Split Plot Designs, Exercises of Design

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Lukas Meier, Seminar für Statistik

Split Plot Designs

 A split plot design is a special case of a factorial treatment structure.  It is used when some factors are harder (or more expensive) to vary than others.  Basically a split plot design consists of two experiments with different experimental units of different “size”.  E.g., in agronomic field trials certain factors require “large” experimental units, whereas other factors can be easily applied to “smaller” plots of land.  Let us have a look at an example…

What is a Split Plot Design? (Oehlert, 2000, Chapter 16.1)

 Randomly assign each irrigation level to 2 of the plots (the so called whole plots or main plots ).  In every of the plots, randomly assign the 4 different corn varieties to the so called split plots.  Two independent randomizations are being performed!  We also call irrigation level the whole-plot factor and corn variety the split-plot factor.

Example I: Irrigation and Corn Variety

4 2 3 1 4 1 3 1 4 2 3 3 1 2 2 4 2 4 1 3 2 1 4 3

 Whole plots (plots of land) are the experimental units for the whole-plot factor (irrigation level).  Split plots (subplots of land) are the experimental units for the split-plot factor.  In the split-plot “world”, whole plots act as blocks.  Basically, we are performing two different experiments in one :  each experiment has its own randomization  each experiment has its own idea of experimental unit

Example I: Irrigation and Corn Variety

 Two piano types (baby grand / concert grand) from each of 4 manufacturers.  40 music students are divided at random into 8 groups (“panels”) of 5 students each.  Two panels are assigned at random to each manufacturer (= 2 panels per manufacturer).  Each panel goes to the concert hall and hears (blindfolded) the sound of both pianos (in random order).  Response: Average rating of the 5 students in the panel (hence, student is “only” measurement unit here).

Example II: Pianos (Oehlert, 2000)

 The whole plots are the 8 panels.  The whole-plot factor is the manufacturer.  The split plots are the two sessions.  The split-plot factor is the piano type (baby vs. concert grand).

Example II: Pianos

1 2 3 4 5 6 7 8 Panel Session 1 Session 2 A B A C D (^) B D C Baby grand Concert grand Manufacturer

 Dataset oats from R-package MASS.  As stated in the help file: The yield of oats from a split-plot field trial using three varieties and four levels of manurial treatment. The experiment was laid out in 6 blocks of 3 main plots, each split into 4 sub-plots. The varieties were applied to the main plots and the manurial treatments to the sub-plots.  Overview of data:  6 different blocks (B)  3 different varieties (V)  4 different nitrogen treatments (N)  Response (Y): Yields (in ¼ lbs per sub-plot, each of area 1 80 acre).  Let us first have a graphical overview of the experimental design.

Example III: Oats

Example III: Oats

I 4 2 3 1 4 1 3 2 1 3 2 4 II 2 1 3 4 1 2 4 3 1 4 2 3 III 3 2 1 4 3 2 4 1 2 3 4 1 IV 1 2 4 3 1 3 2 4 3 2 1 4 V 3 2 4 1 4 1 2 3 3 4 1 2 VI 2 1 4 3 3 4 2 1 4 2 (^13)

 We have an RCB for the whole-plot factor.  The experimental unit on the whole-plot level is given by the combination of block and variety.  We therefore use the model 𝑌𝑖𝑗𝑘 = 𝜇 + 𝛼𝑖 + 𝛾𝑘 + 𝜂𝑖𝑘 + 𝛽𝑗 + 𝛼𝛽 (^) 𝑖𝑗 + 𝜀𝑖𝑗𝑘

Example III: Oats

fixed effect of variety fixed effect of block split-plot error 𝑁 0 , 𝜎𝜂^2 𝑁 0 , 𝜎^2 yield (^) (fixed ) interaction between variety and nitrogen treatment whole-plot error fixed effect of nitrogen treatment

Example III: Oats

 In R we use the lmer function with an extra random effect (error) per combination of block and variety.  We get the following output  Observe that the test for variety uses 2 and 10 degrees of freedom.  Why? Let us a have a closer look at the potential ANOVA table on the whole-plot level.

 This also reveals a problem: We don’t have too many error df’s left to test the whole-plot factor (only 10).  In contrast, we test everything involving the split-plot factor against the residual error , which has 45 df’s.  Remember:  Hence, all effects involving the whole-plot factor are estimated less precisely and tests are less powerful.

Example III: Oats

 Split-plot designs can also arise in (much) more complicated designs.  There can be more than one whole-plot factor. E.g., think of a two-way factorial on the whole-plot level.  In addition, there can be more than one factor on the split- plot level.  To get the correct model we “only” have to follow “the path of randomization”.  For every “level” (whole-plot / split-plot) of the experiment we have to introduce a corresponding random effect (better terminology: error ) which acts as the experimental error on that level.

General Situation

 Experiment studies the effect of  nitrogen (4 levels of nitrogen)  weed (3 levels)  clipping treatments (2 levels: clipping / no clipping) on plant growth in wetlands.  Experiment was performed as follows:  8 trays , whereof each holds three artificial wetlands (rectangular wire baskets)  4 of the trays were placed on a table near the door of the greenhouse  4 of the trays on a table in the center of the greenhouse  On each table , we randomly assign one of the trays to each of the 4 nitrogen treatments.  Within each tray , we randomly assign the 3 weed treatments.  In addition, each wetland is split in half. One half is chosen at random and will be clipped, the other half is not clipped.  After 8 weeks: measure fraction of biomass that is nonweed.

Example IV: Weed Biomass in Wetlands (Oehlert, 2000, Ex. 16.7)

Experimental layout

Example IV: Weed Biomass in Wetlands

Center Door Nitrogen 1 Nitrogen 3 Nitrogen 2 Nitrogen 4 Nitrogen 3 Nitrogen 4 Nitrogen 2 Nitrogen 1 Greenhouse