## Formulating an allocation-based mixed model

(Brien and Bailey, 2006; Brien and Bailey, 2009; Brien and Demétrio, 2009; Bailey and Brien, 2016)

Brien (2017c, Section 5.1) argues for the allocation of factors in an experiment to be based on an anticipated mixed model. Given this, it seems sensible that the mixed model to be used in analyzing responses from the experiment should be based on the allocations used in producing the experimental design, i.e. an allocation-based mixed model. Brien (2017c, Section 5.1) describes a procedure for formulating such a model from the factor-allocation diagram using a three-stage process in which initial, homogeneous and allocation-based models are derived. Brien (2017c, Figure 2) depicts the process. The three-stage process is parallels that of Brien and Demétrio (2009), although the names of the models have been modified. Nonetheless, Brien and Demétrio (2009) contains more detail than presented here and the notation employed here is that of their Table 1.

The derivation of the allocation-based mixed model is related to the process Brien and Bailey (2006, section 7) describe for deriving a randomization-based mixed model differs from an allocation-based mixed model only in that all allocations in the experiment must be by randomization. Bailey and Brien (2016, section 11.2) discuss the place of randomization-based mixed models amongst the classs of mixed models based on factors.

The following diagram illustrates the procedure for formulating an allocation-based mixed model, given that a design for the experiment has been chosen. A description of the procedure follows the diagram or you can go to the description for a particular rectangle by clicking on it. The first two steps of the procedure, which are the same as those for deriving a decomposition table for the design, amount to establishing the factor-allocation description for the design.

Two examples of the derivation of the allocation-based mixed models are available: a two-phase sensory experiment and a two-phase wheat experiment. Brien (2017c) presents additional examples that illustrate the process.

**Sets of objects and observational unit**Firstly, the sets of objects involved in the allocations in the experiment are identified. Then the set of objects that are the observational units is identified. Federer (1975) defines this to be 'the smallest unit on which an observation is made'. An advantage of using the observational unit rather than the experimental unit is that for each response variable there is only one type of observational unit in an experiment whereas it is clear from Federer (1975) that there might be several different types of experimental unit. Thus, it should be easier to identify the observational unit.**Tiers**The crucial feature of the procedure is that the factors are divided into sets or tiers, as described by Brien, Harch, Correll and Bailey (2011), according to their status in the allocations that were performed in designing the experiment. Those factors that are nested within other factors and the factors that nest them also need to be identified. It can be useful to depict the allocations in a factor-allocations diagram, in which there is a panel for each tier. For multitiered experiments there will be at least three tiers. It is vital for determining the tiers that all the factors involved in the experiment are identified.**Initial allocation model**The terms in the symbolic form of the inital allocation model are derived for each tier by forming them from all posible combinations of the factors within a panel, subject to the restriction that terms involving a nested factor must include all its nesting factors. The fixed terms are those that come from the factors in a panel that was only ever allocated; the remaining terms are the random terms. The models can be compressed by using an intratier formula for the fixed terms and the sum of one or more intratier formulae for the random terms. (See Brien and Demétrio, 2009, Table 1 for notation.) A factor-allocation diagram is useful in formulating this model, it containing all the information required to do so.**Homogeneous allocation model**The homogeneous allocation model is derived from the symbolic form of the initial allocation model by (i) adding intertier interactions that are of interest to the researcher (e.g. interactions of factors such as Sites, Sex, Judges and so on with Treatments), (ii) augmenting the model with potentially important terms not taken into account in constructing the experimental design (e.g. sources of variation identified during or after the running of the experiment or that could not be incorporated into the design), and (iii) deciding if any terms need to be swapped from fixed to random and vice versa. The model is homogeneous in the sense that all the fixed terms have a parameter for each level of its generalized factor and the random effects for each term are assumed independently and identically distributed.

**Prior allocation model**The prior allocation model is obtained from the from the homogeneous allocation model by (i) changing the parameterization of those random terms for which it is to be assumed that their effects display unequal variance or are correlated, (ii) consider removing random terms because of potential singularities in the variance matrix, and (iii) reparameterize those fixed for which linear or curved trends are to be fitted.