Select the method or formula of your choice.

The predicted y or ; the mean response value for the given predictor values using the estimated regression equation.

The standard error of the marginal fitted values in the mixed model depend on the test method for the fixed effects. For both methods, the standard errors are the square roots of the diagonal elements of the variance matrix of the fits.

where

where

A residual is the difference between an observed value and a fitted value. This part of the observation is not explained by the fitted model. The residual of an observation is:

When batch is a random factor, Minitab calculates 2 types of residuals. Marginal residuals use the fitted value for a random batch, so the coefficient for batch is not in the equation.

Conditional residuals use the fitted value for a batch that is in the data.

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

y_{i} | i^{th} observed response value |

i^{th} fitted response value | |

the vector of fitted responses | |

X | the design matrix for the fixed effects |

the vector of fixed predictors | |

Z | the design matrix for the random factors |

the vector of the estimated BLUP values |

Standardized residuals are also called "internally Studentized residuals."

where the standard deviation of the residual is the appropriate diagonal square root of the residual variance matrix:

where

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

e_{i} | the i^{th} residual |

Std(e)_{i} | the standard deviation of the i^{th} residual |

The range in which the estimated mean response for a given set of predictor values is expected to fall.

The standard error of the fitted values in the mixed model are the square roots of the diagonal elements of this matrix:

where

The degrees of freedom use this formula when batch is a random factor:

where

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

t_{1-α/2, df} | 1–α/2 quantile from the t distribution with the given degrees of freedom |

standard error of the fitted value | |

X | design matrix, including the constant |

X' | transpose of X |

variance component for error | |

variance component of the i^{th} random factor | |

Z_{i} | n x m matrix of known codings for the _{i}i^{th} random effect in the model |

Z'_{i} | transpose of Z_{i} |

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

x_{i} | predictor values for the fit or prediction |

W | asymptotic variance-covariance matrix of the variance component for error |

c | number of random effects in the model |

The range in which the predicted response for a new observation is expected to fall. The calculation of the prediction interval depends on whether you compute the interval for the marginal fit or for the conditional fit.

where

The degrees of freedom for the t-statistic are given by this formula:

where

where

The degrees of freedom for the t-statistic are:

where

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

1–α/2 quantile from the t distribution with the given degrees of freedom | |

vector of the new values of the random predictors | |

variance component for error | |

vector of new values of the fixed predictors | |

variance component of the i^{th} random factor | |

I_{m} | identity matrix with m rows and columns |

m | number of columns in the design matrix to represent the i^{th} random term in the model |

c | number of random effects in the model |

Z_{i} | n x mdesign matrix for the _{i} i^{th} random effect in the model |

Z'_{i} | transpose of Z_{i} |