Split-plot designs are experimental designs that include at least one hard-to-change factor that is difficult to completely randomize due to time or cost constraints. By making the creation of split-plot experiment designs simple, Minitab makes the benefits of this powerful statistical technique accessible to everyone. Changing this factor is difficult because the enormous ovens the company uses take several hours to attain a stable temperature.

In an ideal world, we would randomize all factors in any experiment. But randomizing all factors is simply not always practical. Changing a variable for full randomization may be too costly, or take too much time, or may not be feasible. Faced with these challenges, people use split-plot designs because they save time and money while collecting the same amount of data.

Consider the example of the bakery. With three variables temperature, flour, and sugareach with two levels a low and a high settingeight combinations of factors are possible. And if the bakery wants to do two replications of the experiment to get better estimates, then 16 cakes need to be baked in total.

The oven can fit four cakes at a time, but randomizing the temperature setting will restrict baking to as few as one cake at a time. If each cake needs to bake for an hour, the experiment could take up to 16 hours, plus the time required for the oven to change temperature.

A completely randomized factorial design requires oven temperature to be adjusted frequently. If we instead use a split-plot design, the flour and sugar amounts can still be varied for each cake with minimal adjustments to the hard-to-change oven temperature. Using a split-plot design, multiple cakes can be baked at the same time, minimizing changes to oven temperature.

In a split-plot experiment, you hold levels of the hard-to-change factor constant for several experimental runs, which are collectively treated as a whole plot. You vary the easy-to-change factors over these runs, each combination of which makes a subplot within the whole plot. Using a split plot design, the bakery can fix the oven temperature at degrees F, and bake four cakes that use the four possible combinations of flour and sugar simultaneously.

Then the bakery changes the oven toand bakes four cakes using all four cake mix combinations again. Two replications of this experiment would take only four hours, plus the time to change the oven temperature—a tremendous savings in time and energy. To evaluate the hard-to-change factor, think of each whole plot of the easy-to-change factor as a set of repeated measures for the given level of your hard-to-change factor. The average of these repeated measures produces one observation for the hard-to-change factor.

The hard-to-change factor is randomized across the whole plots, where each whole plot represents one experimental unit, or one opportunity to collect data at the given level of the hard-to-change factor. Split-plot designs were originally used in agriculture. You can turn to split-plot designs when randomizing a variable would make an experiment cost too much, take too much time, or cause too much difficulty. Temperature is a common hard-to-change variable because heating and cooling often require significant time and expense.

Other common hard-to-change variables include pressure settings and machinery that requires disassembly or recalibration to change settings.These designs were originally developed for agriculture by R.

Fisher and F. Due to their applicability outside agriculture they could also be called split-unit designs. The results from a split-plot experiment are shown in the table below Box, Hunter, and Hunter The positions of the coated steel bars in the furnace were randomized within each heat. In run 1 the heat was and the first position in the furnace had a steel bar with coating 2 the second position had coating 3the third position had coating 1, and the fourth position had coating 4.

But, because the furnace heat was hard to change the heats were run in the systematic order shown. The primary interest was to compare coatings and how they interact with temperature. How does the split-plot design compare with, say, a 3x4 factorial design of coating and temperature? In the factorial design an oven temperature-coating combination would be randomly selected then we would obtain a corrosion resistance measure.

Then randomly select another oven temperature-coating combination and obtain another corrosion resistance measure until we have a resistance measure for all 12 oven temperature-coating combinations. To run each combination in random order would require adjusting the furnace temperature up to 24 times since there were two replicates and would have resulted in a much larger variance. The split plot is like a randomized block design with whole plots as blocks in which the opportunity is taken to introduce additional factors between blocks.

In this design there is only one source of error influencing the resistance. There are two different variances associated with the whole plots and subplots. It would be misleading to treat as if only one error source and one variance. Achieving and maintaining a given temperature in this furnace was very imprecise.

The whole plot variance, measuring variation from one heat to another, was expected to be large. The subplot variance measuring variation from position to position, within a given heat, was expected to be small.

The subplot effects and subplot-main plot interaction are estimated using with the same subplot error. Two considerations important in choosing an experimental design are feasibility and efficiency. In industrial experimentation a split-plot design is often convenient and the only practical possibility.

## Stats for Randomized Complete Block and Split-Plot Designs

This is the case whenever there are certain factors that are difficult to change and others that are easy to change. In this example changing the furnace temperature was difficult to change ; rearranging the positions of the coated bars in the furnace was easy to change.

