Day 13 More practice: Hierarchical (Multilevel) Designs

June 26th, 2025

13.1 Review: Hierarchical Designs

  • Remember the definition of experimental unit? The smallest unit to which a treatment is independently applied.
  • Sometimes we find that there are different sizes of experimental units.
  • In such cases, it is important to identify the different experimental units and the randomization scheme. We may be in front of a multilevel design.
Schematic description of a field experiment with a split-plot design

Figure 13.1: Schematic description of a field experiment with a split-plot design

Schematic description of a swine experiment with a split-plot design

Figure 13.2: Schematic description of a swine experiment with a split-plot design

  • Sometimes, these differences in the sizes of EUs are not that easy to notice.
  • More details in Analysis of Messy Data - Ch5.

13.1.1 Remember our example:

Rows and beds (aka columns) probably looked somewhat like this:

library(tidyverse)
library(agridat)
library(ggpubr)

data("durban.splitplot")
df <- durban.splitplot

theme_set(theme_minimal())

p_blocks <- 
  df %>% 
  ggplot(aes(bed, row))+
  geom_tile(aes(fill = block))+
  geom_tile(color = "black", fill=NA)+
  coord_fixed()

p_wholeplot <-
  df %>% 
  ggplot(aes(bed, row))+
  geom_tile(aes(fill = fung))+
  geom_tile(color = "black", fill=NA)+
  coord_fixed()

p_splitplot <- 
df %>% 
  ggplot(aes(bed, row))+
  geom_tile(aes(fill = gen), show.legend= F)+
  geom_tile(color = "black", fill=NA)+
  coord_fixed()

ggarrange(p_blocks, p_wholeplot, p_splitplot, ncol = 1, nrow = 3)

13.2 Building the ANOVA skeleton using design (aka topographical) and treatment elements

Table 13.1: Constructing the ANOVA skeleton
Table 13.1: Experiment or Topographical
Source df
Block b-1
-
Fungicide(Block) (f-1)*b
-
-
Gen(Fung x Block) (g-1)fb
Total N-1
Table 13.1: Treatment
Source df
- -
Fungicide f-1
-
Genotype g-1
Fung x Gen (f-1)(g-1)
Parallels N-(f*g)
Total N-1
Table 13.1: Combined Table
Source df
Block b-1
Fungicide t-1
Fungicide(Block) (f-1)*b - (t-1)
Genotype g-1
Fung x Gen (f-1)(g-1)
Pens(Block x Trt) error (g-1)* f * b - (g-1 + (f-1)(g-1))
Total N-1

13.4 Tomorrow

  • Zoom classes – we will use the link for the office hours