# Alternative Shortcut Method

The alternative method for finding the expected mean square is described in one of the popular texts in the design of experiments (Montgomery, 2019). In our experience, we have seen that when students are comfortable in developing the expected mean square applying the easy to use the method described earlier (Underwood, Underwood et al. 1997; Kutner, Nachtsheim et al. 2005), this alternative method described in Montgomery text will cut a few steps provided below (Montgomery, 2019).

# Restricted Model

The expected mean square for a model term is the EMS(error) (σ2) plus either the variance component or the fixed effect component for that term, plus those components for all other model terms that contain the effect in question and that involve no interactions with other fixed effects. The coefficient of each variance component or fixed effect is the number of observations at each distinct value of that component (Montgomery, 2019).

Consider a model with A and B fixed factors with a random factor C. To find the expected mean square for C; AC, BC, and ABC are not included because of the presence of fixed factor(s) in the interaction. The total replication for C is given by abn (product of the levels of other factors and the experimental replications). Therefore, the expected mean square for C is given by EMS for all other model terms can be found in Table 7.

# Unrestricted Model

In addition to all terms in the restricted model, include the term for the effect in question, plus all the terms that contain this effect as long as there is at least one random factor (Montgomery, 2019).

Consider a model with A and B fixed factors with a random factor C. To find the expected mean square for C; AC, BC, and ABC are included because of the presence of at least one random factor in the interaction. The replication for AC, BC, ABC, and C are given by bn, an, n, and abn, (product of the levels of other factors and the experimental replications), respectively. Therefore, the expected mean square for C is given by EMS for all other model terms of the design with fixed factors A and B and a random factor C can be found in Table 7.

Table 7

Unrestricted vs. Restricted Mixed Models (A and B fixed factors with a random factor C) 