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 ?

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 ANOVA Main Effect: Fertilizers MSE: 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 design the 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 different levels of A. The levels of B are different for different levels of A. Synonyms indicating nesting: Depends on, different for, within, in, each

Examples - Nested

Examples - Crossed

Examples - Nested

Two-Stage Nested Design Statistical Model and ANOVA

Residual Analysis Calculation of residuals. Calculation of residuals.

m-Stage Nested Design

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

Expected 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 A 1 Level A 2 Level A 1

Split Plot Designs Analysis of Variance Table

Split-Plot Design Split-Plot Experiment : Split-Plot Factor = B Level A 1 Level A 2 Level A 1 B2B2 B1B1 B2B2 B1B1 B1B1 B1B1 B2B2 B2B2 B1B1 B2B2 B2B2 B1B1 B2B2 B1B1 B1B1 B2B2

Split Plot Designs Analysis of Variance Table

Agricultural Field Trial

Insecticide 2 Insecticide 1

Agricultural Field Trial Fert B Fert A Fert B Insecticide 2 Insecticide 1

Agricultural Field Trial Whole Plots = Farms Split Plots = Fields Large Experimental Units Small Experimental Units

Agricultural Field Trial Whole Plots = Farms Split Plots = Fields Large Experimental Units Small Experimental Units Whole-Plot Factor = Insecticide Whole-Plot Error = Whole-Plot Replicates Split-Plot Factor = Fertilizer Split-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

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

Experimental Units in Split Plot Designs Possibilities for executing the example split plot design. Possibilities for executing the example split plot design. Run separate replicates. Each pulp prep method (randomly selected) is tested at four temperatures (randomly selected).Run separate replicates. Each pulp prep method (randomly 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 and then run (in random order) tests with the three pulp preparation methods.If temperature is hard to vary select a temperature at random and then run (in random order) tests with the three pulp 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.

The Split-Plot Design Another way to view a split-plot design is a RCBD with replication. Another way to view a split-plot design is a RCBD with replication. Inferences on the blocking factor can be made with data from replications.Inferences on the blocking factor can be made with data from replications.

The Split-Plot Design Model and Statistical Analysis Sum of squares are computed as for a three factor factorial design without replication.

RCBD Model

The Split-Plot Design Model and Statistical Analysis There are two error structures; the whole-plot error and the subplot error