Penn State STAT 503 Lesson 7: Confounding and Blocking in $$2^k$$ Factorial Designs

A comprehensive set of vocabulary flashcards covering the principles of blocking, deliberate confounding, and specialized experimental designs based on Penn State STAT 503 Lesson 7.

Blocking

A design strategy used to remove or account for extraneous sources of variation by grouping similar experimental units.

Confounding

A situation in which two effects are inseparable from the data because they are represented by the same comparison.

Completely randomized replicated $2^k$ design

A design where $n$ replications per treatment combination exist and all $n \times 2^k$ experimental units are randomly assigned to the $2^k$ treatment combinations.

Statistical cost of blocking

The loss of block degrees of freedom from the error term.

Statistical benefit of blocking

If blocks explain substantial nuisance variation, the $MSE$ decreases and tests of treatment effects gain power.

Yates treatment set for $2^2$

The set of treatments denoted as (1), a, b, and ab.

Rule for choosing an effect to confound

Choose an effect of least scientific interest, which is usually the highest-order interaction.

Confounded effect in $2^2$ example

The $AB$ interaction, which is the highest-order interaction in a $2^2$ design.

Confounded effect in $2^3$ design with 2 blocks

The $ABC$ interaction is confounded with blocks.

Number of defining effects for four blocks

$p = 2$ selected effects are required to define the four combinations.

Generalized interaction

The product of the selected defining effect columns that is automatically confounded along with the defining effects.

Cancellation rule

In effect algebra, a repeated factor cancels because its coded column squared is $+1$ (e.g., $A \times A = I$).

Defining effects

The $p$ effects deliberately selected to construct blocks in a $2^k$ design.

Block degrees of freedom ($df$)

For $b$ blocks, the degrees of freedom is calculated as $b - 1$.

Number of confounded effects in $2^p$ blocks

The number of confounded effects is $2^p - 1$.

Effect hierarchy assumption

The assumption that higher-order interactions are generally less important than lower-order interactions and main effects.

Blocks nested within replicates

A structure in replicated confounded designs where each replicate contains its own set of blocks, denoted as $Block(Rep)$.

Pseudo-factor in Minitab

A factor created (e.g., $ABC$) by multiplying the coded levels of individual factors (e.g., $A$, $B$, and $C$) using Minitab's Calculator before fitting a model.

Split-plot connection

A $2^k$ factorial design in which a main effect is confounded with blocks.

Whole plots

The large fields or blocks receiving the levels of the whole-plot factor (the factor confounded with blocks).

Subplots

The smaller experimental units within each whole plot that receive combinations of the remaining factors.

Block size formula

The number of treatment combinations per block calculated as $2^{k-p}$.

Confounded set in Attempt 3 (Example 7.1)

$ABC$, $BCD$, and $AD$; preferred because it confounds two three-way interactions and only one two-way interaction.

Mod 2 arithmetic

Arithmetic where values are reduced to their remainder after division by 2 (e.g., $1 + 1 = 0$).

0/1 Method for block assignment

An algebraic method using 0 for low levels and 1 for high levels to assign treatments to blocks using linear combinations reduced modulo 2.

Partial confounding

A strategy using different interaction effects as the block-defining effect in different replicates to recover information on every interaction.

Relative information

The fraction of full information retained for an effect under partial confounding, expressed as the ratio of replicates where the effect is unconfounded ($q$) to total replicates ($r$).