Select the method or formula of your choice.

Enforce parameter constraints by transforming the parameters.^{1}

If | Then |
---|---|

a < θ | θ = a + exp( φ ) |

θ < b | θ = b - exp( φ ) |

a < θ < b | θ = a +((b - a) / (1 + exp( -φ ))) |

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

a and b | numeric constants |

θ's | parameters |

φ | transformed parameters |

Minitab performs these transforms, and displays the results in terms of the original parameters.

- Bates and Watts (1988). Nonlinear Regression Analysis and Its Applications. John Wiley & Sons, Inc.

The approximate standard error of the estimate of θ_{p} is S times the square root of diagonal element p of , which is written as:
where **e**_{p} is a P by 1 vector with element p equal to 1 and all other elements equal to 0. Minitab computes:
by back-solving:

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

n | n^{th} observation |

N | total number of observations |

p | number of free (unlocked) parameters |

R | the (upper triangular) R matrix from the QR decomposition of V^{i} for the final iteration |

V_{0} | gradient matrix = ( ∂f(x_{n}, θ) / ∂θ_{p}), the P by 1 vector of partial derivatives of f(x_{0}, θ), evaluated at θ* |

S |

The approximate variance-covariance matrix of the parameter estimates is:
The approximate correlation between the estimates of θ_{p} and θ_{q} is:
Because **R** is triangular, Minitab can obtain its inverse by back-solving rather than by a general-purpose inversion algorithm.

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

R | the (upper triangular) R matrix from the QR decomposition of V^{i} for the final iteration |

P | number of free (unlocked) parameters |

v_{0} | gradient matrix = ( ∂f(x_{n}, θ) / ∂θ p), the P by 1 vector of partial derivatives of f( x_{0}, θ), evaluated at θ* |

θ's | parameters |

Let θ = (θ_{1}, . . . . θ_{p}) * with *θ** being the final iteration for θ.

The likelihood-based 100 (1 - α) % confidence limits satisfy:

where S( *θ*_{p} ) is the SSE obtained when holding *θ*_{p} fixed and minimizing over the other parameters.^{1} This is equivalent to solving:

S(*θ*_{p}) = S(** θ***) + (

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

θ's | parameters |

n | n^{th} observation |

N | total number of observations |

P | number of free (unlocked) parameters |

t_{α/2} | upper α/2 point of the t distribution with N - P degrees of freedom |

S(θ) | Sum of the squared error |

MSE | mean squared error |

- Bates and Watts (1988). Nonlinear Regression Analysis and Its Applications. John Wiley & Sons, Inc.