Formulating the allocation-based mixed modelfor a duplicated wheat experiment

The sense in which the allocation is used is that it determines, via the tiers, the terms that will be included in the mixed model.

Example 9 of Brien and Bailey (2006) is an experiment that consists of a field phase and a laboratory phase. In the field phase 49 lines of wheat are investigated using a randomized complete-block design with four blocks. Here the laboratory phase is modified by supposing that the procedure described by Brien and Bailey (2006) is repeated on a second occasion. That is, two samples will be obtained from each plot and one of them processed on the first occasion and the other on the second occasion. The figure below gives the factor-allocation diagram for the modified experiment that, because all allocation in the modified experiment was by randomization, is also a randomization diagram. Recall that a 7 x 7 balanced lattice square design with four replicates is used to assign the blocks, plots and lines to four intervals in each occasion. In each interval on one occasion there are seven runs at which samples are processed at seven consecutive times. Pseudofactors are introduced for lines and plots in order to define the design of the second phase.

Randomization diagram for wheat experiment

An experimental design can be generated for this example and the decomposition table exhibiting its anatomy computed using the following R script; its output is here.

Formulating the model

The following diagram illustrates the procedure for formulating the allocation-based mixed model. Derivation of the mode for the example follows the diagram or you can go to the derivation for a particular rectangle by clicking on it.

Mixed model derivation
  1. Sets of objects and obervational unit The set of objects for this experiment are the lines, the samples and the analyses. Of these, the observational units are the analyses. It is the entity from which a single value of the response variable is obtained.

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  3. Tiers The tiers for this experiment correspond to the factors that were (i) allocated in the field phase, (ii) the recipient factors in the field phase (and allocated in the laboratory phase), and (iii) the recipient factors in the laboratory phase. That is, there are 3 tiers as follows:
    Recipient laboratory
    {Occasions, Intervals, Runs, Times}
    Recipient field, allocated laboratory
    {Blocks, Plots, Samples}
    Allocated field
    {Lines}
    Also, Plots are nested within Blocks and Samples are nested within Blocks and Plots; Intervals are nested within Occasions and both Runs and Times are nested within Occasions and Intervals. This nesting reflects the randomizations performed for the design.

    Note that the tiers form disjoint sets of factors, the sets differing in their status in the allocation. Factors in different tiers have been associated by allocation, those within have not.

    Now, if desired, the factor-allocation diagram for this example can be constructed. The first step in doing so is to place each of these 3 tiers in a panel, adding the nesting factors to those factors that are nested within others. Then the arrows, lines and symbols that detail the allocation would be added.

    More explanation

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  5. Initial allocation model The symbolic form of the inital allocation model for this example, using first-letters for the factors, is:

    L | B + B^P + B^P^S + O + O^I + O^I^R + O^I^T + O^I^R^T

    Note that, because of the nesting of the factors, Plots only occurs with Blocks, Samples with Plots (and Blocks), Intervals with Occasions and Runs and Times with Intervals (and Occasions).

    More explanation

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  7. Homogeneous allocation model Perhaps Blocks and Occasions are to be changed from random to fixed, because of the small number ) of them and the difficulty of estimating a canonical component in these circumstances. Modifying the initial allocation model to effect these changes produces the following homogeneous allocation model:

    B + O + L | B^P + B^P^S + O^I + O^I^R + O^I^T + O^I^R^T

    More explanation

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  9. Prior allocation model It can be seen from the skeleton analyis-of-variance table for this example that all canonical components are estimable. There does not seem to be a need to alter the parameterization of the terms in the model and so the prior allocation model is the same as the homogeneous allocation model:

    B + O + L | B^P + B^P^S + O^I + O^I^R + O^I^T + O^I^R^T

    More explanation

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