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Mixed effects models contain both fixed and random effects. The general form of the mixed effects model is:

**y **=** Xβ **+** Z**_{1}**μ**_{1}** **+** Z**_{2}**μ**_{2} + ... + **Z**_{c}**μ**_{c} + **ε**

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

y | the n x 1 vector of response values |

X | the n x p design matrix for the fixed effect terms, p ≤ n |

β | a p x 1 vector of unknown parameters |

the n x m design matrix for the random term in the model_{i} | |

μ_{i} | an m x 1 vector of independent variables from N(0, )_{i} |

ε | an n x 1 vector of independent variables from N(0, ) |

n | the number of observations |

p | the number of parameters in |

c | the number of random terms in the model |

Based on the model assumption for the general form of the mixed effects model, the response vector, **y**, has a multivariate normal distribution with mean vector **Xβ** and the following variance-covariance matrix:

V(**σ**^{2}) = V(σ^{2}, σ^{2}_{1}, ... , σ^{2}_{c}) = σ^{2}I_{n} + σ^{2}_{1}**Z**_{1}**Z'**_{1} + ... + σ^{2}_{c}**Z**_{c}**Z'**_{c}

where

**σ**^{2} = (σ^{2}, σ^{2}_{1}, ... , σ^{2}_{c})'

σ^{2}, σ^{2}_{1}, ... , σ^{2}_{c} are called variance components.

By factoring from the variance, you can find a representation of H(**θ**), which is in the computation of the log-likelihood of mixed effects models.

V(**σ**^{2}) = **σ**^{2}H(**θ**) = **σ**^{2}[**I**_{n} + *θ*_{1}**Z**_{1}**Z'**_{1} + ... + *θ*_{c}**Z**_{c}**Z'**_{c}]

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

θ_{i} | , the ratio of the variance of the random term over the error variance |

When the model contains a random factor, by default the unknown parameter estimates come from minimizing twice the negative of the restricted log-likelihood function. The minimization is equivalent to maximizing the restricted log-likelihood function. Minitab uses an iterative algorithm to minimize the restricted log-likelihood function. The function to minimize is:

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

H | I_{n} + θ_{1}Z_{1}Z'_{1} + ... + θ_{c}Z_{c}Z'_{c} |

|H| | the determinant of H |

H^{-1} | the inverse of H |

m_{i} | the number of levels for the random term |

the error variance component | |

I_{n} | the identity matrix with n rows and columns |

By default, Minitab calculates parameter estimates that maximize the restricted likelihood function, which is equivalent to minimizing the following function:

To minimize the function, Minitab differentiates the function with respect to **β**, *σ*^{2}, and *θ*_{i} and sets the differentials equal to 0:

where

Algebraic rearrangement of the first two equations to solve for the estimated parameters with respect to the differentiation give the following equations:

The derivative with respect to cannot be explicitly solved for the . Minitab uses Newton's method to estimate with the following steps:

The converged solutions for are the variance ratio estimates. The variance component for the random term is as follows:

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

tr(·) | the trace of the matrix |

X' | the transpose of X |