Detailed STAT 503 Lesson 9 - 3-Level and Mixed-Level Factorials

Vocabulary practice flashcards for STAT 503 Lesson 9 covering 3-level designs, blocking, confounding, and mixed-level factorials.

3-level factorial design

A factorial design in which each factor has three levels, usually low, middle, and high.

Primary role of 3-level designs

Move beyond screening toward understanding the shape of the response function.

Typical factor type in 3-level designs

Factors are generally quantitative because the design is often used to study curvature and response surfaces.

2-level versus 3-level response information

Two levels can support a straight-line effect; three levels include a middle point and can reveal curvature or quadratic behavior.

$3^k$ design

A full factorial with $k$ factors, each at three levels, containing $3^k$ treatment combinations.

Growth of $3^k$ designs

The number of runs increases very quickly; for example, four factors require $3^4 = 81$ treatment combinations.

Preferred 3-level coding

Levels are commonly coded $0$, $1$, and $2$.

Alternative 3-level coding

Low, middle, and high may also be represented as $-$, $0$, and $+$.

Design region with two factors

The treatment combinations fill a square region at three values of each axis.

Design region with three factors

The design region becomes a cube.

Design region with four or more factors

The design region is a hypercube that cannot be drawn easily.

Lesson objective: factorial structure

Apply 3-level factorial designs and understand interaction components and their degrees of freedom.

Lesson objective: blocking

Block 3-level designs efficiently and choose effects to confound with blocks.

Lesson objective: partial confounding

Explain partial confounding in replicated blocked designs and why it is useful.

Lesson objective: fractional factorials

Construct reasonable 3-level fractions and interpret their alias structure.

Lesson objective: classical designs

Recognize Latin squares and Graeco-Latin squares as special cases of 3-level fractional factorial designs.

Lesson objective: mixed levels

Describe mixed-level factorial designs and their applications.

Main-effect df in a 3-level factor

$3 - 1 = 2$ df.

Two-way interaction df (3-level)

$(3 - 1) \times (3 - 1) = 4$ df.

Three-way interaction df (3-level)

$(3 - 1)^3 = 8$ df.

General p-way interaction df in a 3-level design

$2^p$ df.

Two-factor $3 \times 3$ design treatment df

$A$ has $2$ df, $B$ has $2$ df, and $A \times B$ has $4$ df, totaling $8$ treatment df.

Two-factor $3 \times 3$ design with $n$ replicates

There are $9n$ observations and $9n - 1$ total df.

Error df for replicated $3 \times 3$ factorial

$9(n - 1)$.

Reason 3-level confounding is complicated

An interaction has multiple degrees of freedom and must be partitioned into orthogonal components before selected pieces can be confounded.

Two-way interaction partition

A $4$-df two-way interaction can be partitioned into two orthogonal $2$-df pseudo-interaction components.

Three-way interaction partition

An $8$-df three-way interaction can be partitioned into four orthogonal $2$-df pseudo-interaction components.

Pseudo-factor

A three-level constructed factor representing one $2$-df component of a larger interaction.

Purpose of pseudo-factors

They allow a $2$-df main effect or interaction component to be confounded with a three-level block effect.

Polynomial partition note

Interaction sums of squares can also be split into $1$-df polynomial pieces, though mainly treated as polynomial regression.

Mod 3 arithmetic

Arithmetic in which values are reduced to remainders $0$, $1$, or $2$ after division by $3$.

Why mod 3 is used

It creates balanced three-level pseudo-factor columns from combinations of factor levels coded $0$, $1$, and $2$.

AB interaction components

The $4$-df $A \times B$ interaction is represented by two orthogonal pseudo-components, each with three levels and $2$ df.

Balance of pseudo-components

Each pseudo-component column contains equal numbers of $0$s, $1$s, and $2$s.

Orthogonality of pseudo-components

The two $2$-df pieces partition the $4$ df of the $A \times B$ interaction without overlap.

Component-label convention

The first factor is kept to exponent $1$; equivalent forms with higher exponents can be reduced by modular arithmetic.

Main conceptual takeaway on components

Interactions can be split into orthogonal $2$-df pieces for efficient confounding.

Single $3 \times 3$ replicate

Contains nine treatment combinations.

Goal of blocking the $3 \times 3$ design

Partition the nine runs into three blocks of three runs each.

Block df with three blocks

$3 - 1 = 2$ df.

Effect chosen for confounding (3x3)

One $2$-df pseudo-component of $A \times B$ can be confounded with blocks.

Why the pseudo-component fits blocks

Both the block effect and the selected interaction component have $2$ df and three levels.

Block assignment rule

Treatment combinations are grouped according to whether the selected pseudo-component has level $0$, $1$, or $2$.

Consequence of confounding

The selected $2$-df interaction component cannot be estimated independently of block differences within that replicate.

Unconfounded interaction component

The other $2$-df component of $A \times B$ remains estimable within blocks.

