8.5 Split-Plot Designs

  • Typically used in the case that you have two factors where one needs much larger units than the other.

Example:

A: 3 levels (large units)

B: 2 levels (small units)

  • A and B levels are randomized into 4 blocks.
  • But it differs from Randomized Block Designs. In each block, both have one of the 6 (3x2) treatment combinations. But Randomized Block Designs assign in each block randomly, while split-plot does not randomize this step.
  • Moreover, because A needs to be applied in large units, factor A is applied only once in each block while B can be applied multiple times.

Hence, we have our model

If A is our factor of interest

\[ Y_{ij} = \mu + \rho_i + \alpha_j + e_{ij} \]

where

  • i = replication (block or subject)
  • j = level of Factor A
  • \(\mu\) = overall mean
  • \(\rho_i\) = variation due to the i-th block
  • \(e_{ij} \sim N(0, \sigma^2_e)\) = whole plot error

If B is our factor of interest

\[ Y_{ijk} = \mu + \phi_{ij} + \beta_k + \epsilon_{ijk} \]

where

  • \(\phi_{ij}\) = variation due to the ij-th main plot
  • \(\beta_k\) = Factor B effect
  • \(\epsilon_{ijk} \sim N(0, \sigma^2_\epsilon)\) = subplot error
  • \(\phi_{ij} = \rho_i + \alpha_j + e_{ij}\)

Together, the split-plot model

\[ Y_{ijk} = \mu + \rho_i + \alpha_j + e_{ij} + \beta_k + (\alpha \beta)_{jk} + \epsilon_{ijk} \]

where

  • i = replicate (blocks or subjects)
  • j = level of factor A
  • k = level of factor B
  • \(\mu\) = overall mean
  • \(\rho_i\) = effect of the block
  • \(\alpha_j\) = main effect of factor A (fixed)
  • \(e_{ij} = (\rho \alpha)_{ij}\) = block by factor A interaction (the whole plot error, random)
  • \(\beta_k\) = main effect of factor B (fixed)
  • \((\alpha \beta)_{jk}\) = interaction between factors A and B (fixed)
  • \(\epsilon_{ijk}\) = subplot error (random)

We can approach sub-plot analysis based on

  • the ANOVA perspective

    • Whole plot comparisons

      • Compare factor A to the whole plot error (i.e., \(\alpha_j\) to \(e_{ij}\))

      • Compare the block to the whole plot error (i.e., \(\rho_i\) to \(e_{ij}\))

    • Sub-plot comparisons:

      • Compare factor B to the subplot error (\(\beta\) to \(\epsilon_{ijk}\))

      • Compare the AB interaction to the subplot error (\((\alpha \beta)_{jk}\) to \(\epsilon_{ijk}\))

  • the mixed model perspective

\[ \mathbf{Y = X \beta + Zb + \epsilon} \]

8.5.1 Application

8.5.1.1 Example 1

\[ y_{ijk} = \mu + i_i + v_j + (iv)_{ij} + f_k + \epsilon_{ijk} \]

where

  • \(y_{ijk}\) = observed yield
  • \(\mu\) = overall average yield
  • \(i_i\) = irrigation effect
  • \(v_j\) = variety effect
  • \((iv)_{ij}\) = irrigation by variety interaction
  • \(f_k\) = random field (block) effect
  • \(\epsilon_{ijk}\) = residual
  • because variety-field combination is only observed once, we can’t have the random interaction effects between variety and field
library(ggplot2)
data(irrigation, package = "faraway")
summary(irrigation)
##      field   irrigation variety     yield      
##  f1     :2   i1:4       v1:8    Min.   :34.80  
##  f2     :2   i2:4       v2:8    1st Qu.:37.60  
##  f3     :2   i3:4               Median :40.15  
##  f4     :2   i4:4               Mean   :40.23  
##  f5     :2                      3rd Qu.:42.73  
##  f6     :2                      Max.   :47.60  
##  (Other):4
head(irrigation, 4)
##   field irrigation variety yield
## 1    f1         i1      v1  35.4
## 2    f1         i1      v2  37.9
## 3    f2         i2      v1  36.7
## 4    f2         i2      v2  38.2
ggplot(irrigation,
       aes(
         x = field,
         y = yield,
         shape = irrigation,
         color = variety
       )) +
  geom_point(size = 3)

