Deriving the decomposition table using the allocation for the meatloaf tasting experiment

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

Brien and Bailey (2006, Section 4.1) give the two-phase sensory experiment in the following factor-allocation diagram that, because all allocation in this experiment was by randomization, is also a randomization diagram. The first phase was a production phase and the second phase the sensory phase. The design in the second phase consists of a pair of 6 x 6 Latin squares in each session.

Factor-allocation diagram for sensory 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.

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 treatments, the meatloaves and the tastings. Of these, the observational units are the tastings. It is the entity from which a single value of the response variable, a score, is obtained.

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  3. Tiers The tiers for this experiment correspond to the factors that were (i) allocated in the production phase, (ii) the recipient factors in the production phase (and allocated in the sensory phase), and (iii) the recipient factors in the sensory phase. That is, there are 3 tiers as follows:
    Recipient sensory
    {Sessions, Panellists, Time-orders}
    Recipient production, allocated sensory
    {Blocks, Meatloaves}
    Allocated production
    {Rosemary, Irradiation}
    Also, Meatloaves is nested within Blocks and both Panellists and Time-orders are nested within Sessions, because of 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. Intratier formulae Given the nesting shown in the panels of the factor-allocation diagram, the intratier formulae for this experiment are as follows:
    3 Sessions / (12 Panellist * 6 Time-orders)
    3 Blocks / 6 Meatloaves
    2 Rosemary * 3 Irradiation
    More explanation

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  7. Analysis formulae It may be that the researcher is interested in investigating the interaction of the Panellists with the treatment terms, although the design for the experiment did not take into account the need to estimate these interactions. Because factors from different tiers are involved, these are intertier interactions. Their effect on the analysis could be investigated by adding Panellists to the third intratier formulae. This yields the analysis formula 2 Rosemary * 3 Irradiation * Panellists. We will proceed without including the intertier interactions, leaving their inclusion as an exercise for those interested.

    More explanation

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  9. Derive the decomposition table The decomposition 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.
      S / (P * T) = S + S^P + S^T + S^P^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 / M = B + B^M. 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 Blocks and Sessions are aligned as these are confounded, as are Meatloaves[Blocks] and Panelists#Time-orders[Sessions]. Further, a Residual line was added for Panelists#Time-orders[Sessions].
    3. The third formula expands as follows: R * I = R + I + R^I. The sources and their degrees of freedom are listed in the third major column of the following decomposition table:
      Third column of decomposition table
      All three sources from this tier are confounded with Meatloaves[Blocks]. Clearly, an advantage of the decomposition table is that it displays the confounding in the experiment.

  10. Categorize terms as fixed or random One division of factors into fixed and random is as follows:
    Sessions, Panellists, Time-orders, Blocks, Meatloaves
    Rosemary, Irradiation
    More explanation

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  12. Derive the expected mean squares and add to form the skeleton analysis-of-variance table Let the canonical components from the first formula be φS, φSP, φST and φSPT (with φt for the Mean). Similarly those from the second formula are φB and φBM (with φm for the Mean). Also let q(R), q(I) and q(RI) be the quadratic forms for the sources corresponding to the three fixed terms, respectively. Then the expected mean squares are given in the last major column of the skeleton analysis-of-variance table. Note the multiplier of 12 for the φs from the meatloves tier, this being the replications of the meatloaves amongst the tastings.

  13. More explanation
    E[MSq] in skeleton analysis-of-variance table