# How is a coefficient for a main effect calculated?

In general, coefficients can be calculated from this formula:

TermDescription
Xthe design matrix
Ythe response vector

For a balanced, orthogonal design with no covariates, coefficients for main effects have a simple relationship to factor means. The following table of data means lets you calculate the coefficients for this simple case:

Rows: Experience Columns: RoadType Dirt Gravel Paved All Advanced 15.00 7.00 7.50 9.83 Beginner 20.00 18.00 10.00 16.00 All 17.50 12.50 8.75 12.92 Cell Contents Correction Time : Mean

There is one main effect coefficient for each level of each factor. In the case that the factor coding in the design matrix is (-1, 0, 1), the coefficient is the mean for that level minus the overall mean. For example, the mean of the observations for RoadType: Dirt is 4.58 from the overall mean, while the mean of the observations for RoadType: Gravel is - 0.42 from the overall mean. The vertical lines in the plots show the coefficients for the driving data.

The coefficients are summarized in the following table:
Factor/Level Mean Effect
Experience: Advanced 16 16 – 12.92 = 3.08
Experience: Beginner 9.83 9.83 – 12.92 = -3.08
RoadType: Dirt 17.5 17.5 - 12.92 = 4.58
RoadType: Gravel 12.5 12.5 - 12.92 = -0.42
RoadType: Paved 8.75 8.75 - 12.92 = -4.17

Notice that the effects for each factor add up to 0, except for rounding error.

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