Select the method or formula of your choice.

Minitab uses least squares estimation to calculate the coefficients.

In matrix terms, the least squares estimates of the coefficients are:

**b** = (**X'X**)^{-1}**X'y**

For more information on coefficients of higher order models, see Cornell^{1}.

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

X | design matrix |

y | response column |

- J.A. Cornell (1990). Experiments With Mixtures: Designs, Models, and the Analysis of Mixture Data, John Wiley & Sons.

For simple linear regression, the standard error of the coefficient is:

The standard errors of the coefficients for multiple regression are the square roots of the diagonal elements of this matrix:

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

x_{i} | i^{th} predictor value |

mean of the predictor | |

X | design matrix |

X' | transpose of the design matrix |

s^{2} | mean square error |

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

test statistic for the coefficient | |

estimated coefficient | |

standard error of the estimated coefficient |

The two-sided p-value for the null hypothesis that a regression coefficient equals 0 is:

The degrees of freedom are the degrees of freedom for error, as follows:

*n* – *p*

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

The cumulative distribution function of the t distribution with degrees of freedom equal to the degrees of freedom for error. | |

t_{j} | The t statistic for the j^{th} coefficient. |

n | The number of observations in the data set. |

p | The sum of the degrees of freedom for the terms. |

The VIF can be obtained by regressing each predictor on the remaining predictors and noting the R^{2}value.

For predictor *x _{j}*, the VIF is:

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

R^{2}( x)_{j} | coefficient of determination with x as the response variable and the other terms in the model as the predictors_{j} |