Minitab calculates variance components only for random factors. A model with two random factors is used to present the formulas.
where, αi, βj , (αβ)ij, and εijk are independent random variables. The variables are normally distributed with mean zero and variances given by these formulas:
These variances are the variance components. In this case, test the hypothesis that the variance components are equal to zero.
For a restricted mixed model with two factors, the model is:
where αi is a fixed effect and βj is a random effect, (αβ)ij, is a random effect, and εijk is random error. The Σαi = 0 and Σ(αβ)ij = 0 for each j. The variances are V(βj) = σ2β,V[(αβ)ij] =[(a - 1)/a]σ2αβ, and V(εijk) = σ2. σ2β, σ2αβ, and σ2 are variance components. Summing the interaction component over the fixed factor equals zero, which indicates this is the restricted mixed model.
For an unrestricted mixed model with a fixed factor, A, and a random factor, B, this formula describes the model:
where αi are fixed effects and βj, (αβ)ij and εijk are uncorrelated random variables having zero means and these variances:
These variances are the variance components. The Σα i = 0 and Σ(αβ)ij = 0 for each j.
This information is for balanced models. For information on unbalanced or more complex models, see Montgomery1 and Neter2.
- D.C. Montgomery (1991). Design and Analysis of Experiments, Third Edition. John Wiley & Sons.
- J. Neter, W. Wasserman and M.H. Kutner (1985). Applied Linear Statistical Models, Second Edition. Irwin, Inc.