Detailed 503 Lesson 14: Nested and Split Plot Designs

Practice flashcards for STAT 503 Lesson 14: Nested and Split Plot Designs, covering definitions, formulas, and examples.

Nested design

A multifactor design in which the levels of one factor occur only within particular levels of another factor.

Split-plot design

A multifactor design with restricted randomization and different sizes of experimental units for different treatment factors.

Historical origin of split plots

Agricultural experiments in which fields were divided into whole plots and then into smaller subplots receiving additional treatments.

Experimental unit

The smallest unit that can independently receive a particular treatment.

Lesson 14 Objective 1

Understand nesting of factors within other factors.

Lesson 14 Objective 2

Understand two-stage nested designs with fixed and/or random factors.

Lesson 14 Objective 3

Understand split-plot designs, especially when one factor is hard to change.

Lesson 14 Objective 4

Understand the two main approaches for analyzing split-plot designs and the basis of each.

Lesson 14 Objective 5

Understand split-split-plot designs as an extension of split plots.

Lesson 14 Objective 6

Understand strip-plot/split-block designs and how they differ from ordinary split plots.

B nested in A

The levels of B do not have the same identity or meaning under different levels of A.

Nested notation B(A)

Read as 'B nested in A.'

School/teacher nested example

If A = school and B = teacher, teacher 1 at one school is a different person from teacher 1 at another school; teacher is nested within school.

Crossed school/teacher requirement

For school and teacher to be crossed, the same teachers would need to appear at every school.

Supplier/batch example

If material is purchased from several suppliers and several batches are sampled from each supplier, batches are nested within supplier because the batches from one supplier are not the same as those from another.

Two-stage nested model formula

$$y_{ijk} = \mu + \tau_i + \beta_{j(i)} + \epsilon_{k(ij)}$$

$$\beta_{j(i)}$$

Effect of the j-th level of B nested within the i-th level of A.

$$\epsilon_{k(ij)}$$

Error/replicate k nested within the A-B(A) treatment structure.

Why there is no A x B interaction in nested designs

Not every B level occurs with every A level, so a conventional crossed A x B interaction is not defined.

Random B(A) as replication for A

When B is random and nested within A, the B(A) units act as the replicates for factor A.

Correct error for testing A when B(A) is random

$MS_{B(A)}$, whether A itself is fixed or random.

ANOVA df for A (two-stage nested)

$$a - 1$$

ANOVA df for B(A) (two-stage nested)

$$a(b - 1)$$

ANOVA df for Error (two-stage nested)

$$ab(n - 1)$$

ANOVA total df (two-stage nested)

$$abn - 1$$

Supplier experimental unit

Batch is the experimental unit for supplier.

Effect of more measurements per batch

More measurements improve precision for estimating a batch's response but do not create additional independent experimental units for supplier.

Appropriate supplier variability

Variability among batches within suppliers is the appropriate error variation for comparing suppliers when batches are random.

Supplier question

Is material purity the same across suppliers?

Supplier factor status

Suppliers are fixed, batches are random samples nested within suppliers, and observations are random.

Supplier experiment design particulars

Randomly select four batches from each of three suppliers and make three purity determinations on each batch.

Important factors in supplier selection

Both the supplier mean purity and the variability among batches.

Choosing b versus n factors

The number of batches and number of measurements per batch depend on the relative costs and relative variance components.

Minimum replication for measurement variability

At least two measurements per batch are needed to estimate measurement variability.

Minimum replication for batch variability

At least two batches per supplier are needed to estimate batch variability.

Minitab GLM behavior

Uses expected mean squares to determine the appropriate error term for F tests.

Supplier F-test denominator

Variation among batches is used as the error for testing supplier because Batch is random and nested in Supplier.

Supplier F-test degrees of freedom

Uses an F distribution with 2 numerator and 9 denominator degrees of freedom.

Variance components estimation (A random)

The ANOVA method can estimate variance due to A, B(A), and residual variance by equating observed MS to EMS.

Supplier result (Example 14.1)

No significant difference in purity among suppliers; $p = 0.416$.

Batch-within-supplier result (Example 14.1)

Significant variation in purity among batches within suppliers; $p = 0.017$.

Practical implication of supplier consistency

Even if average supplier purity is similar, consistency among batches may still be an important quality issue.

Important residual plot (Nested analysis)

Residuals versus supplier are important because equal variability among suppliers is an ANOVA assumption.

Measurement-error assumption

Measurement-error variability is assumed not to depend on batch quality or mean.

Batch-variance assumption

The variability among batches is assumed to be the same for all suppliers.

Why equal batch-variance assumption matters

A supplier could be preferable specifically because it has lower batch-to-batch variability.

Incorrect crossed-analysis supplier row (Example 14.1)

SS = 15.06, df = 2, MS = 7.53, F = 1.02, p = 0.42.

Incorrect crossed-analysis batch row (Example 14.1)

SS = 25.64, df = 3, MS = 8.55, F = 3.24, p = 0.04.

