Coefficients for the reference level in general linear model (GLM) in Minitab

General Linear Model (GLM) uses a regression approach to fit your model. After, GLM codes the factor levels as indicator variables it uses them to calculate the coefficients for all terms. The interpretation of the coefficients depends on whether the indicator variables use (-1,0,+1) coding or (1,0) coding. With (-1,0,+1) coding, the coefficients represent the distance between the factor levels and the overall mean. With (1,0) coding, the coefficients represent the difference between the other factor levels and the reference level for the factor.

For both types of coding, one of the levels is the reference level. By default, Minitab does not list the coefficient for the reference level in the coefficients table. Sometimes, you may want to know the reference level coefficient to understand how the reference value compares in size and direction to the overall mean.

How to display the coefficient for the reference level

In GLM, the coefficient for the reference level is in the single regression equation. To display the equation in the coefficients table, follow these steps:

Choose Stat > ANOVA > General Linear Model > Fit General Linear Model.
Click Results.
For Coefficients, choose Full set of coefficients.
Click OK in each dialog box.

How to calculate the coefficient for the reference level

Suppose you perform a general linear model test with 2 factors. Factor 1 has 3 different settings (35, 44, and 52). Factor 2 is 2 different times (1 and 2). Minitab uses (-1,0,+1) coding. The factors and their indicator variables are in the tables that follow:

Factor 1 has 3 levels, so Factor 1 has 2 indicator variables. When the setting is 35, the first indicator variable is 1 and the second indicator variable is 0. When the setting is 44, the first indicator variable is 0 and the second indicator variable is 1. When the setting is 52, both indicator variables are -1. The level where the setting is 52 is the reference level.

Factor 2	Indicator 1	Indicator 2
52	-1	-1
35	1	0
44	0	1
52	-1	-1
44	0	1
35	1	0

For Factor 2, when the time is 1 the indicator variable is also 1. When the time is 2, the indicator variable is -1. The level where time is 2 is the reference level.

Factor 1	Indicator
1	1
1	1
2	-1
2	-1
1	1
2	-1

You obtain the following table of coefficients:

Coefficients Term Coef SE Coef T-Value P-Value VIF Constant 68.22 1.28 53.36 0.000 Setting 35 -27.64 1.81 -15.29 0.000 1.33 44 4.86 1.81 2.69 0.011 1.33 Time 1 -0.50 1.28 -0.39 0.698 1.00

The ANOVA model is:

Regression Equation Thickness = 68.22 - 27.64 Setting_35 + 4.86 Setting_44 + 22.78 Setting_52 - 0.50 Time_1 + 0.50 Time_2

Notice that the table does not include the coefficients for 52 (Factor 1) or 2 (Factor 2), which are the reference levels for each factor. However, you can easily calculate these values by subtracting the overall mean from each level mean. The constant term is the overall mean.

To view the mean for each level in Minitab, follow these steps:

Choose Stat > Basic Statistics > Display Descriptive Statistics.
In Variables, enter the response variable.
In By variables (optional), enter the factor.
Click OK.

Repeat the steps for each factor.

The means for the example data follow:

Overall = 68.22
Setting 35 (Factor 1) = 40.583
Setting 44 (Factor 1) = 73.08
Setting 52 (Factor 1) = 91
Time 1 (Factor 2) = 67.72
Time 2 (Factor 2) = 68.72

The coefficients are calculated as the level mean − overall mean. Thus, the coefficients for each level are:

Setting 35 (Factor 1) = 40.58 – 68.22 = –27.64
Setting 44 (Factor 1) = 73.08 − 68.22 = 4.86
Setting 52 (Factor 1) = 91 − 68.22 = 22.78 (not shown in the coefficients table)
Time 1 (Factor 2) = 67.72 − 68.22 = –0.5
Time 2 (Factor 2) = 68.72 − 68.22 = 0.5 (not shown in the coefficients table)

Note

A quick way to obtain the coefficients for the reference level is to add the level coefficients for a factor (excluding the intercept) and multiply by −1. For example, the coefficient for Setting 52 = −1 * [(−27.64) + (4.86)] = 22.78.

If you add a covariate or have unequal sample sizes within each group, coefficients are based on weighted means for each factor level instead of the arithmetic mean (sum of the observations divided by n).