sp_model <- lmerTest::lmer(yield ~ irrigation * variety + (1 | field), irrigation)
summary(sp_model)
## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## Formula: yield ~ irrigation * variety + (1 | field)
##    Data: irrigation
## 
## REML criterion at convergence: 45.4
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -0.7448 -0.5509  0.0000  0.5509  0.7448 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  field    (Intercept) 16.200   4.025   
##  Residual              2.107   1.452   
## Number of obs: 16, groups:  field, 8
## 
## Fixed effects:
##                        Estimate Std. Error     df t value Pr(>|t|)    
## (Intercept)              38.500      3.026  4.487  12.725 0.000109 ***
## irrigationi2              1.200      4.279  4.487   0.280 0.791591    
## irrigationi3              0.700      4.279  4.487   0.164 0.877156    
## irrigationi4              3.500      4.279  4.487   0.818 0.454584    
## varietyv2                 0.600      1.452  4.000   0.413 0.700582    
## irrigationi2:varietyv2   -0.400      2.053  4.000  -0.195 0.855020    
## irrigationi3:varietyv2   -0.200      2.053  4.000  -0.097 0.927082    
## irrigationi4:varietyv2    1.200      2.053  4.000   0.584 0.590265    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) irrgt2 irrgt3 irrgt4 vrtyv2 irr2:2 irr3:2
## irrigation2 -0.707                                          
## irrigation3 -0.707  0.500                                   
## irrigation4 -0.707  0.500  0.500                            
## varietyv2   -0.240  0.170  0.170  0.170                     
## irrgtn2:vr2  0.170 -0.240 -0.120 -0.120 -0.707              
## irrgtn3:vr2  0.170 -0.120 -0.240 -0.120 -0.707  0.500       
## irrgtn4:vr2  0.170 -0.120 -0.120 -0.240 -0.707  0.500  0.500
anova(sp_model,ddf = c("Kenward-Roger"))
## Type III Analysis of Variance Table with Kenward-Roger's method
##                    Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
## irrigation         2.4545 0.81818     3     4  0.3882 0.7685
## variety            2.2500 2.25000     1     4  1.0676 0.3599
## irrigation:variety 1.5500 0.51667     3     4  0.2452 0.8612

Since p-value of the interaction term is insignificant, we consider fitting without it.

library(lme4)
## Loading required package: Matrix
sp_model_additive <- lmer(yield ~ irrigation + variety + (1 | field), irrigation)
anova(sp_model_additive,sp_model,ddf = "Kenward-Roger")
## refitting model(s) with ML (instead of REML)
## Data: irrigation
## Models:
## sp_model_additive: yield ~ irrigation + variety + (1 | field)
## sp_model: yield ~ irrigation * variety + (1 | field)
##                   npar    AIC    BIC  logLik deviance  Chisq Df Pr(>Chisq)
## sp_model_additive    7 83.959 89.368 -34.980   69.959                     
## sp_model            10 88.609 96.335 -34.305   68.609 1.3503  3     0.7172

Since p-value of Chi-square test is insignificant, we can’t reject the additive model is already sufficient. Looking at AIC and BIC, we can also see that we would prefer the additive model

Random Effect Examination

exactRLRT test

  • \(H_0\): Var(random effect) (i.e., \(\sigma^2\))= 0
  • \(H_a\): Var(random effect) (i.e., \(\sigma^2\)) > 0
sp_model <- lme4::lmer(yield ~ irrigation * variety + (1 | field), irrigation)
library(RLRsim)
exactRLRT(sp_model)
## 
##  simulated finite sample distribution of RLRT.
##  
##  (p-value based on 10000 simulated values)
## 
## data:  
## RLRT = 6.1118, p-value = 0.0088

Since the p-value is significant, we reject \(H_0\)