Deriving the decomposition table using the randomization for the meatloaf tasting 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.

Brien and Bailey (2006, Section 4.1) give the two-phase sensory experiment in the following 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.

Randomization diagram for sensory 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 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.

  2. Top
  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 sensory
    {Sessions, Panellists, Time-orders}
    Unrandomized production, randomized sensory
    {Blocks, Meatloaves}
    Randomized production
    {Rosemary, Irradiation}
    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

  4. Top
  5. Intratier formulae Given the nesting shown in the panels of the randomization 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

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

  8. Top
  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.
      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

  11. Top
  12. Derive the expected mean squares and add to table Let the stratum stratum variance components from the first formula be ξS, ξSP, ξST and ξSPT (with ξ0 for the Mean). Similarly those from the second formula are ηB and ηBM (with η0 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 columns of the table. Note the multiplier of 12 for the ηs, this being the replications of the meatloaves amongst the tastings. More explanation

  13. E[MSq] in decomposition table
    In terms of variance components (φs), the stratum variance components are: ξS = φSPT + 6φSP + 12φST + 72φS, ξSP = φSPT + 6φSP, ξST = φSPT + 12φST, ξSPT = φSPT, ηB = φBM + 6φB and ηBM = φBM