STAT 503 Lesson 14 - Nested and Split Plot Designs

Flashcards covering Lesson 14: Nested, Split-Plot, Split-Split-Plot, and Strip-Plot designs, including model structures, degrees of freedom, and error term logic.

Nested design

A multifactor design where the levels of one factor (e.g., Factor B) occur only within particular levels of another factor (e.g., Factor A).

Split-plot design

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

Experimental unit

The smallest unit that can independently receive a particular treatment; essential to identify for each factor in split-plot designs.

Nested notation $B(A)$

Notation read as "B nested in A," signifying that Factor B levels do not have the same identity across different levels of Factor A.

Two-stage nested model equation

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

Effect $\beta_{j(i)}$

The effect of the $j$-th level of factor B nested within the $i$-th level of factor A.

Reason for no $A \times B$ interaction in nested designs

Because every level of B does not occur with every level of A, a conventional crossed interaction cannot be defined.

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

The Mean Square for B nested in A: $MS_{B(A)}$.

ANOVA degrees of freedom ($df$) for factor A

$$a - 1$$

ANOVA degrees of freedom ($df$) for $B(A)$

$$a(b - 1)$$

ANOVA degrees of freedom ($df$) for Error (two-stage nested)

$$ab(n - 1)$$

ANOVA degrees of freedom ($df$) for Total (two-stage nested)

$$abn - 1$$

Supplier/Batch experimental unit

Batch is the experimental unit for the supplier factor.

Impact of more measurements ($n$) per batch

Improves precision for estimating a batch's response but does not increase independent experimental units for the supplier.

Minimum replication for nested variance estimation

At least two measurements per batch and two batches per supplier.

General m-stage nested design

A hierarchical design containing multiple completely nested factors.

Three-stage nested model equation

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

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

$$ab(c - 1)$$

Hard-to-change factor

A factor whose levels are inconvenient or costly to change frequently, usually assigned to the whole-plot level in split-plot designs.

Whole plot

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

Split plot / subplot

The smaller experimental unit within a whole plot receiving the easier-to-change treatment factor.

Confounding of whole-plot treatment

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

Traditional randomization-based approach

Constructing separate whole-plot and subplot error terms based on randomization restrictions, rather than relying solely on EMS.

Whole-plot error in blocked split-plot

$\text{Blocks} \times \text{Whole-Plot Treatment}$ (e.g., $Blocks \times Method$).

Subplot error construction

Pooling the interactions involving blocks that belong to the subplot randomization level.

Paper example whole-plot error $df$

$$4$$

Paper example subplot error $df$

$$18$$

Method F-calculation (Paper example)

$$F = \frac{MS_{\text{Method}}}{MS_{\text{WP Error}}}$$

Temperature F-calculation (Paper example)

$$F = \frac{MS_{\text{Temp}}}{MS_{\text{SP Error}}}$$

Split-split-plot design

An extension of split-plots with three sizes of experimental units: whole plot, split plot, and split-split plot.

Antibiotic example stages

Technician (Whole Plot), Dosage (Split Plot), and Wall Thickness (Split-Split Plot).

Strip-plot design (split-block design)

A design where two treatment factors are applied to large strips in perpendicular directions.

Strip-plot experimental unit for $A \times B$

The subplot formed at the intersection of an A strip and a B strip.

Strip-plot $df$ for $A$ error

$$(r - 1)(a - 1)$$

Strip-plot $df$ for $B$ error

$$(r - 1)(b - 1)$$

Strip-plot $df$ for $AB$ error

$$(r - 1)(a - 1)(b - 1)$$

Core exam rule for split-plot error

Never use one common residual error; match each effect to the error from its specific randomization level.