 

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		11.542
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		11.542
	Std. Error		3.286
	Sig.		.001
	95% Confidence Interval for Difference	Lower Bound	5.061
		Upper Bound	18.022
a Based on the user-specified contrast coefficients (L’) matrix: level 1 versus level 2

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	1004.785	1	1004.785	12.336	.001
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-9.800
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-9.800
	Std. Error		3.388
	Sig.		.004
	95% Confidence Interval for Difference	Lower Bound	– 16.481
		Upper Bound	– 3.119
a Based on the user-specified contrast coefficients (L’) matrix: level 2 versus level 3

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	681.574	1	681.574	8.368	.004
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		5.855
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		5.855
	Std. Error		2.153
	Sig.		.007
	95% Confidence Interval for Difference	Lower Bound	1.610
		Upper Bound	10.101
a Based on the user-specified contrast coefficients (L’) matrix: level 3 versus level 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	602.550	1	602.550	7.398	.007
Error	15964.717	196	81.453

 

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

 

Method 2: GLM with /CONTRAST

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

 

Method 3: Regression

 

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

BACKWARD DIFFERENCE regression coding

 

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

 

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

BACKWARD DIFFERENCE regression coding

 

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

 

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

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

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

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

 

 

 

Coefficients(a)

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

 

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

 

5.4 Helmert Coding

 

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

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

Method 1: GLM with /LMATRIX

 

HELMERT contrast coding

 

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

 

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

 

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

 

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

 

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

 

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-6.960
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-6.960
	Std. Error		2.175
	Sig.		.002
	95% Confidence Interval for Difference	Lower Bound	-11.250
		Upper Bound	-2.670
a Based on the user-specified contrast coefficients (L’) matrix: group 1 versus groups 2 3 and 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	833.927	1	833.927	10.238	.002
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		6.872
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		6.872
	Std. Error		2.926
	Sig.		.020
	95% Confidence Interval for Difference	Lower Bound	1.101
		Upper Bound	12.644
a Based on the user-specified contrast coefficients (L’) matrix: group 2 versus groups 3 and 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	449.240	1	449.240	5.515	.020
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-5.855
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-5.855
	Std. Error		2.153
	Sig.		.007
	95% Confidence Interval for Difference	Lower Bound	-10.101
		Upper Bound	-1.610
a Based on the user-specified contrast coefficients (L’) matrix: group 3 versus group 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	602.550	1	602.550	7.398	.007
Error	15964.717	196	81.453

 

Method 2: GLM with /CONTRAST

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

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

 

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

 

 

Contrast Coefficients (L’ Matrix)

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

 

Contrast Results (K Matrix)

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

Method 3: Regression

 

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

HELMERT regression coding

 

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

 

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

 

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

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

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

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

 

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

 

 

 

Coefficients(a)

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

 

5.5 Reverse Helmert Coding

 

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

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

Method 1: GLM with /LMATRIX

 

REVERSE HELMERT contrast coding

 

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

 

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

 

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

 

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

 

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		11.542
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		11.542
	Std. Error		3.286
	Sig.		.001
	95% Confidence Interval for Difference	Lower Bound	5.061
		Upper Bound	18.022
a Based on the user-specified contrast coefficients (L’) matrix: group 2 versus group 1

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	1004.785	1	1004.785	12.336	.001
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-4.029
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-4.029
	Std. Error		2.602
	Sig.		.123
	95% Confidence Interval for Difference	Lower Bound	-9.161
		Upper Bound	1.103
a Based on the user-specified contrast coefficients (L’) matrix: group 3 versus groups 1 and 2

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	195.254	1	195.254	2.397	.123
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		3.169
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		3.169
	Std. Error		1.488
	Sig.		.034
	95% Confidence Interval for Difference	Lower Bound	.235
		Upper Bound	6.104
a Based on the user-specified contrast coefficients (L’) matrix: group 4 versus groups 1 2 and 3

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	369.460	1	369.460	4.536	.034
Error	15964.717	196	81.453

 

Method 2: GLM with /CONTRAST

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

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

 

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

 

 

Contrast Coefficients (L’ Matrix)

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

 

Contrast Results (K Matrix)

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

Method 3: Regression

 

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

REVERSE HELMERT regression coding

 

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

 

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

 

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

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

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

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

 

 

 

Coefficients(a)

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

 

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

 

5.6 Deviation Coding

 

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

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

Method 1: GLM with /LMATRIX

 

DEVIATION contrast coding

 

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

 

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

 

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

 

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

 

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-5.220
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-5.220
	Std. Error		1.631
	Sig.		.002
	95% Confidence Interval for Difference	Lower Bound	-8.437
		Upper Bound	-2.003
a Based on the user-specified contrast coefficients (L’) matrix: group 1 versus groups 1 2 and 3

