Factorial Designs and Interactions

Upon Completing This Topic

You will be able to distinguish between the main effects and interactions of two factors. You will express situations where the effect of one factor depends on the level of another factor using tables and research questions, and explain the risks of interpreting data under the assumption that no interaction exists.

Designs That Do Not Examine Factors in Isolation

If factors A and B each have two levels, four condition combinations arise. A factorial design asks about "the average difference of A," "the average difference of B," and "whether the effect of A varies by B" simultaneously through the combination of each factor.

A main effect is the difference averaged across the levels of the other factor. An interaction is the relationship where the effect of one factor varies depending on the level of another factor. When an interaction exists, it is difficult to explain all combinations by looking only at main effects.

Data Structure

unit_id | factor_a | factor_b | outcome
U01     | low      | control  | ...
U02     | high     | control  | ...
U03     | low      | treated  | ...
U04     | high     | treated  | ...

Verify the number of independent units per combination and the allocation process. Simply increasing the number of measurements per combination does not constitute a factorial design.

Translating Interactions into Questions

"Does B increase the outcome?" is not an interaction question. "Does the difference between the levels of B vary across the levels of A?" is an interaction question. While you can explore whether the average differences between combinations of the two factors are non-parallel in a line graph, a single graph showing intersecting lines does not confirm the cause or biological mechanism of the interaction.

Four Values to Read in a 2 × 2 Design

In a 2 × 2 design, verify the outcomes of the four combinations: A_low/B_control, A_high/B_control, A_low/B_treated, and A_high/B_treated. To examine the effect of B, you can calculate the difference between the levels of B when A is low and compare it with the corresponding difference when A is high. If these two differences differ, you can consider the direction of the interaction.

At this time, do not look only at the mean per combination; also verify the number of independent units and dispersion per combination. If there is missing data or batch bias in one combination, a pattern that looks like an interaction may be linked to design imbalance.

Interpretation Sequence

First, verify the design and the unit of analysis, and examine the observations per combination. Next, select a model including interactions or pre-defined contrasts. If interactions are important, you may report simple effects at specific factor levels, but you must record the number of comparisons and the family.

Returning to Research Questions for Interpretation

The advantage of factorial designs lies in managing combinations of multiple conditions within a single question framework. However, do not assume that all factor combinations have the same quality, sample size, or allocation structure. Report the magnitude and uncertainty of interactions, along with the number of units per combination.

Common Failures

• Do not omit interactions by reporting only main effects.
• Do not use the same term for the fact that means per combination differ and for interaction.
• Do not select only favorable contrasts after observing the data.
• Do not hide imbalances in factor combination frequencies or missing data.

Key Takeaways

• Factorial designs ask about main effects and interactions simultaneously through factor combinations.
• An interaction is a relationship where the effect of one factor varies depending on the level of another factor.
• You must verify the independent units per combination and the family of comparisons.

Next Topic

In the next section, we expand the hierarchical structure using repeated measures, clusters, and pseudoreplication.

References

• Bioconductor OSTA experimental design: https://bioconductor.org/books/release/OSTA/pages/bkg-exp-design.html
• NIST/SEMATECH e-Handbook: https://www.itl.nist.gov/div898/handbook/

The factorial schema and scenarios in this section were written independently by BioStatPy.