Blocking Factorial Experiments

A full factorial may not fit into one batch, day, or session. Splitting it into smaller blocks can protect important comparisons from nuisance variation, but the split can make some treatment effects indistinguishable from block effects.

The goal is to choose what information is sacrificed, rather than let scheduling accidentally confound an important factor. The sign-column calculations build on Two-Level Factorial Designs.

Complete Blocks vs. Confounded Blocks¶

Here, every factorial treatment combination is still run somewhere in the experiment. This differs from a fractional factorial, where some combinations are not run at all.

Confounded / aliased columns: two model columns are identical or differ only by sign over the observed design. Their separate coefficients cannot be identified from those runs.

A 2² Design in Two Blocks¶

Suppose A and B each have low/high levels, but only two runs fit in a block. Use the AB product column to form blocks:

Both A and B vary within each block. Their main effects remain orthogonal to the block contrast, but AB is identical to that contrast.

Writing $z=x_Ax_B$ for the coded block column,

Only the sum $\beta_{AB}+\gamma$ is identifiable. One cannot claim that an observed difference between blocks is specifically an AB interaction, or specifically a block effect.

A 2³ Design in Two Blocks¶

A common choice is to confound the highest-order interaction ABC with block, protecting main effects and two-way interactions from the additive block effect.

Each block contains four runs. Every main-effect and two-way sign column is balanced within a block.

Let $Z$ denote the block sign column. Since $Z=ABC$,

Here $I$ is the all-+1 column, not an experimental run. The treatment label $(1)$ means all factors low; it is a different object.

Multiplying Columns to Find Aliases¶

With $\pm1$ columns, each letter cancels when it occurs twice:

Under an additive block model, terms such as AZ and BCZ are assumed absent. If treatment effects change by block, these aliases matter: an estimated AB term can contain a C-by-block effect. Confounding with the highest-order treatment interaction does not protect against every possible block interaction.

Degrees of Freedom & Error¶

For one unreplicated 2³ experiment split into two blocks, fitting the intercept, A, B, C, AB, AC, BC, and block uses all eight available columns. There is no residual df.

Blocking does not create replication or pure-error information. To make conventional significance tests, obtain independent error information through suitable replication or make explicit, defensible assumptions about negligible effects.

If only selected treatment terms are fitted, leftover variation may serve as model residual error, but it can include omitted treatment effects. It is not automatically pure error.

Randomize which physical block receives each block subset when feasible, then randomize run order within each block. Keep records of the nuisance conditions that motivated blocking.

More Blocks: Include Products of the Generators¶

To divide a regular $2^k$ design into $2^q$ equal blocks, use $q$ independent block-generating columns. Their sign combinations identify the blocks.

The full block space includes all nonidentity products of the generators: $2^q-1$ columns. All corresponding treatment effects are confounded with block effects.

For a 2³ design in four blocks, choose $Z_1=AB$ and $Z_2=AC$. Their product is

Therefore AB, AC, and BC, not just the first two, are confounded with blocks.

Each pair consists of opposite corners, so all three main effects vary within every block.

Partial Confounding Across Replications¶

Repeating the same split can improve precision for estimable effects, but an interaction confounded with blocks in every replication stays confounded with the corresponding block contrasts.

Partial confounding changes the block generator across complete factorial replications:

Each listed interaction is confounded in one replication and available from within-block comparisons in the other three. Main effects remain unconfounded with additive blocks throughout.

Analyze the actual structure, including replication and blocks within replication. Do not average all raw treatment responses as if they came from a completely randomized experiment. Different effects can have different amounts of information because they are recovered from different subsets of the runs.

This cell builds the factorial table, verifies block balance, and lists the four-block groups. It does not manufacture an error estimate for an unreplicated design.