Recall the Randomized Complete Block Design (RCBD) we discussed in lesson 7. In RCBD, general blocks are formed such that the experimental units are expected to be homogenous within a block and heterogeneous between blocks. 

For example, suppose we are studying the effect of irrigation amount (\(I_1\) and \(I_2\)) and fertilizer type (\(A\) and \(B\)) on crop yield. We have 4 treatments in this experiment. Suppose we want to have at least 2 replicates and two large lands that can be used for the experiment. In RCBD, we can split each land into 4 fields and can apply the 4 treatments randomly to each field. Here lands are blocks and fields are the experimental units. 

In this example, we have assumed that managing levels of irrigation and fertilizer require the same effort. Now suppose varying the level of irrigation is difficult on a small scale and it makes more sense to apply irrigation levels to larger areas of land. 

In such situations, we can divide each land into two large fields (whole plots) and apply irrigation amounts to each field randomly. And then divide each of these large fields into smaller fields (subplots) and apply fertilizer randomly within the whole plots. 

In this strategy, each land contains two whole plots and the irrigation amount is assigned to each whole plot randomly using RCBD (i.e. lands are treated as blocks, and the irrigation amount is assigned randomly within each block to the whole plots). Each whole plot contains two subplots and fertilizer type is assigned to each subplot using RCBD (i.e. whole plots are treated as blocks and fertilizer type is assigned randomly within each whole plot to the subplots).

When some factors are more difficult to vary than others at the level of experimental units, it is more efficient to assign the difficult-to-change factors to larger units (whole plots) and then apply the easier-to-change factor to smaller units (subplots). This is known as the split-plot design. 

The data (Dye Time Data) were:

It is important to notice that in a split-plot design, randomization is a two-stage process. Levels of one factor (say, factor A) are randomized over the whole plots within each block, and the levels of the other factor (say, factor B) are randomized over the subplots within each whole plot. This restriction in randomization results in two different error terms: one appropriate for comparisons at the whole plot level and one appropriate for comparisons at the subplot level. 

The appropriate error for whole plot level in split-plot RCBD is \(\text{whole plot factor} \times \text{block}\) interaction. In other words, the analysis at the whole plot level is essentially a one-way ANOVA with blocking (i.e. one observation per block-treatment combination). From the perspective of the whole plot, the subplots are simply subsamples and it is reasonable to average them when testing the whole plot effects (i.e. factor A effects).

The subplot factor (i.e. factor B) is always compared within the whole plot factor. 

The statistical model associated with the split-plot design with whole plots arranged as RCBD is 

\(Y_{ijk} = \mu + \alpha_i + \gamma_k + (\alpha\gamma)_{ik} + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}\)

where \(\gamma_k\) for \(k=1,...,r\) are block effects, \(\alpha_i\) for \(i=1,...,a\) are factor A effects, and \(\beta_j\) for \(j=1,...,b\) are factor B effects.

Therefore, the ANOVA table for the example above would look like this:

Recall the irrigation amount and fertilizer type example we discussed in the previous section. We had two large lands and managing the irrigation amount was harder on a smaller scale, so we assigned the irrigation amount within each land to whole plots using an RCBD.

Now suppose in this case, instead of two large lands, we had 4 large fields. Irrigation amount is still a factor that is difficult to control. In that case, we can assign the irrigation amount randomly using a CRD for the 4 whole plots (fields). Then each whole plot can be divided into smaller fields (subplots) and we can assign fertilizer type randomly within each whole plot. 

In split-plot designs, the subplots are always arranged in an RCBD within the whole plot. The difference between split-plot in RCBD and split-plot in CRD is how the whole plot factor is randomized. 

In the first step, the levels of the irrigation treatment are applied to four experimental (fields) to end up with 2 replications:

Following that, the fields are split into two subplots and a level of Factor B is randomly applied to subplots within each application of the Irrigation treatment:

In this design, the whole plot treatments (i.e. factor A, irrigation) are arranged in a CRD and the subplot treatments (i.e. factor B, seed type) are arranged within whole plots in an RCBD.

If we carefully think about this, we see that the replicates (i.e. fields) are nested within the whole factor levels. For example, fields 2 and 3 are nested within level \(A_1\), and fields 1 and 4 are nested within level \(A_2\). So the variability due to replicates is nested within the whole factor. 

