Split Plot Experiments
Norberto E. Milla, Jr.
2023-11-29
Basic ideas
a two-factor experiment wherein levels of one of the factors require large plot size for execution and also show large differences in their effects
the experiment will consist of a set of large plots called main plots in which levels for the main plot factor are assigned
each main plot is divided into subplots to which the second factor, called the subplot factor, are assigned
the precision for measuring the effects of the main-plot factor is sacrificed to improve that of the subplot factor
the more important factor of which a higher precision is desired must be used as the sub-plot factor
involves two-stage randomization process
in the first stage the levels of factor A are randomized over the main plots and in the second stage the levels of factor B are randomized over the subplots within each main plot
this randomization scheme results in two distinct error terms, one appropriate for the main plots and another for the subplots
the error term for the main plots is expected to be larger than the subplot error which implies that a higher precision is expected in the subplots than in the main plots
Uses of split-plot experiments
When the level of one factor requires larger experimental units than the other factor
When greater precision is desired for comparisons among levels of one factor than that desired for another factor
When it is known that larger variation can be expected among the levels of one factor than those expected on levels of the other factor
When an additional factor is added to the experiment to increase the scope of the experiment
Example layout: Split Plot in RCBD
Linear model (CRD)
\[ Y_{ijk} = \mu + \alpha_i + \delta_{ij} + \beta_k + (\alpha \beta)_{ik} +\epsilon_{ijk} \] where:
\(Y_{ijk} =\) observation from subplot given the \(k^{th}\) level of B in the \(j^{th}\) whole plot given the \(i^{th}\) level of A
\(\mu\) = overall mean
\(\alpha_i\) = effect of the \(i^{th}\) level of A
\(\delta_{ij}\) = random error associated with the \(j^{th}\) whole plot given the \(i^{th}\) level of A
\(\beta_k\) = effect of the \(k^{th}\) level of B
\((\alpha \beta)_{ik}\) = interaction effect between the \(i^{th}\) level of A and the \(k^{th}\) level of B
\(\epsilon_{ijk}\) = random error associated with the subplot given the \(k^{th}\) level of B in the \(j^{th}\) whole plot given the \(i^{th}\) level of A
Test of hypothesis
The test procedure for split plot experiments is similar to factorial experiments
- Test of interaction effect
- Test of main effects, if interaction effect is not significant; to be followed with post analysis if main effects are significant
- Post hoc analysis for simple effects (effect of A at each level of B, or vice versa), if interaction effect is significant
Test statistic:
- F ratio for the effect of the main plot factor (A) is \(F_A = \frac{MS_A}{MS_{A:Rep}}\)
Split plot experiment in CRD: an example
An experiment is conducted to evaluate effects of grass species (G1 and G2) and stocking density (20 and 24) on the daily gain of sheeps kept on a pasture. Six 1-hactare pastures were planted with the two species of grass. Each grass species is randomly assigned to three pastures (replicates). Then each pasture is split into two where different numbers of sheeps (20 and 24) is randomly assigned. At the end of the experiment the daily gains in weight were determined.
grassden <- read.csv("GrassStockingDensity.csv")
head(grassden)## Rep Grass Density weight
## 1 R1 G1 D2 310
## 2 R1 G1 D1 290
## 3 R1 G2 D1 310
## 4 R1 G2 D2 330
## 5 R2 G2 D2 400
## 6 R2 G2 D1 380spcrd.out <- aov(weight ~ Grass + Grass:Rep + Density + Grass:Density,
data = grassden)
anova(spcrd.out)## Analysis of Variance Table
##
## Response: weight
## Df Sum Sq Mean Sq F value Pr(>F)
## Grass 1 7500.0 7500.0 112.5 0.0004472 ***
## Density 1 1200.0 1200.0 18.0 0.0132356 *
## Grass:Rep 4 8266.7 2066.7 31.0 0.0028687 **
## Grass:Density 1 33.3 33.3 0.5 0.5185185
## Residuals 4 266.7 66.7
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1We need to correct the F ratio for Grass
The revised/corrected ANOVA table is given below.
| SoV | df | SS | MS | F | p |
|---|---|---|---|---|---|
| Grass (A) | 1 | 7500.0 | 7500.0 | 3.63 | 0.1295 |
| Error A | 4 | 8266.7 | 2066.7 | ||
| Density (B) | 1 | 1200.0 | 1200.0 | 18.0 | 0.0132 |
| A*B | 1 | 33.3 | 33.3 | 0.5 | 0.5185 |
| Residual or Error B | 4 | 266.7 | 66.7 |
- There is no interaction effect between grass species and stocking density.
