3 TAPS - some learning

3.1 What is special about TAPS designs?

Designed experiments in TAPS are typically arranged in a randomized complete block design (RCBD).

Schematic representation of a randomized complete block design.

Figure 3.1: Schematic representation of a randomized complete block design.

Some TAPS designed experiments are arranged in a split-plot design.

Schematic representation of a split-plot design randomized complete block arrangement.

Figure 3.2: Schematic representation of a split-plot design randomized complete block arrangement.

  • Designs are D-Optimal - best to estimate the team’s yield/profit precisely.
Prioritizing our research questions often seems like a balancing act between resources allocated to the competition, and resources allocated to additional research.

Figure 3.3: Prioritizing our research questions often seems like a balancing act between resources allocated to the competition, and resources allocated to additional research.

3.2 An applied example

A manager is designing a competition with 24 teams signed up, and needs to decide whether to use 4 replicates or 3 replicates and allocate more resources to the research question.

Some questions:

  • What is the trade-off between #reps and #points?
  • How do we assign the new treatments?

Consider the following options:

Table 3.1: Design Options
# Teams # Reps per team # Plots assigned to other treatments Total plots available
24 4 12 + 48 108
24 3 12 + 24 108
24 2 12 108

Mathematically, we can infer a few points:

  • Recall that \(se(\hat{\mu_j}) =\sqrt\frac{\sigma^2}{r}\).
Table 3.2: SE of the team’s means for the different design options
# Teams # Reps per team # Plots assigned to other treatments SE(Teams mean)
24 4 12 + 48 sigma2/sqrt(4) = 0.5 sigma
24 3 12 + 24 sigma2/sqrt(3) = 0.58 sigma
24 2 12 sigma2/sqrt(2) = 0.71 sigma

How are the estimates of N and Irr affected?

  • Recall that \(se(\hat{\boldsymbol\beta}) =\sqrt{\sigma^2 (\mathbf{X}^\top\mathbf{X})^{-1}}\) and thus \(se(\hat{\beta_j}) =\sqrt{\sigma^2 (\mathbf{X}^\top\mathbf{X})^{-1}_{jj}}\).

Essentially,

\[se(\hat\beta_j) = \sqrt{\sigma^2 (r\mathbf{X}_{teams}^\top \mathbf{X}_{teams} + \mathbf{X}_{extra}^\top \mathbf{X}_{extra})^{-1}}\]

We can also consider different \(\sigma^2\) for treatment versus extra.

\[se(\hat\beta_j) = \sqrt{(\frac{1}{\sigma^2_t} r\mathbf{X}_{teams}^\top \mathbf{X}_{teams} + \frac{1}{\sigma^2_{extra}} \mathbf{X}_{extra}^\top \mathbf{X}_{extra})^{-1}_{jj}}\]

3.2.1 Comparing designs side by side

Table 3.3: Table 3.4: Evaluation of Standard Errors (Raw Agronomic Units)
# reps Extra plots SE(trt mean) SE(Intercept) SE(N) SE(Irr) SE(N^2) SE(Irr^2) SE(NxIrr) SE(OPT N) SE(OPT Irr)
Option 1 4 0 30.000 126.996 1.881 30.585 0.009 1.572 0.252 20.201 1.441
Option 2 4 12 30.000 115.235 1.185 14.039 0.004 0.652 0.072 14.010 0.987
Option 3 3 36 34.641 109.026 1.003 12.986 0.003 0.560 0.045 12.677 0.879
Option 4 2 60 42.426 108.776 0.969 12.882 0.003 0.555 0.041 12.294 0.768
Design spaces in raw units. Collinear team behavior is shown alongside the extra treatments offset to map unexplored regions.

Figure 3.4: Design spaces in raw units. Collinear team behavior is shown alongside the extra treatments offset to map unexplored regions.

3.3 Planning Colby 2026

An applied scenario & discussion.

  • How many repetitions?
  • Should we combine competitions?
  • Discuss tradeoffs on the whiteboard.