Nested and Split Plot Designs

Nested and Split-Plot Designs

These are multifactor experiments that address common economic and practical constraints encountered in experimentation with real systems.

Nested and split-plot designs frequently involve one or more random factors.

There are many variations of these designs.

Fertilizers can be applied to individual fields;Insecticides must be applied to an entire farm

from an airplane

Fertilizers can be applied to individual fields;Insecticides must be applied to an entire farm

from an airplane

Agricultural Field Trial

Investigate the yield of a new variety of crop Factors

• Insecticides• Fertilizers

Experimental Units• Farms• Fields within farms

Experimental Design ?Experimental Design ?

Agricultural Field Trial

Insecticides applied to farms One-factor ANOVA

Main effect: Insecticides MSE: Farm-to-farm

variability

Farms

Agricultural Field Trial

Fertilizers applied to fields

One-factor ANOVAMain Effect: FertilizersMSE: Field-to-field

variability

Fields

Agricultural Field Trial

Insecticides applied to farms, fertilizers to fields

Two sources of variability Insecticides subject to

farm-to-farm variability Fertilizers and

insecticides x fertilizers subject to field-to-field variability

Farms

Fields

Two-Stage Nested Design

Nested designthe levels of one factor (B) are similar to, but not identical to each other at different levels of another factor (A).

Consider a company that purchases material from three suppliers• The material comes in batches.• Is the purity of the material uniform?

Experimental design • Select four batches at random from each

supplier.• Make three purity determinations from each

batch.

Two-Stage Nested Design

Nested Design

A factor B is considered nested in another factor, A if the levels of factor B differ for different levels of factor A.

The levels of B are different for The levels of B are different for different levels of A.different levels of A.

Synonyms indicating nesting:• Depends on, different for, within, in,

each

Examples - Nested

Examples - Nested

Examples - Crossed5-1 The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. The yield data follow:

Pressure Temperature 200 215 230

150 90.4 90.7 90.2 90.2 90.6 90.4

160 90.1 90.5 89.9 90.3 90.6 90.1

170 90.5 90.8 90.4 90.7 90.9 90.1

Examples - Crossed

5-2 An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. She selects three feed rates and four depths of cut. She then conducts a factorial experiment and obtains the following data:

Depth of Cut (in) Feed Rate (in/min) 0.15 0.18 0.20 0.25

74 79 82 99 0.20 64 68 88 104

60 73 92 96 92 98 99 104

0.25 86 104 108 110 88 88 95 99 99 104 108 114

0.30 98 99 110 111 102 95 99 107

Examples - Nested

Two-Stage Nested DesignStatistical Model and ANOVA

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Test statistics depend on the type of Test statistics depend on the type of factors and the expected mean squares.factors and the expected mean squares.• Random.Random.• Fixed.Fixed.

Expected Mean SquaresExpected Mean Squares

Assume that fixtures and layouts are fixed, operators are random – gives a mixed model

(use restricted form).

Alternative Analysis

If the need detailed analysis is not available, start with multi-factor ANOVA and then combine sum of squares and degrees of freedom.

Applicable to experiments with only nested factors as well as experiments with crossed and nested factors.Sum of squares from interactions are combined with the sum of squares for a

nested factor – no interaction can be determined from the nested factor.

Alternative Analysis

Split-Plot Design

Two hierarchically nested factors,with additional crossed factors occurring within

levels of the nested factor

Two sizes of experimental units,one nested within the other, with crossed

factors applied to the smaller units

An Experiment Can Have Either of these Features

Split-Plot Design

Whole-Plot Experiment : Whole-Plot Factor = A

Level A1

Level A2

Level A2

Level A1

Split Plot DesignsAnalysis of Variance Table

Source dfWhole-Plot Analysis

Factor A a-1Whole-Plot Error a(r-1)

Split-Plot DesignSplit-Plot Design

Split-Plot Experiment : Split-Plot Factor = B

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Split Plot DesignsSplit Plot DesignsAnalysis of Variance TableAnalysis of Variance Table

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Split-Plot AnalysisFactor B b-1A x B (a-1)(b-1)Split-Plot Error a(b-1)(r-1)Total abr-1

