Day 18 Analysis and inference for a split-plot design

July 7th, 2025

18.1 Announcements

  • Homework 3 is due this Friday (July 11).
  • Homework 2 grades will be posted this evening.
  • Semester project:
    • An example is posted.
    • Wednesday July 23: send project for peer review.
    • Schedule for somewhere between July 21-August 1 (anywhere between 8am-5pm): 15 min presentation + 15 min Q&A.
    • Submit final version of your report by August 1. Include ANOVA, stat model, mock R code, discussion of strengths and weaknesses of the experiment design.

18.2 Review of our experiment

  • Research question: can we include more banana than the original recipe? Will the optimum temperature change depending on that recipe?
  • Treatment structure: 3 \(\times\) 2 factorial, with 3 levels for temperature (250, 400, 500) and 2 levels for recipe (Control, Extra Banana)
  • Design structure: split-plot in an CRD, with temperature in the whole plot.
Muffin experiment

Figure 18.1: Muffin experiment

With that treatment structure, the statistical model will always begin with \(y_{ijk} = \mu + T_i + R_j +(TR)_{ij}\), followed by the random effects. The random effects will depend on the design structure.

18.3 ANOVA tables

18.3.1 Split-plot in a CRD

Table 18.1: ANOVA without subsampling
Table 18.1: Design or Topographical Sources of Variability
Source df
-
Error(oven) t*r-1 = 9-1 = 8
-
-
Error(oven) (b-1)* t * r = (2-1) * 3 * 3 = 9
Total N-1 = 17
Table 18.1: Treatment Sources of Variability
Source df
Temperature t-1 = 3-1 = 2
-
Banana b-1 = 2-1 = 1
TxB (t-1)*(b-1) = 2
Parallels N-tb = 18 - (3*2) = 12
Total N-1 = 17
Table 18.1: Combined Table of the Sources of Variability
Source df
Temperature t-1 = 3-1 = 2
Error(oven) t*r-1 - (t-1)= 8 - 2 = 6
Banana b-1 = 2-1 = 1
TxB (t-1)*(b-1) = 2
Error(oven x day) (b-1)* t * r - (b-1) - (t-1)*(b-1) = 9 - 1 -2 = 6
Total N-1 = 17

18.3.2 Other possible designs

18.3.2.1 Split-plot in an RCBD

Table 18.2: ANOVA without subsampling
Table 18.2: Design or Topographical Sources of Variability
Source df
Day (block) r-1 = 3-1 = 2
-
Error(oven) (t-1)r = (3-1)3 = 6
-
-
Error(oven x day) (b-1)* t * r = (2-1) * 3 * 3 = 9
Total N-1 = 17
Table 18.2: Treatment Sources of Variability
Source df
-
Temperature t-1 = 3-1 = 2
-
Banana b-1 = 2-1 = 1
TxB (t-1)*(b-1) = 2
Parallels N-tb = 18 - (3*2) = 12
Total N-1 = 17
Table 18.2: Combined Table of the Sources of Variability
Source df
Day r-1 = 3-1 = 2
Temperature t-1 = 3-1 = 2
Error(oven) (t-1)*r - (t-1)= 6 -2 = 4
Banana b-1 = 2-1 = 1
TxB (t-1)*(b-1) = 2
Error(oven x day) (b-1)* t * r - (b-1) - (t-1)*(b-1) = 9 - 1 -2 = 6
Total N-1 = 17

18.3.2.2 RCBD

Table 18.3: ANOVA with subsampling
Table 18.3: Design or Topographical Sources of Variability
Source df
Day (block) r-1 = 3-1 = 2
-
Error(oven) (t-1)r = (3-1)3 = 6
-
-
Error(oven x day) (b-1)* t * r = (2-1) * 3 * 3 = 9
Error(recipe x oven x day) (m-1) * b * t * r = (3-1) * 2 * 3 * 3 = 36
Total N-1 = 53
Table 18.3: Treatment Sources of Variability
Source df
-
Temperature t-1 = 3-1 = 2
-
Banana b-1 = 2-1 = 1
TxB (t-1)*(b-1) = 2
-
Parallels N-tb = 54 - (3*2) = 48
Total N-1 = 17
Table 18.3: Combined Table of the Sources of Variability
Source df
Day r-1 = 3-1 = 2
Temperature t-1 = 3-1 = 2
Error(oven) (t-1)*r - (t-1)= 6 -2 = 4
Banana b-1 = 2-1 = 1
TxB (t-1)*(b-1) = 2
Error(oven x day) (b-1)* t * r - (b-1) - (t-1)*(b-1) = 9 - 1 -2 = 6
Error(recipe x oven x day) (m-1) * b * t * r - 0 = 36
Total N-1 = 53

18.4 Applied analysis in R

Get code from Thursday here.