σ2, σ21, ... , σ2c 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 models.
V(σ2) = σ2H(θ) = σ2[In + θ1Z1Z'1 + ... + θcZcZ'c]
When batch is a random factor, 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. The function to minimize is:
the number of observations
the number of parameters in β, 2 for stability studies
the error variance component
the design matrix ––for the fixed terms, the constant and time
In + θ1Z1Z'1 + ... + θcZcZ'c
the identity matrix with n rows and columns
the ratio of the variance of the ith random term over the error variance
the n x mi matrix of known codings for the ith random effect in the model
the number of levels for the ith random effect
the number of random effects in the model
the determinant of H(θ)
the transpose of X
the inverse of H(θ)
Box-Cox transformation selects lambda values, as shown below, which minimize the residual sum of squares. The resulting transformation is Yλ when λ ≠ 0 and ln(Y) when λ = 0. When λ < 0, Minitab also multiplies the transformed response by −1 to maintain the order from the untransformed response.
Minitab searches for an optimal value between −2 and 2. Values that fall outside of this interval might not result in a better fit.
Here are some common transformations where Y′ is the transform of the data Y:
Lambda (λ) value
λ = 2
Y′ = Y2
λ = 0.5
λ = 0
Y′ = ln(Y )
λ = −0.5
λ = −1
Y′ = −1 / Y
Random batch model selection
The model selection determines whether the shelf life depends on batch and whether the effect of time depends on the batch. Minitab considers the following three models in sequence:
Time + Batch + Batch*Time (unequal slopes and intercepts for batches)
Time + Batch (equal slopes and unequal intercepts for batches)
Time (equal slopes and intercepts for batches)
If the Batch*Time interaction is significant, the analysis fits the first model. If the interaction is not significant but the Batch term is significant in the second model, the analysis fits the second model. Otherwise, the analysis fits the third model.
The test for whether to pool batches is slightly different from the test to include batch, although both depend on the chi-square distribution. The formulas for the test statistics and p-values are as follow.