Day 16 Review: Applied split-plot

July 2nd, 2025

16.1 Announcements

  • Planning to miss >2 classes in July? survey
  • Watch last week’s classes (especially days 3+4)
  • Homework 2 due today.
  • Homework 3 is posted and due next Friday (July 11).

16.2 Background

Yesterday, we designed an experiment with an RCBD to figure out the best temperature to bake the muffins. Check out the original recipe here.

Yesterday we agreed on the following model:

\[y_{ik} = \mu + T_i + b_k + \varepsilon_{ik},\]

\[b_k \sim N(0, \sigma_b^2), \\ \varepsilon_{ik} \sim N(0, \sigma_{\varepsilon}^2),\]

and design:

Table 16.1:
Table 16.1: Design Sources of Variability
Source df
Day (block) r-1 = 3-1 = 2
-
Error(oven) (t-1)r = (3-1)*3 = 6
Total N-1 = 8
Table 16.1: Treatment Sources of Variability
Source df
-
Temperature t-1 = 3-1 = 2
Parallels N-t = 9-3 = 6
Total N-1 = 8
Table 16.1: 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) = 2*3 - 2 = 4
Total N-1 = 8

Dry ingredients:

  • 1 1/2 cups all-purpose flour (7 1/2 ounces)
  • 1 1/2 teaspoons baking powder
  • 1/4 teaspoon baking soda
  • 1/2 teaspoon salt
  • 1/4 teaspoon ground cinnamon
  • 1/4 teaspoon ground nutmeg
  • 2/3 cup finely chopped pecans, toasted

Wet ingredients:

  • 1/2 cup light brown sugar (3 3/4 ounces)
  • 1/4 cup granulated sugar (2 ounces)
  • 2 large eggs
  • 1 stick unsalted butter, browned and cooled (4 ounces)
  • 1 1/2 cups mashed bananas, from very ripe bananas that have been peeled and mashed well with a fork (12 3/4 ounces or 361 gr.)
  • 2 tablespoons full-fat sour cream
  • 1 teaspoon pure vanilla extract

16.2.1 Research question

What is the best temperature to bake the muffins?

  • 300 °F
  • 400 °F
  • 500 °F

How much banana?

  • 1 1/2 cups (12.75 oz, or 361 gr.)
  • 2 cups (17 oz, or 482 gr.)

16.3 What is the best design?

16.3.1 Option A: another RCBD

Corresponding model:

\[y_{ik} = \mu + T_i + B_j + (TB)_{ij} + b_k + \varepsilon_{ik},\]

\[b_k \sim N(0, \sigma_b^2), \\ \varepsilon_{ik} \sim N(0, \sigma_{\varepsilon}^2).\]

Table 16.2:
Table 16.2: Design Sources of Variability
Source df
Day (block) r-1 = 3-1 = 2
-
-
-
Error(oven) (tb-1)r = (6-1)3 = 15
Total N-1 = 17
Table 16.2: 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 16.2: Combined Table of the Sources of Variability
Source df
Day r-1 = 3-1 = 2
Temperature t-1 = 3-1 = 2
Banana b-1 = 2-1 = 1
TxB (t-1)*(b-1) = 2
Error(oven) (tb-1)r - (t-1) - (b-1) - (t-1)(b-1) = 15 - 2 -1 -2 = 10
Total N-1 = 17

16.3.2 Option B: split-plot design

In the model:

  • Cooking days are blocks.
  • Oven are “mini-blocks” for banana recipe.

Statistical model:

\[y_{ik} = \mu + T_i + B_j + (TB)_{ij} + b_k + w_{i(k)} + \varepsilon_{ik},\]

\[b_k \sim N(0, \sigma_b^2), \\ w_{i(k)} \sim N(0, \sigma^2_w) , \\ \varepsilon_{ik} \sim N(0, \sigma_{\varepsilon}^2).\]

ANOVA table for this split-plot design

Table 16.3:
Table 16.3: 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
Total N-1 = 17
Table 16.3: 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 = 18 - (3*2) = 12
Total N-1 = 17
Table 16.3: 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
Total N-1 = 17

16.4 Treatment means and confidence intervals for the split-plot design

The treatment mean for the \(i\)th temperature and \(j\)th banana level is \(\mu_{ij} = \mu + T_i + B_j +(TB)_{ij}\). That mean won’t change under different design structures. What may change is the confidence interval around the mean difference.

  • First, recall the formula for a CI: \(\theta \pm t_{df, \frac{\alpha}{2}} \cdot se(\hat{\theta})\)
#get test t
qt(p = .975, df = 4) # df are df error(oven)
## [1] 2.776445
  • For example, the CI for the differences between means for 300F and 400F \(\mu_{1 \cdot} - \mu_{2 \cdot}\) is \((\mu_{1 \cdot} - \mu_{2 \cdot}) \pm 2.78 \cdot se(\widehat{\mu_{1 \cdot} - \mu_{2 \cdot}})\)
  • \(se(\widehat{\mu_{1 \cdot} - \mu_{2 \cdot}}) = \sqrt{\frac{2 (\sigma^2_{\varepsilon} + b \cdot \sigma^2_w)}{b \cdot r}}\)

\[\mu_i \pm 2.78 \cdot \sqrt{\frac{2 (\sigma^2_{\varepsilon} + b \cdot \sigma^2_w)}{b \cdot r}}\]

#get test t
qt(p = .975, df = 6) # df are df error(oven)
## [1] 2.446912
  • The CI for the differences between means for normal and high banana \(\mu_{\cdot 1} - \mu_{\cdot 2}\) is \((\mu_{\cdot 1} - \mu_{\cdot 2}) \pm 2.44 \cdot se(\widehat{\mu_{\cdot 1} - \mu_{\cdot 2}})\)
  • \(se(\widehat{\mu_{\cdot 1} - \mu_{\cdot 2}}) = \sqrt{\frac{2 \sigma^2_{\varepsilon}}{t \cdot r}}\)

\[\mu_i \pm 2.44 \cdot \sqrt{\frac{2 \sigma^2_{\varepsilon}}{r}}\]

16.5 Tomorrow:

  • Measure heights of the muffins and analyze the data!