Deriving the decomposition table using the randomization for a duplicated wheat experiment

The sense in which the randomization is used is that it determines, via the tiers, the sources that will be included in the decomposition table.

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 randomization diagram for the modified experiment. 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

Deriving the decomposition table

The following diagram illustrates the procedure for deriving the decomposition table. Derivation of the table for the example follows the diagram or you can go to the derivation for a particular rectangle by clicking on it.

Decomposition table 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 correspond to the factors in the same panel in the randomization diagram for the experiment. There are 3 tiers as follows:
    Unrandomized laboratory
    {Occasions, Intervals, Runs, Times}
    Unrandomized field, randomized laboratory
    {Blocks, Plots, Samples}
    Randomized field
    {Lines}
    Note that the tiers form disjoint sets of factors, the sets differing in their status in the randomization. Factors in different tiers have been associated by randomization, those within have not. More explanation

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  5. Intratier formulae Given the nesting shown in the panels of the randomization diagram and the pseudofactors shown, the intratier formulae for this experiment are as follows:
    2 Occasions / 4 Intervals / (7 Runs * 7 Times)
    (4 Blocks / (49 Plots // (7 P1 + 7 P2)) / 2 Samples) // (S1 / Blocks / (P1 + P2))
    49 Lines // (7 L1 + 7 L2 + 7 L3 + 7 L4 + 7 L5 + 7 L6 + 7 L7 + 7 L8)
    More explanation

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  7. Analysis formulae There are no intertier interactions thought relevant for this experiment. More explanation

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  9. Derive the decomposition table The deomposition table is derived by performing a circuit of the loop in the diagram as follows:
    1. Derive the terms and sources from each formula in turn More explanation
    2. Incorporate current sources and their degrees of freedom into the decomposition table. More explanation
    1. Using first-letters for the factors, the first formula expands as follows.
      O / I / (R * T) = O + O^I + O^I^R + O^I^T + O^I^R^T
      The sources and their degrees of freedom are listed in the following first major column of the decomposition table:
      First column of decomposition table
    2. The second formula expands as follows: (B / (P // (P1 + P2)) / S) // (S1 / B / (P1 + P2)) = B + B^P1 + B^P2 + B^P + S1 + B^S1 + B^S1^P1 + B^S1^P2 + B^P^S. The sources and their degrees of freedom are listed in the second major column of the following decomposition table:
      Second column of decomposition table
      Note that different parts of Samples[Blocks^Plots], specified by the pseudofactors, are confounded with each of the analyses sources. The last source has a horizontal ⊥ as a subscript indicating that it is the part of the Samples[Blocks^Plots] source that is residual to the S1 pseudofactor sources. Further, different parts of Plots[Blocks] are confounded with the Runs[O^I], Times[O^I] and Runs#Times[O^I} sources.
    3. The third formula expands as follows: L // (L1 + L2 + L3 + L4 + L5 + L6 + L7 + L8) = L1 + L2 + L3 + L4 + L5 + L6 + L7 + L8 + L = LinesR + LinesT + L where LinesR and LinesT are is the sums of the odd-numbered and even-numbered L pseudofactors respectively. The sources and their efficiency factors and degrees of freedom are listed in the third major column of the following decomposition table:
      Third column of decomposition table
      Here Lines is confounded with Plots[Blocks], which is spread over three analyses sources as mentioned in the discussion of the second formula. Clearly, an advantage of the decomposition table is that it displays the confounding in the experiment.

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  10. Categorize terms as fixed or random One division of factors into fixed and random is as follows:
    Random
    Occasions, Ontervals, Runs, Times, Blocks, Block, Samples
    Fixed
    Lines
    More explanation

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  12. Derive the expected mean squares and add to table Let the stratum stratum variance components from the first formula be ξO, ξOI, ξOIR, ξOIT and ξOIRT (with ξ0 for the Mean). Similarly those from the second formula are ηB, ηBP and ηBPS (with η0 for the Mean). Also let q(LR) and q(LT) be the quadratic forms for the sources corresponding to the fixed Lines term. Then the expected mean squares are given in the last major columns of the table. More explanation

  13. E[MSq] in decomposition table
    In terms of variance components (φs), the stratum variance components are: ξO = φOIRT + 7φOIR + 7φOIT + 49φOI + 96φO, ξOI = φOIRT + 7φOIR + 7φOIT + 49φOI, ξOIR = φOIRT + 7φOIR, ξOIT = φOIRT + 7φOIT, ξOIRT = φOIRT, ηB = φBPS + 2φBP + 98φB and ηBPS = φBPS
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