The numerical calculations for the ANOVA of a split-plot design are the same as for other balanced designs designs where all treatment combinations have the same number of observations and can be performed in R or with other statistical software. Experimenters sometimes have difficulty identifying appropriate error terms.

The whole plot effects are replication and replication:heats. The whole plot mean square error is This measures the differences between the replicated heats at the three different temperatures. The sum of squares for the subplot error is the sum of interaction between replicate and coating replication:coating and the three way interaction of replication, heats and coating replication:heats:coating.

The subplot error measures to what extent the coatings give dissimilar results within each of the replicated temperatures. The whole plot effects are under the heading Error: heats:replication and the subplot effects are under the heading Error: Within.

The subplot effects of coating and the interaction of temperature and coating can be tested by forming F statistics using the subplot mean square error. There are statistically significant differences between coatings and the interaction between temperature and coating.

Estimates of the whole plot and sub plot standard deviations are. The estimated standard deviation of furnace heats is approximately three times as large as the standard deviation for coatings. Would you reach the same conclusions?Thank you for visiting nature. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

A Nature Research Journal. We have already seen that varying two factors simultaneously provides an effective experimental design for exploring the main average effects and interactions of the factors 1. However, in practice, some factors may be more difficult to vary than others at the level of experimental units.

Factorial Designs

For example, drugs given orally are difficult to administer to individual tissues, but observations on different tissues may be done by biopsy or autopsy. When the factors can be nested, it is more efficient to apply a difficult-to-change factor to the units at the top of the hierarchy and then apply the easier-to-change factor to a nested unit. This is called a split plot design. It is instructive to review completely randomized design CRD and randomized complete block design RCBD 2 and show how these relate to split plot design. Suppose we are studying the effect of irrigation amount and fertilizer type on crop yield. We have access to eight fields, which can be treated independently and without proximity effects Fig. Each field is composed of two whole plots, each composed of two subplots. If our land is divided into two large fields that may differ in some way, we can use the field as a blocking factor Fig.

Each combination of irrigation and fertilizer is balanced within the large field. So far, we have not considered whether managing levels of irrigation and fertilizer require the same effort. If varying irrigation on a small scale is difficult, it makes more sense to irrigate larger areas of land than in Figure 1a and then vary the fertilizer accordingly to maintain a balanced design. If our land is divided into four fields whole plotseach of which can be split into two subplots Fig.

Within a whole plot, fertilizer would be distributed across subplots using RCBD, randomly and balanced within whole plots with a given irrigation level.Example One of the most common mixed models is the split-plot design. The split-plot design involves two experimental factors, A and B.

Levels of A are randomly assigned to whole plots main plotsand levels of B are randomly assigned to split plots subplots within each whole plot.

The design provides more precise information about B than about Aand it often arises when A can be applied only to large experimental units.

## DOE: Handling Hard-to-Change Factors with Split-Plot Designs in Minitab

An example is where A represents irrigation levels for large plots of land and B represents different crop varieties planted in each large plot. Consider the following data from Stroup awhich arise from a balanced split-plot design with the whole plots arranged in a randomized complete-block design. The variable A is the whole-plot factor, and the variable B is the subplot factor. The R matrix is also diagonal and contains the residual variance. The SAS code produces Output REML is used to estimate the variance components, and the residual variances are profiled out of the optimization.

You can check this table to make sure that the data are correct. The REML estimates are the values that maximize the likelihood of a set of linearly independent error contrasts, and they provide a correction for the downward bias found in the usual maximum likelihood estimates. The minimization method is the Newton-Raphson algorithm, which uses the first and second derivatives of the objective function to iteratively find its minimum.

The "Iteration History" table records the steps of that optimization process. For this example, only one iteration is required to obtain the estimates. The Evaluations column reveals that the restricted likelihood is evaluated once for each of the iterations.

A criterion of 0 indicates that the Newton-Raphson algorithm has converged. Akaike's and Schwarz's criteria can be used to compare different models; the ones with larger values are preferred. Consider the following examples: estimate 'a1 mean narrow' intercept 1 A 1 B. Chapter Contents. The Mixed Procedure. Model Information.It is often inconvenient, costly, or even impossible to perform a factorial design in a completely randomized fashion.