Agricultural Field Trial

Agricultural Field TrialAgricultural Field Trial

Insecticide 2Insecticide 2

Insecticide 2

Insecticide 1 Insecticide 1

Insecticide 1

Agricultural Field TrialAgricultural Field Trial

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Insecticide 2Insecticide 2

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Agricultural Field TrialAgricultural Field Trial

Whole Plots = Farms

Split Plots = Fields

Large Experimental Units

Small Experimental Units

Agricultural Field TrialAgricultural Field Trial

Whole Plots = Farms

Split Plots = Fields

Large Experimental Units

Small Experimental Units

Whole-Plot Factor = InsecticideWhole-Plot Error = Whole-Plot Replicates

Split-Plot Factor = FertilizerSplit-Plot Error = Split-Plot Replicates

The Split-Plot Design

The split-plot is a multifactor experiment where it is not practical to completely randomize the order of the runs.

Example – paper manufacturing• Three pulp preparation methods.• Four different temperatures.• The experimenters want to use three

replicates.• How many batches of pulp are required?

The Split-Plot Design

Pulp preparation method is a hard-to-change factor.

Consider an alternate experimental design:• In replicate 1, select a pulp preparation

method, prepare a batch.• Divide the batch into four sections or samples,

and assign one of the temperature levels to each.

• Repeat for each pulp preparation method.• Conduct replicates 2 and 3 similarly.

The Split-Plot Design

Each replicate has been divided into three parts, called the whole plots.• Pulp preparation methods is the whole plot

treatment. Each whole plot has been divided into four

subplots or split-plots.• Temperature is the subplot treatment.

Generally, the hard-to-change factor is assigned to the whole plots.

This design requires 9 batches of pulp (assuming three replicates).

The Split-Plot Design

Tensile StrengthRep (Day) 1 Rep (Day) 2 Rep (Day) 3

Pulp Prep Method 1 2 3 1 2 3 1 2 3

Temperature200 30 34 29 28 31 31 31 35 32

225 35 41 26 32 36 30 37 40 34

250 37 38 33 40 42 32 41 39 39

275 36 42 36 41 40 40 40 44 45

The Split-Plot Design

There are two levels of randomization restriction.• Two levels of experimentation

Tensile StrengthRep (Day) 1 Rep (Day) 2 Rep (Day) 3

Pulp Prep Method 1 2 3 1 2 3 1 2 3

Temperature200 30 34 29 28 31 31 31 35 32

225 35 41 26 32 36 30 37 40 34

250 37 38 33 40 42 32 41 39 39

275 36 42 36 41 40 40 40 44 45

Experimental Units in Split Plot Designs

Possibilities for executing the example split plot Possibilities for executing the example split plot design.design.• Run separate replicates. Each pulp prep method (randomly Run separate replicates. Each pulp prep method (randomly

selected) is tested at four temperatures (randomly selected).selected) is tested at four temperatures (randomly selected). Large experimental unit is four pulp samples.Large experimental unit is four pulp samples. Smaller experimental unit is a an individual sample.Smaller experimental unit is a an individual sample.

• If temperature is hard to vary select a temperature at random If temperature is hard to vary select a temperature at random and then run (in random order) tests with the three pulp and then run (in random order) tests with the three pulp preparation methods.preparation methods.

Large experimental unit is three pulp samples.Large experimental unit is three pulp samples. Smaller experimental unit is a an individual sample.Smaller experimental unit is a an individual sample.Tensile Strength

Rep (Day) 1 Rep (Day) 2 Rep (Day) 3Pulp Prep Method 1 2 3 1 2 3 1 2 3

Temperature200 30 34 29 28 31 31 31 35 32

225 35 41 26 32 36 30 37 40 34

250 37 38 33 40 42 32 41 39 39

275 36 42 36 41 40 40 40 44 45

The Split-Plot Design

Another way to view a split-plot Another way to view a split-plot design is a RCBD with replication.design is a RCBD with replication.• Inferences on the blocking factor can be Inferences on the blocking factor can be

made with data from replications.made with data from replications.

The Split-Plot Design Model The Split-Plot Design Model and Statistical Analysisand Statistical Analysis

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There are two error structures; the whole-plot error and the subplot error