Factorial Design
One trial that tests TWO interventions at once by randomising each independently — two questions for roughly the price of one.
A trial that randomises participants to two (or more) interventions simultaneously. A 2×2 factorial allocates each person to A or not-A and to B or not-B, creating four groups (A+B, A only, B only, neither). It answers both questions in one population — and can formally test whether the two interventions interact.
| B given | B not given | |
|---|---|---|
| A given | A + Bboth | A only |
| A not given | B only | neitherplacebo |
One trial answers two questions (assuming no interaction).
Pool both “A given” cells vs both “A not given” — the whole trial powers each question.
Four randomised groups from two independent coin-flips. The shaded cells are the four allocations; the main effect of A is the two top cells versus the two bottom cells, and the main effect of B is the two left cells versus the two right cells.
To judge intervention A, ignore B and compare everyone who got A against everyone who did not; do the mirror image for B. Each comparison uses the entire trial population, which is why two questions cost barely more than one — provided A’s effect does not depend on whether B was also given.
Factorial designs are efficient: a single recruited cohort, two answers. But the efficiency rests on the no-interaction assumption. If A and B genuinely interact, the pooled main effects are misleading and you must analyse the four cells separately — a comparison the trial is usually underpowered to make.
- 2×2 →
four groups(A+B, A, B, neither) - Main effect of A =
pool A-given vs A-not-given - Valid only if
no interaction(effects independent)
ISIS-2 (Lancet 1988) — 17,187 patients with suspected acute MI, randomised in a 2×2 factorial to streptokinase × aspirin (vs matching placebos). Aspirin alone cut 5-week vascular mortality 11.8% → 9.4% and streptokinase alone 12.0% → 9.2%; the combination was best (13.2% → 8.0%, a ~42% odds reduction vs neither). Two landmark answers from one trial. The streptokinase and aspirin effects were broadly additive (no important interaction), which is exactly what licenses pooling each main effect across the whole trial — the assumption the factorial design depends on.
- Unrecognised interaction — reporting pooled main effects when A’s effect actually depends on B.
- Treating a non-significant interaction test as proof of no interaction — the trial is usually underpowered for it.
- Underestimating the added operational complexity (two consents, two placebos, more protocol deviations).
Quick check
What assumption lets a factorial trial test two treatments efficiently?
Answer: That the two interventions do not interact — their effects are independent, so each main effect can be estimated by pooling across the whole trial.
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