In general, coefficients can be calculated from this formula:

Term | Description |
---|---|

X | the design matrix |

Y | the 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.