Split-plot

Today’s goals

Explore key concepts in split-plot:

  1. Different sizes of EU
  2. Treatment randomization
  3. The effects model
  4. The ANOVA table

Motivational example - Treatment design

  • 2-way factorial
  • N fertilizer rates: 0, 100, 200 kg N/ha
  • K fertilizer rates: 0, 30, 60 kg K/ha
  • 3 x 3 = 9 treatment combinations

When is split-plot needed?

  1. When levels of one treatment factor need more/less experimental material than the levels of other treatment factors

For ex., K treatments are applied with a 12-row spreader, and N rates are applied with a 4-row coulter.

When is split-plot needed?

  1. Don’t have enough experimental material to have all plots be the size of the requirements of treatment factor needing more resources

  • In the example above, we could, potentially, decide to have all plots of size 12 rows, where a given K treatment is applied with one pass on the plot, and a given N treatment is applied with 3 passes on the plot.

  • In that, we would need 36 plots x 12 rows/plot = 432 rows total

When is split-plot needed?

  1. Don’t have enough experimental material to have all plots be the size of the requirements of treatment factor needing more resources
  • However, that requires more land than we anticipated (we planed for 4-row plots), and we only have space for 36 plots x 4 rows = 144 rows total

When is split-plot needed?

Motivational example - Treatment design in a split-plot

  • 2-way factorial in a split-plot
  • Whole-plot treatment factor: K fertilizer rates: 0, 30, 60 kg K/ha
  • Split-plot treatment factor: N fertilizer rates: 0, 100, 200 kg N/ha
  • 3 x 3 = 9 treatment combinations that will be assigned to different sizes of experimental units!

Experimental design

A split-plot can be used with different experimental designs like CRD or RCBD

Assuming we have heterogeneous experimental material (e.g., different soil type, topography, etc.)

  • Thus, we should use a randomized complete block design (RCBD)

Treatment randomization - Split-plot in RCBD

  1. We first randomize the levels of the whole-plot treatment factor within each of our blocks.

Treatment randomization - Split-plot in RCBD

  1. Then we randomize the levels of the split-plot treatment factor within each of our whole-plot treatment factor plots.

The effects model

\[ y_{ijk} = \mu + \rho_{k} + \alpha_{i} + (\alpha\rho)_{ik} + \beta_{j} + (\alpha\beta)_{ij} + e_{ijk} \]

  • \(y_{ijk}\) is the observation on the kth block from ith K rate and jth N rate
  • \(\mu\) is the overall mean 
  • \(\rho_{k}\) is the random effect of kth block
  • \(\alpha_{i}\) is the differential effect of ith K rate
  • \((\alpha\rho)_{ik}\) is the whole-plot random error
  • \(\beta_{j}\) is the differential effect of jth N rate
  • \((\alpha\beta)_{ij}\) is the differential effect of the combination of the ith N rate and ith K rate
  • \(e_{ijk}\) is the residual corresponding to the block k of K rate i and N rate j.

The ANOVA table

In the following ANOVA table…

  • n is number of levels in N rate = 3
  • k is number of levels in K rate = 3
  • r is number of blocks = 4
  • N is total number of obserevations or EUs = 3 x 3 x 4 = 36

The ANOVA table

Source of variation

df

SS

MS

F ratio

Block

dfb =
r - 1 =
4 - 1 = 3

SSb

K rate

dfk =
k - 1 =
3 - 1 = 2

SSk

MSk =
SSk / dfk

MSk / MSwpe

WP error (K x Block)

dfwpe =
(k - 1) x (r - 1) =
(3 - 1) x (4 - 1) = 6

SSwpe

MSwpe =
SSwpe / dfwpe

N rate

dfn =
n - 1 =
3 - 1 = 2

SSn

MSn =
SSn / dfn

MSn / MSspe

N x K

dfnk =
(n - 1) x (k - 1) =
(3-1) x (3-1) = 4

SSnk

MSnk =
SSnk / dfnk

MSnk / MSspe

SP error

dfspe =
k(r - 1) x (n - 1) =
3(4 - 1) x (3 - 1) =
9 x 2 = 18

SSspe

MSspe =
SSspe / dfspe

TOTAL

dft =
N -1 = 35

SSt

  • Notice how we have 2 error sources: the WP and the SP.

  • What would happen if we ran this mistakenly as full RCBD with a single EU size?

ANOVA table - motivational example

Source of variation

Chisq

Df

Pr(>Chisq)

(Intercept)

913.7660083

1

<0.001

krate_kgha

0.9183707

2

0.632

nrate_kgha

4.5029460

2

0.105

krate_kgha:nrate_kgha

33.5440880

4

<0.001

  • What is significant here (say at \(\alpha = 0.05\))?
  • ANOVA table only provides numerator df! Denominator df is important to know as well to check that proper errors are being used to test different effects.

The split-plot trade-off

Compared to a RCBD where each of the 3 N x 3 K combinations would have been randomly assigned to same-sized EUs:

  • The whole-plot treatment factor has power decreased
  • The split-plot treatment factor has power increased
  • Overall, split-plot designs have relatively lower power than their same-sized EU counterpart

Summary

  • Split-plot designs are used when different treatment factors require a different size/amount of experimental material

  • Split-plots have one treatment factor allocated to the whole-plot experimental units (larger), and another treatment factor allocated to the split-plot experimental units (smaller)

Summary

  • Due to randomization happening at two different levels, that creates 2 different experimental errors, one for each experimental unit size. Need to ensure that proper model is specified so proper errors are used.

  • Although split-plot treatment factor has increased power, overall split-plot designs have less power compared to their one-EU-size counterpart.