- Gain in weight is not significantly affected by grass species (p > 0.05)
- Gain in weight is significantly affected by stocking density (p < 0.05)
HSD.test(spcrd.out,
trt = "Density",
group = TRUE,
console = TRUE)##
## Study: spcrd.out ~ "Density"
##
## HSD Test for weight
##
## Mean Square Error: 66.66667
##
## Density, means
##
## weight std r Min Max
## D1 333.3333 37.77124 6 290 380
## D2 353.3333 42.26898 6 310 410
##
## Alpha: 0.05 ; DF Error: 4
## Critical Value of Studentized Range: 3.926503
##
## Minimun Significant Difference: 13.08834
##
## Treatments with the same letter are not significantly different.
##
## weight groups
## D2 353.3333 a
## D1 333.3333 bLinear model (RCBD)
\[ Y_{ijk} = \mu + \alpha_i + \rho_j + \delta_{ij} + \beta_k + (\alpha \beta)_{ik} +\epsilon_{ijk} \] where:
\(Y_{ijk} =\) observation from subplot given the \(k^{th}\) level of B in the \(j^{th}\) whole plot given the \(i^{th}\) level of A
\(\mu\) = overall mean
\(\alpha_i\) = effect of the \(i^{th}\) level of A
\(\rho_j\) = effect of the \(j^{th}\) block
\(\delta_{ij}\) = random error associated with the \(i^{th}\) level of A in the \(j^{th}\) block
\(\beta_k\) = effect of the \(k^{th}\) level of B
\((\alpha \beta)_{ik}\) = interaction effect between the \(i^{th}\) level of A and the \(k^{th}\) level of B
\(\epsilon_{ijk}\) = random error associated with the subplot given the \(k^{th}\) level of B in the \(j^{th}\) whole plot given the \(i^{th}\) level of A
Test of hypothesis
- The test procedure for split plot experiments is similar to factorial experiment
- Test statistic for block effect: \(F_{Block} = \frac{MS_{Block}}{MS_{A:Block}}\)
- Test statistic for main plot effect: \(F_A = \frac{MS_A}{MS_{A:Block}}\)
Split plot experiment in RCBD: an example
An experiment was conducted in order to investigate four different treatments of pasture and two mineral supplements on milk yield. The experiment was designed as a split-plot, with pasture treatments (factor A) assigned to the main plots and mineral supplements (factor B) assigned to sub-plots. The experiment was replicated in three blocks.
- Let us load the data and convert the variables with numeric codes to factors.
supp <- read.csv("mineral_supplement.csv")
supp$Block <- factor(supp$block)
supp$Pasture <- factor(supp$Pasture)
supp$Mineral <- factor(supp$Mineral)sprbd.out <- aov(Milk.Yield ~ Block + Pasture + Block:Pasture + Mineral + Pasture:Mineral, data = supp)
anova(sprbd.out)## Analysis of Variance Table
##
## Response: Milk.Yield
## Df Sum Sq Mean Sq F value Pr(>F)
## Block 2 212.583 106.292 47.2407 3.713e-05 ***
## Pasture 3 71.167 23.722 10.5432 0.003742 **
## Mineral 1 8.167 8.167 3.6296 0.093224 .
## Block:Pasture 6 26.083 4.347 1.9321 0.190897
## Pasture:Mineral 3 5.833 1.944 0.8642 0.498130
## Residuals 8 18.000 2.250
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1- We need to recompute the F ratio for Block:
\[\begin{align} F_{Block} &= \frac{MS_{Block}}{MS_{A:Block}} \notag \\ &= \frac{106.292}{4.347} \notag \\ &= 24.45\notag \\ p-value &= 0.0013 \end{align}\]
- Likewise, we need to recompute the F ratio for Pasture:
\[\begin{align} F_{Pasture} &= \frac{MS_{Pasture}}{MS_{A:Block}} \notag \\ &= \frac{23.722}{4.347} \notag \\ &= 5.46\notag \\ p-value &= 0.0377 \end{align}\]
- The revised/corrected ANOVA table is given below.
| SoV | df | SS | MS | F | p |
|---|---|---|---|---|---|
| Block | 2 | 212.583 | 106.292 | 24.45 | 0.0013 |
| Pasture (A) | 3 | 71.167 | 23.722 | 5.46 | 0.0377 |
| Error A | 6 | 26.083 | 4.347 | ||
| Mineral (B) | 1 | 8.167 | 8.167 | 3.6296 | 0.0932 |
| A*B | 3 | 5.833 | 1.944 | 0.8642 | 0.4981 |
| Residual or Error B | 8 | 18.000 | 2.250 |
There is significant block effect (\(p<0.01\)) \(\Rightarrow\) blocking strategy is successful
There is no significant interaction effect of Pasture and Mineral (\(p>0.05\))
Mineral supplement has no significant effect on milk yield (\(p>0.05\))
Pasture has significant effect on milk yield (\(p<0.05\))
- We perform post hoc analysis on pasture means
#Modified HSD.test()
hsd.test(sprbd.out,
trt = "Pasture",
group = TRUE,
console = TRUE)##
## Study: sprbd.out ~ "Pasture"
##
## HSD Test for Milk.Yield
##
## Mean Square Error: 4.347
##
## Pasture, means
##
## Milk.Yield std r Min Max
## 1 29.66667 3.204164 6 25 34
## 2 30.66667 3.777124 6 26 37
## 3 30.00000 3.949684 6 24 33
## 4 34.00000 3.741657 6 29 38
##
## Alpha: 0.05 ; DF Error: 6
## Critical Value of Studentized Range: 4.895599
##
## Minimun Significant Difference: 4.167015
##
## Treatments with the same letter are not significantly different.
##
## Milk.Yield groups
## 4 34.00000 a
## 2 30.66667 ab
## 3 30.00000 ab
## 1 29.66667 b