Four replicated blocked experiments observations

Four reps of the $3 \times 3$ design produce $36$ observations.

Rep df with four reps

$4 - 1 = 3$.

Blocks within rep

Each replicate contains three smaller blocks.

Inter-block variation df

Variation among the $12$ total blocks is represented by $11$ df.

Inter-block total df calculation

$12$ blocks give $12 - 1 = 11$ df.

Inter-block components (same-confounding)

Rep contributes $3$ df, the confounded block component contributes $2$ df, and Rep x block contributes $6$ df.

Intra-block main effects

$A$ and $B$ each retain $2$ df.

Intra-block interaction information (same-confounding)

Only the unconfounded $2$-df component of $A \times B$ is directly estimable.

Error df in the four-rep example (same-confounding)

$18$ df.

Total df in the four-rep example

$36 - 1 = 35$ df.

Error interpretation (replicated blocks)

Error can be viewed as treatment-by-replicate variation for effects estimable within blocks.

Two levels of blocking terminology

Replicates act like larger super-blocks, while smaller blocks occur within each replicate.

Partial confounding

A replicated blocked design in which different replicates confound different components of an interaction.

Purpose of partial confounding

Preserve some information about every interaction component rather than losing one component in every replicate.

Two-component $A \times B$ strategy

Confound one $A \times B$ component in some reps and the other component in the remaining reps.

Four-rep partial confounding example

Two reps confound one $A \times B$ component and two reps confound the other.

Information retained (partial confounding)

Each interaction component is estimable from the replicates in which it is not confounded.

Overall $A \times B$ information (partial confounding)

The full $A \times B$ interaction can be estimated, but with reduced information compared with a completely unconfounded design.

Partial confounding information amount

With each component confounded in half the replicates, the design provides about half the information on the interaction.

Benefit over complete confounding

No interaction component is lost from all replicates.

Inter-block df under partial confounding

The $12$ blocks still account for $11$ inter-block df.

Main effects under partial confounding

$A$ and $B$ remain estimable with $2$ df each.

$A \times B$ df under partial confounding

The full interaction again has $4$ df.

Error df in partial-confounding example

$16$ df.

General partial-confounding principle

Rotate which high-order component is confounded across replicates when those interaction components may still be scientifically useful.

Confounding a main effect

Using the levels of a main factor rather than an interaction component to define blocks.

Example main effect for blocking

Factor $A$ is used to define the three blocks.

Agricultural interpretation of main-effect blocking

$A$ represents irrigation method, $B$ crop variety, and blocks are whole plots receiving an irrigation method.

Resulting design type from blocking a main effect

A split-plot design.

Reason split-plot arises

The hard-to-change whole-plot factor $A$ is confounded with whole plots/blocks.

Whole-plot factor

The main factor whose levels are applied to the large blocks or whole plots.

Subplot factor

The factor varied within whole plots; in the example, $B$.

Inter-block part of split-plot analysis

Contains reps, $A$, and Rep x $A$.

Intra-block part of split-plot analysis

Contains $B$ and $A \times B$ plus the appropriate error.

Key connection of split-plot

A split-plot design can be understood as deliberately confounding a main effect with blocks.

Full $3^3$ design

Three factors at three levels each, giving $27$ treatment combinations.

Total df in one $3^3$ replicate

$27 - 1 = 26$.

Main-effect df in $3^3$

$A$, $B$, and $C$ each have $2$ df.

Two-way interaction df in $3^3$

$AB$, $AC$, and $BC$ each have $4$ df.

Three-way interaction df in $3^3$

$ABC$ has $8$ df.

Preferred effect to confound in $3^3$

A higher-order interaction component is typically chosen rather than an important lower-order effect.

$ABC$ component count

The $8$-df three-way interaction can be partitioned into four orthogonal $2$-df components.

Block structure for $3^3$

The $27$ runs can be partitioned into three blocks of nine runs.

Confounded component in $3^3$

One $2$-df $ABC$ pseudo-component defines the three blocks.

Practical scheduling example (blocks)

Nine runs can be performed per day, with the three blocks completed over three days for one replicate.

Four-rep scheduling example (blocks)

Four complete replicates would require $12$ such nine-run blocks.

Four-rep observations ($3^3$)

$27 \times 4 = 108$.

Four-rep total df ($3^3$)

$108 - 1 = 107$.

Rep df ($3^3$ four-rep)

$4 - 1 = 3$.

Confounded block-component df ($3^3$)

$2$ df.

Rep x confounded-component df

$3 \times 2 = 6$.

Inter-block total df ($3^3$)

$3 + 2 + 6 = 11$ df among the $12$ blocks.

Unconfounded main effects df ($3^3$)

Each remains at $2$ df.

Unconfounded two-way interactions df ($3^3$)

Each remains at $4$ df.

Remaining $ABC$ df

$6$ df because one $2$-df $ABC$ component is confounded with blocks.