As discussed in section 8.1, from the perspective of whole plots, the subplots are simply subsamples and it is reasonable to average them when testing the whole plot effects (i.e. Factor A). If the values of the subplots within each whole plot are averaged, the resulting design is CRD. The error term in a simple CRD is \(\text{replication}(\text{whole factor})\). Therefore, for split-plot in CRD, the whole plot errors are computationally equivalent to \(\text{replication}(\text{whole factor})\), but in order to use it, we must explicitly extract it from the error term and put it in the model.  

Again, the subplot factor (i.e. factor B) is always compared within the whole plot factor. 

The statistical model for the design is:

\(Y_{ijk} = \mu + \alpha_i + \gamma_{k(i)} + \beta_j + (\alpha\beta)_{ij} + \epsilon_{ijk}\)

where \(i=1, 2, \dots, a\), \(j=1, 2, \dots, b\) and \(k=1,2,\dots, r\), where \(a\) is the number of levels in factor A, \(b\) is the number of levels in factor B and \(r\) is the number of replicates within each level of factor A (e.g. \(r=2\) in the example above).

Therefore, the ANOVA table for the example above would look like this:

The idea of split-plots can easily be extended to multiple splits. In a 3-factor factorial, for example, it is possible to assign Factor A to whole plots, then Factor B to subplots within the applications of Factor A, and then split the experimental units used for Factor B into sub-subplots to receive the levels of Factor C. 

For a fixed effect factorial treatment design in an RCBD (with blocks for the levels of Factor A, levels of Factor B, and levels of Factor C), the split-split-plot would produce the following table. (Hover over the error rows to see where the source or df comes from):

The model is specified as we did earlier for the split-plot in RCBD, retaining only the interactions involving replication where they form denominators for F-tests for factor effects. For the model above, we would need to include the block, block × A, and block × A × B terms in the random statement in SAS. In SAS, Block × A × B would automatically include the Block × B effect SS and df.  All other interactions involving replications and factor C would be included in the residual error term. The block × A term is often referred to as ‘Error A’ (‘Whole plot error’ in the table), the Block × A × B term as ‘Error B’ (‘Subplot error’ in the table), and the residual error as ‘Error C’ (‘Sub-subplot error’ in the table) because of their roles as the denominator in the F-tests.

1. An experimenter wants to compare the yield of three varieties of oats at four different levels of manure. Suppose 6 farmers agree to participate in the experiment and each farmer will designate 3 fields from their farms for the experiment.

   1. What is the treatment design?
   2. What is the randomization design?

   Solution

   1. Treatment design: 3X4 factorial with oat variety and manure levels as factors having 3 and 4 levels respectively.
   2. Randomization design: Three oats varieties will be randomly assigned to the 3 fields from each farm using RCBD with farms as blocks. Four manure levels are then randomized within each field using an RCBD. So the randomization design is a split-plot in RCBD.
2. In an agricultural setting, an experimenter is applying one of two irrigation methods randomly to 6 plots where all plots are similar in moisture, soil type, slope, fertility, etc. Each plot is then subdivided into 5 portions and 5 levels of nitrogen fertilizer are applied randomly to these portions.
   1. What is the treatment design?
   2. What is the randomization design?

   Solution

   1. Treatment design: 2X5 factorial with irrigation method and fertilizer levels as factors having 2 and 5 levels, respectively.
   2. Randomization design: Split-plot in CRD with the whole factor as irrigation method and subplot factor as fertilizer level.
3. A survey was conducted to study whether the financial aid package amounts at a certain university differ between male and female athletes. The university offers 3 different types of aid packages: tuition reduction for 4 years, tuition reduction for the first year, or partial coverage of room and board. Within each package type, the package amount for a randomly selected male and female athlete was recorded.
   1. What is the treatment design?
   2. What is the randomization design?

   Solution

   1. Treatment design: A single factor study with 2 levels - the factor of interest is gender.
   2. Randomization design: RCBD with package type as the blocking factor.

In this lesson, we discussed split-plot designs with the special feature of having two types of experimental units: whole plots into which the whole plot treatments are assigned and the subplots into which the subplot treatments are assigned.

The whole plot assignment can be either according to a CRD or an RCBD, and depending on this design type, the overall design is either called a split-plot in CRD or RCBD. Note that in either case, the denominator of the F-statistic for testing the whole plot factor is not the MSE, but equals the MS of \(\text{replicate}(\text{A})\) and MS of \(\text{block} \times \text{A}\), respectively.