Math 558 Lecture #32
Split plot designs Introduction
The factorial designs are used to choose the factors and factor
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combinations that have significant effect on response. We take all factors
and their combinations at all the levels of interest for this purpose which
we can call treatments. These treatments are applied to the experiments
units either as a completely randomised design or as a blocked designs.
In some situations the levels of one factor are more difficult to change as
compared to the other. This difficulty can be related to the technical
issues or the financial resources.
In this situation, complete randomization of factor-level combi- nations to experimental units could make the experiment much more time consuming or perhaps impossible to conduct. This is the case frequently in process improvement studies where the process consists of several steps, and some factors in an exper- iment relate to one process step while others relate to a later process step.(Lawson, p. 307)
Split plot designs Example
Also when we have some hard to vary factors and some easy to vary factors, the response from the experimental units can get confounded1 with the treatment effects. The designs used to address this situation are called split plot designs. Split plot designs are considered a generalization of the factorial designs. In this lecture we will mainly focus on the terminology and the randomization process in the split-plot designs.
1A confounding variable is a variable whose presence affects the variables being studied so that the results do not reflect the actual relationship
Experiment: A researcher wants to study four irrigation methods and two fertilizer types. There are 24 fields available for the study. The researchers finds irrigation methods hard to change as compared to the fertilizer levels. So he decides to group the fields in four groups and applies a different irrigation method to each group. Within each group the fertilizers are applied to individual plots. the design layout is given in the next slides.
Example2: Consider a paper manufacturing experiment in which the manufacturer wants to compare three different pulp mixtures and four different cooking temperatures. The difference in mixtures is due to the amount of hard wood used. The response of interest is the tensile strength of the paper. The manufacturer wants to run three replications of the experiment. Each replicate of a factorial experiment consists of 12 observations.
The complete experiment will require a total of 36 runs. One of the options is a completely randomized design on 36 units. The second can be taking each replicate a block (should we block on the pulp mixtures?) However, the experimenter decides to conduct the experiment as follows.
1. A batch of pulp mixture is produced by one of the three methods the manufacturer wants to compare. Pulp mixture can be considered factor A. Three different pulp mixtures (or preparation methods) are three levels of this factor.
2. The next step is to divide batch into four parts , and cook it at one of the four temperatures. The temperature is the second factor that can be denoted by B. Factor B has four levels. This means that we are treating level 1 of factor A at all levels of factor B.
3 This process is repeated for other two levels of factor A. Hence the manufacturer achieves all 36 runs through this process.
This design is a split plot design.
This planning is different from a factorial experiment planning. In a factorial experiment each response is obtained by taking a combination of factors A and B from the set of all the possible combinations (12 in total). For that the experimenter has to prepare each combination individually, like
1. method 1, temperature 1
2. method 2 temperature 1
and so on. This process will not only require 36 batches of paper pulp.
In the manufacturer’s design which is a slpit plot design nine batches are produced using each pulp producing method (three batches per method). Each batch is then divided into 4 parts to receive each temperature level. The nine batches are called nine whole plots, and the preparation methods are called the whole plot or main treatments.
The parts of the batches called subplots (or split-plots). Temperature is called the subplot treatment. Note that if there are undesigned factors present and if they vary as the pulp preparation methods are changed, then any effect of the undesigned factors will be confounded with the methods. However, the subplot treatments will not be confounded with the whole plot treatments. Therefore it is advisable to use the levels of the factor we are most interested in as subplot treatments.
Also notice that this experiment is conducted in two steps. In step 1 the batches are produced which are whole plots(factor A is applied). In step 2 the material is divided into subplots and cooked (factor B is applied).
Linear Model
The linear model for the split-plot design is
yijk = μ+τi +βj +(τβ)ij +γk +(τγ)ik + (βγ)jk + (τβγ)ijk + εijk
i = 1, 2, ….r
j = 1, 2……t k = 1, 2…….b
Linear Model
τi corresponds to the replicates
βj corresponds to factor A
(τβ)ij corresponds to replicates × factor A interactions γk corresponds to treatment (factor B)
(τγ)ik corresponds to replicates × factor A interaction (βγ)jk corresponds to factor A × factor B interaction (τβγ)ijk corresponds to replicates ×AB
prep rep 1 rep 2 rep 3
method 123123123
Temperature oF 200
30 34 29 35 41 26 37 38 33 36 42 36
28 31 31 32 36 30 40 42 32 41 40 40
31 35 32 37 40 34 41 39 39 40 44 45
Statistical Analysis
Note that factor A effects will be estimated using the whole plots and factor B and the A*B interaction effects will be estimated using the split plots.The difference in the size of the whole plot and split plots will effect the precision of the effect estimates. Therefore, in the statistical analysis of split-plot designs, we must take into account the presence of two different sizes of experimental units.
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