Incorrect crossed-analysis interaction row (Example 14.1)

SS = 44.28, df = 6, MS = 7.38, F = 2.80, p = 0.03.

Incorrect crossed-analysis error (Example 14.1)

SS = 63.33, df = 24, MS = 2.64.

Why crossed analysis is inappropriate for supplier/batch

Batch main effects and Supplier x Batch interaction are not meaningful because batches are not the same across suppliers.

Relationship between wrong and correct SS

Batch SS plus Supplier x Batch SS and their degrees of freedom combine to form the correct Batch(Supplier) source.

General m-Stage Nested Design

A hierarchy with multiple completely nested factors.

Three-stage alloy example factors

Two alloy formulations, three heats within each formulation, and two ingots within each heat.

Three-stage model structure formula

$$y_{ijkl} = \mu + \tau_{i} + \beta_{j(i)} + \gamma_{k(ij)} + \epsilon_{l(ijk)}$$

Factor A (Alloy example)

Alloy formulation.

Factor B(A) (Alloy example)

Heat nested within alloy formulation.

Factor C(B(A)) (Alloy example)

Ingot nested within heat and alloy formulation.

Error (Alloy example)

The usual NID measurement error for repeated hardness measurements within ingot.

Three-stage df for A

$$a - 1$$

Three-stage df for B(A)

$$a(b - 1)$$

Three-stage df for C(B(A))

$$ab(c - 1)$$

Three-stage error df

$$abc(n - 1)$$

Three-stage total df

$$abcn - 1$$

Course-note correction for textbook SS formulas

The A means and B means should be subtracted, respectively, rather than the overall mean.

Testing in general nested designs

Expected mean squares are used to determine the appropriate F-test denominators.

Split-Plot Design trigger

Complete randomization may be impractical because one or more treatment factors are difficult, expensive, or cumbersome to change.

Hard-to-change factor

A factor whose level is inconvenient or costly to change frequently; commonly assigned at the whole-plot level.

Paper example factors

A = pulp preparation method; B = cooking temperature.

Paper example response

Paper tensile strength.

Paper example replication structure

Three replicates/days, each containing 12 runs from 3 methods x 4 temperatures.

Why preparation method is whole-plot factor

Preparing a pulp blend is cumbersome, so one batch is made and divided for temperature treatments.

Whole plot

The larger experimental unit receiving the hard-to-change whole-plot treatment.

Split plot/subplot

A smaller experimental unit within a whole plot that receives the easier-to-change subplot treatment.

Paper whole plots

Within each replicate/day, the three preparation-method units are the whole plots.

Paper split plots

Each preparation-method whole plot is divided into four samples, one for each temperature.

Confounding of whole-plot treatment

The whole-plot treatment is confounded with whole plots, leading to less precision than the subplot treatment.

Factor of greatest scientific interest in split-plots

Preferably assigned to split plots because split-plot comparisons are generally more precise.

Split-plot experimental-unit sizes

Whole plots are used for factor A; split plots are used for factor B and A x B.

Precision consequence in split-plots

Because whole plots and split plots are different sizes/error structures, treatment effects have different precisions.

Book/EMS analysis approach

Use expected mean squares from the model to construct F tests.

Disadvantage of EMS-only approach

Does not directly account for the randomization restrictions that generated the split-plot structure.

Traditional randomization-based approach

Construct separate whole-plot and subplot error terms based on randomization restrictions.

Course emphasis for split-plot analysis

Emphasis is on the traditional/randomization-restriction approach as it is more widely accepted.

Whole-plot effects (traditional analysis)

Factor A is estimated and tested at the whole-plot level.

Subplot effects (traditional analysis)

Factor B and A x B are estimated and tested at the subplot level.

Whole-plot section as RCBD

With blocks, the whole-plot portion can be viewed as an RCBD with Method as the treatment factor.

Whole-plot section without blocks

Without blocks, the whole-plot portion could be viewed as a completely randomized design.

Whole-plot error

Blocks x Method.

Why Blocks x Method is error

In the RCBD interpretation, treatment-by-block interaction is part of experimental error.

Subplot error construction

Pool interactions involving Block that belong to the subplot randomization level.

Paper-example subplot error pool

Block x Temp + Block x Method x Temp.

Paper-example subplot error df

$2 \times 3 + 2 \times 2 \times 3 = 18$ df.

Paper-example block result

DF = 2, SS = 77.556, MS = 38.78, F = 4.28, p = 0.1014.

Paper-example Method result

DF = 2, SS = 128.389, MS = 64.19, F = 7.08, p = 0.0485.

Paper-example whole-plot error (MS)

Blocks x Methods: DF = 4, SS = 36.278, MS = 9.07.

Paper-example Temperature result

DF = 3, SS = 434.083, MS = 144.69, F = 36.43, p = 0.0000.

Paper-example Method x Temperature result

DF = 6, SS = 75.167, MS = 12.53, F = 3.15, p = 0.0272.

Paper-example subplot error (MS)

DF = 18, SS = 71.5, MS = 3.97.

Paper-example total SS and DF

DF = 35, SS = 822.973.