An alternative to a completely randomized design is a split-plot design. The use of split-plot designs started in agricultural experimentation, where experiments were carried out on different plots of land.

Classical agricultural split-plot experimental designs were full factorial designs but run in a specific format. The key feature of split-plot designs is that levels of one or more factors are assigned to entire plots of land referred to as whole plots or main plots, whereas levels of other factors are assigned to parts of these whole or main plots.

These parts are called subplots or split-plots. Split-plot designs thus have two types of experimental units, whole plots Show page numbers Download PDF.

Search form icon-arrow-top icon-arrow-top. Page Site Advanced 7 of Edited by: Neil J. Buy in print. Looks like you do not have access to this content. Entries Per Page:. Methods Map Research Methods. Explore the Methods Map. Related Content.

Back to Top. Find content related to this author.Sometimes multi-factor experiments use multiple different experimental units for the various factors in the experiment. To visualize this, think of applying multiple treatments in a sequence.

The levels of the first factor are applied to experimental units using some form of randomization. Following that, the levels of second Factor are applied to sub-units within the application of the first factor. In other words, the experimental unit used for the application of the first factor has been split, forming the experimental units for the application of the levels of the second treatment.

Split plot designs are extremely common, and typically result from logistical restrictions, practicality, or efficiency. Sometimes split plots are difficult to recognize, and it emphasizes the absolute necessity of determining what the experimental unit s are in setting up an ANOVA. Split plots can be extended to accommodate multiple splits.

For example, it is not uncommon to see a split-split-plot experimental design being used. In this case, there are three experimental units involved and three stages of randomization of treatment levels. The split plot design is often employed in a randomized complete block design, where one factor is applied to whole plots forming a complete block, and then the second factor is applied to sub-plots within the whole plots within each block.

As an example adapted from Hicks,consider an experiment where an electrical component is subjected to different temperatures for different amounts of time.

### design.split

However, it was impractical to run individual treatment combinations, and instead they opted to set the oven a temperature and then take out randomly selected components at different times. This process was repeated for each temperature until a replication was complete. The data Bake Time Data were:. If repeat observations are made within the split plots, then a separate error term can be estimated.

However, it is important to keep in mind that tests of replication effects are not of interest, but are being isolated in the ANOVA to reduce the error variance. This has the effect of combining these interactions with the true error variance, for a working error term. Notice that in Minitab all effects have to appear in the Model box, and that we are not specifying all the random effects in the Random factors box.

Unlike SAS, where we do specify all the random effects in the random statement, in Minitab we only specify the simple factor name in the Random factors box.

See Textbook Section Split plot designs came out of agricultural field experiments and our text uses an example of an agricultural experiment to illustrate the principles of split plot design. Two factors are of interest, Irrigation Factor A at 2 levels and Fertilizer Factor B at 2 levels and they are crossed to form a factorial treatment design. The fertilizer treatment can be easily applied at a small scale, but the irrigation treatment is problematic.

Irrigating one plot may influence neighboring plots, and furthermore, the irrigation equipment is most efficiently used in a large area. 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 sub-plots within each application of the Irrigation treatment:. This design can now be viewed as 2 replications of the CRD with the split-plot sequence of randomization of the factors.

The consequence of this design is seen in the error terms used in constructing F tests for the two factors. The important thing to see here is that the test for the treatment applied to whole plots requires a different error term than the other factors.

The random effect we need to serve as the denominator for the F test for Factor A as a split plot in a CRD is the Experimental unit Whole plot factorand in this case that is the field Factor A.SARE's mission is to advance—to the whole of American agriculture—innovations that improve profitability, stewardship and quality of life by investing in groundbreaking research and education.

SARE's vision is For on-farm research projects comparing three or more treatments, a more complex analysis is required than the t-test.

You could potentially compare your treatments two at a time using the t-test. For example, in an experiment with three treatments, you could calculate the LSD to compare treatment one and treatment two, two and three, and one and three. Note, however, that you normally would not calculate all of the possible comparisons because doing so will increase your chance of coming to a wrong conclusion. There is a statistical correction that needs to be made in this case. Similarly, in a split-plot experiment, a simple t-test can provide an LSD for comparing main treatments.

But this can be quite cumbersome to do by hand, so we recommend using statistical software see Resources. When discussing your project with your cooperating researcher or Extension agent, make sure to ask them about getting assistance with statistical analysis. See Figure 8 for an example of how the results of three treatments might relate to one another in terms of LSD.

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