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	833.927	1	833.927	10.238	.002
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		6.322
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		6.322
	Std. Error		2.160
	Sig.		.004
	95% Confidence Interval for Difference	Lower Bound	2.061
		Upper Bound	10.582
a Based on the user-specified contrast coefficients (L’) matrix: group 2 versus groups 1 3 and 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	697.475	1	697.475	8.563	.004
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-3.478
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-3.478
	Std. Error		1.732
	Sig.		.046
	95% Confidence Interval for Difference	Lower Bound	-6.895
		Upper Bound	-6.203E-02
a Based on the user-specified contrast coefficients (L’) matrix: group 3 versus groups 1 2 and 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	328.405	1	328.405	4.032	.046
Error	15964.717	196	81.453

 

Method 2: GLM with /CONTRAST

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

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

 

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

 

 

Contrast Coefficients (L’ Matrix)

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

 

Contrast Results (K Matrix)

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

Method 3: Regression

 

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

DEVIATION regression coding

 

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

 

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

 

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

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

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

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

 

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

 

 

 

Coefficients(a)

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

 

5.7 Orthogonal Polynomial Coding

 

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

POLYNOMIAL

 

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

 

Method 1: GLM with /LMATRIX

 

glm write by race
 /lmatrix "linear" race -.671 -.224 .224 .671
 /lmatrix "quadratic" race .5 -.5 -.5 .5
 /lmatrix "cubic" race -.224 .671 -.671 .224.

 

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		2.902
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		2.902
	Std. Error		1.535
	Sig.		.060
	95% Confidence Interval for Difference	Lower Bound	-.125
		Upper Bound	5.930
a Based on the user-specified contrast coefficients (L’) matrix: linear

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	291.104	1	291.104	3.574	.060
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-2.843
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-2.843
	Std. Error		1.964
	Sig.		.149
	95% Confidence Interval for Difference	Lower Bound	-6.717
		Upper Bound	1.031
a Based on the user-specified contrast coefficients (L’) matrix: quadratic

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	170.665	1	170.665	2.095	.149
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		8.277
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		8.277
	Std. Error		2.316
	Sig.		.000
	95% Confidence Interval for Difference	Lower Bound	3.709
		Upper Bound	12.846
a Based on the user-specified contrast coefficients (L’) matrix: cubic

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	1040.029	1	1040.029	12.769	.000
Error	15964.717	196	81.453

 

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

 

Method 2: GLM with /CONTRAST

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

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

Contrast Coefficients (L’ Matrix)

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

Contrast Results (K Matrix)

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

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

Method 3: Regression

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

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

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

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

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

 

 

 

Coefficients(a)

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

 

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

 

5.8 User Defined Coding

 

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

 

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

 

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

 

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

 

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

 

Method 1: GLM with /LMATRIX

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

 

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

 

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

 

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		-1.742
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		-1.742
	Std. Error		2.732
	Sig.		.525
	95% Confidence Interval for Difference	Lower Bound	-7.131
		Upper Bound	3.647
a Based on the user-specified contrast coefficients (L’) matrix: compare group 1 to group 3

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	33.092	1	33.092	.406	.525
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		7.743
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		7.743
	Std. Error		2.897
	Sig.		.008
	95% Confidence Interval for Difference	Lower Bound	2.030
		Upper Bound	13.457
a Based on the user-specified contrast coefficients (L’) matrix: compare group 2 to groups 1 and 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	581.833	1	581.833	7.143	.008
Error	15964.717	196	81.453

 

Contrast Results (K Matrix)(a)

			Dependent Variable
Contrast			writing score
L1	Contrast Estimate		1.102
	Hypothesized Value		0
	Difference (Estimate – Hypothesized)		1.102
	Std. Error		1.964
	Sig.		.576
	95% Confidence Interval for Difference	Lower Bound	-2.772
		Upper Bound	4.975
a Based on the user-specified contrast coefficients (L’) matrix: compare groups 1 and 2 to groups 3 and 4

 

Test Results Dependent Variable: writing score

Source	Sum of Squares	df	Mean Square	F	Sig.
Contrast	25.618	1	25.618	.315	.576
Error	15964.717	196	81.453

 

Method 2: GLM with /CONTRAST

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

 

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

 

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

 

 

Contrast Coefficients (L’ Matrix)

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

 

Contrast Results (K Matrix)

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

Method 3: Regression

 

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

 

 

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

 

 

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

 

matrix.

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

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

print x.

end matrix.

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

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

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

 

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

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

if race = 1 x3 = 1.5.
if race = 2 x3 = -.5.
if race = 3 x3 = 1.5.
if race = 4 x3 =-2.5.
execute.

 

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

 

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

 

get file = "d: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.

 

 

Coefficients(a)

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

5.9 Summary

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

5.10 For more information

Here are some additional resources.

• 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