Select the method or formula of your choice.

To simplify the calculation of the shelf life, consider whether the model includes the time*batch interaction or not.

When the batch effect and the batch*time interaction are in the model, the fit for the *i*^{th} batch at time *x _{ij}* uses this model:

where

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

the estimated fixed mean intercept for all batches | |

the estimated mean slope for all batches | |

the estimated random intercept for the i^{th} batch | |

the estimated slope for the i^{th} batch |

For this model, the (1 - *p*)^{th} percentile of the response is:

We assume that these equalities are true:

The variance of the estimated percentile is:

where

To calculate a meaningful shelf life, Minitab evaluates two conditions. First, Minitab determines whether the estimated fixed mean intercept for all batches is positive:

Second, Minitab determines whether the estimated mean slope for all batches is negative:

To find the shelf life, set the equation that follows equal to the lower specification limit and solve for time (*x*):

Minitab uses an iterative algorithm to find a solution between 0 and 10 times the largest value in the time variable.

When the batch*time interaction is not in the model, the slopes are the same for every batch. The fit for the *i*^{th} batch at time *x*_{ij} uses this model:

where

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

the estimated fixed mean intercept for all batches | |

the estimated mean slope for all batches | |

the estimated random intercept for the i^{th} batch |

For this model, the (1 - *p*)^{th} percentile of the response is:

We assume that these equalities are true:

The variance of the estimated percentile is:

where

Substituting these values into the formula for the variation of the percentile yields:

where

To calculate a meaningful shelf life, Minitab evaluates three conditions. First, Minitab determines whether the estimated fixed mean intercept for all batches is positive:

Second, Minitab determines whether the estimated mean slope for all batches is negative:

Third, Minitab determines whether the square root portion of the quadratic equation has a real number solution:

where

To find the shelf life, set the equation that follows equal to the lower specification limit and solve for time (*x*):

For this model, the solution for *x* simplifies to this formula:

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

the estimated fixed mean intercept for all batches | |

the estimated mean slope for all batches | |

Z | the value of the inverse cumulative distribution function |

L | the lower specification limit |

x | the shelf life |

p | the proportion of the product above the lower specification limit |

X | the design matrix |

When the batch effect and the batch*time interaction are in the model, the fit for the *i*^{th} batch at time *x _{ij}* uses this model:

where

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

the estimated fixed mean intercept for all batches | |

the estimated mean slope for all batches | |

the estimated random intercept for the i^{th} batch | |

the estimated slope for the i^{th} batch |

For this model, the *p*^{th} percentile of the response is:

We assume that these equalities are true:

The variance of the estimated percentile is:

where

To calculate a meaningful shelf life, Minitab evaluates two conditions. First, Minitab determines whether the estimated fixed mean intercept for all batches is positive:

Second, Minitab determines whether the estimated mean slope for all batches is negative:

To find the shelf life, set the equation that follows equal to the lower specification limit and solve for time (*x*):

Minitab uses an iterative algorithm to find a solution between 0 and 10 times the largest value in the time variable.

When the batch*time interaction is not in the model, the slopes are the same for every batch. The fit for the *i*^{th} batch at time *x*_{ij} uses this model:

where

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

the estimated fixed mean intercept for all batches | |

the estimated mean slope for all batches | |

the estimated random intercept for the i^{th} batch |

For this model, the *p*^{th} percentile of the response is:

We assume that these equalities are true:

The variance of the estimated percentile is:

where

Substituting these values into the formula for the variation of the percentile yields:

where

To calculate a meaningful shelf life, Minitab evaluates three conditions. First, Minitab determines whether the estimated fixed mean intercept for all batches is positive:

Second, Minitab determines whether the estimated mean slope for all batches is negative:

Third, Minitab determines whether the square root portion of the quadratic equation has a real number solution:

where

For this model, the solution for *x* simplifies to this formula:

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

the estimated fixed mean intercept for all batches | |

the estimated mean slope for all batches | |

Z | the value of the inverse cumulative distribution function |

L | the lower specification limit |

x | the shelf life |

p | the proportion of the product above the lower specification limit |

X | the design matrix |

Minitab first determines whether a meaningful estimate of the shelf life exists. A meaningful estimate of the shelf life exists when the mean response is between the upper and lower specification limits at time 0:

If a meaningful estimate exists, then Minitab determines whether to estimate the shelf life relative to the upper specification limit or the lower specification limit.

If the response decreases with time, then Minitab calculates the shelf life relative to the lower specification limit. This formula gives the condition when the response decreases:

If the response increases with time, then Minitab calculates the shelf life relative to the upper specification limit. This formula gives the condition when the response increases:

For details on the calculation of the shelf life for each case, go to the section that describes the calculation when there is only one specification limit. To calculate the shelf life when the calculations have two limits, change *Z*_{cl} and *Z*_{0.95} to *Z*_{0.5+cl/2}

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

the estimated fixed mean intercept for all batches | |

the estimated mean slope for all batches | |

cl | the confidence level |