17 Factorial Designs and Matched-Tuple Randomization
Prerequisites: Chapters 3, 5, and 16.
17.1 Learning objectives
By the end of this chapter you should be able to:
- Describe a factorial trial, state what it estimates, and explain the efficiency argument for it.
- Identify the assumption a factorial design relies on and what happens when it fails.
- Explain matched-tuple randomization and how it generalizes matched pairs to \(k\) arms.
- State the variance of a factorial contrast under the matched-tuple randomization distribution and the degrees-of-freedom cost.
- Decide when matched-tuple randomization is worth its complexity.
17.2 Orientation
A factorial trial randomizes each patient to a combination of two or more interventions simultaneously. With two binary factors A and B, patients are assigned to one of four arms: neither, A alone, B alone, both. The main effect of A is estimated by comparing everyone who got A to everyone who did not, averaging across B, and vice versa.
The efficiency claim is striking: two questions answered for the price of one trial. It holds when the two treatments do not interact, and it is the basis for some of the largest and most informative trials ever run.
The second half of the chapter takes up a distinct question. A factorial design with \(k\) arms needs balance across \(k\) groups, and the standard tools (stratification, minimization) are strained by \(k > 2\). Matched-tuple randomization is the natural generalization of the matched pair: group patients into tuples of size \(k\) matched on baseline covariates, and randomly assign the \(k\) treatment combinations within each tuple. Every factorial contrast is then exactly balanced within every tuple.
17.3 Provenance
The matched-tuple methodology, variance formulas, and simulation results come from the research compendium 16-factorial-matched-random (project factorialrandom), which develops matching algorithms using Mahalanobis and prognostic-score distances, derives exact and asymptotic variance formulas for factorial effects under the matched-tuple randomization distribution, and evaluates efficiency in simulations motivated by Alzheimer disease prevention trials with trial-ready cohorts.
17.4 The statistician’s contribution
(Judgment 1.) Whether the no-interaction assumption is defensible. A factorial trial powered for main effects is under-powered for interaction by a factor of about four. If a substantial interaction is plausible on mechanistic grounds, the design either needs to be powered for it, which is expensive, or should not be factorial.
(Judgment 2.) Whether the pool is available in advance. Matched-tuple randomization requires the subjects to be matchable, which requires a pool of characterized candidates. In sequential enrollment this is impossible; in a trial-ready cohort, where participants are screened and characterized before any trial opens, it is natural.
(Judgment 3.) The degrees-of-freedom accounting. Matching constrains the randomization distribution, and the variance estimator is based on \(M - 1\) tuples rather than \(N - k\) patients. In a small trial with large \(k\) this can cost more than the matching gains.
17.5 Factorial designs
Consider two factors, each at two levels, with \(N\) patients allocated equally to the four cells. Write the outcome model \[ Y = \mu + \alpha A + \beta B + \gamma AB + \epsilon, \] with \(A, B \in \{-1, +1\}\) coded as contrasts.
The main effect of A is estimated by \(\hat\alpha = (\bar Y_{A+} - \bar Y_{A-})/2\), using all \(N\) patients. Its variance is \(\sigma^2 / N\), exactly as if the trial had studied A alone with \(N\) patients. The same holds for B. This is the efficiency claim: two main effects at the cost of one trial’s sample size.
The interaction is estimated by a contrast that pits the diagonal cells against the off-diagonal ones, and its variance is four times that of a main effect. Hence the standard result that detecting an interaction of the same magnitude as a main effect requires four times the sample size.
What happens when there is an interaction. The main effect of A is then an average over the levels of B, and whether that average is meaningful depends on the question. If B is a treatment that will be used in half of practice, the averaged effect may be exactly what a health system wants. If A works only in the presence of B, the averaged main effect describes no patient’s situation.
Practical examples. The Physicians’ Health Study (aspirin and beta-carotene) and ISIS-2 (streptokinase and aspirin) are canonical, and the latter found no interaction between two treatments acting through different mechanisms, which is the typical case. Yusuf et al. (1991) argue that factorial designs are under-used for exactly this reason: interactions between mechanistically distinct interventions are usually small.
A factorial trial that finds a significant interaction is usually reporting noise. With the design powered for main effects, an interaction test at conventional significance has power around 20 to 25% for an interaction equal to the main effect, and much less for smaller ones. A significant result at that power is more likely to be a false positive than a true one when interactions are a priori improbable. Report the interaction test; do not build the conclusion on it.
17.6 The balance problem with \(k\) arms
Chapter 3’s tools were developed for two arms. With \(k = 4\) or \(k = 8\) factorial cells, they strain:
- Stratified blocks need block sizes that are multiples of \(k\), so a block of size 8 in a \(2^3\) design, which makes within-stratum balance coarse when strata are small.
- Minimization generalizes but the imbalance metric across \(k\) groups is less clean and the assignment probabilities more arbitrary.
- Simple randomization leaves substantial imbalance across four or eight groups at realistic trial sizes.
17.7 Matched-tuple randomization
The construction. Given a pool of \(N_{\text{pool}}\) characterized candidates, form \(M\) tuples of size \(k\) by optimal matching on baseline covariates, using Mahalanobis distance or a prognostic score. Within each tuple, assign the \(k\) treatment combinations by drawing uniformly from the \(k!\) permutations. Tuples are independent, so the randomization space contains \((k!)^M\) equally likely allocations.
The consequence is exact: within every tuple, each treatment combination is represented once, so every factorial contrast is perfectly balanced within every tuple, and the covariate distributions across arms are as similar as the matching allows.
Estimation. For a contrast defined by weights \(w_z\) over the \(k\) combinations with \(\sum_z w_z = 0\), the estimator is the average across tuples of the contrast-weighted outcomes, \[ \hat\tau_c = \frac{1}{M}\sum_{m=1}^{M} \sum_z w_z Y_m(z), \] which is unbiased under the matched-tuple randomization distribution.
Variance. The exact variance is \(\mathrm{Var}(\hat\tau_c) = M^{-2}\sum_m V_m(c)\), where \(V_m(c)\) depends on the within-tuple potential outcomes. A conservative estimator replaces \(V_m(c)\) by the within-tuple sample variance of the contrast-weighted outcomes, which is the natural analogue of the paired \(t\)-test variance.
Regression adjustment. Following the Lin-style argument of Chapter 16, the difference-in-means can be augmented with a regression adjustment on residual covariate differences, which recovers efficiency when the matching is imperfect and cannot hurt asymptotically.
# Sketch: form tuples by optimal matching, then permute
# treatments within tuple.
library(nbpMatching)
d <- dist_mahalanobis(pool[, covariates])
tp <- form_tuples(d, k = 4) # M tuples of size 4
assign <- unlist(lapply(split(tp$id, tp$tuple),
function(ids) sample(1:4)))17.8 Efficiency and its limits
The compendium’s simulation, motivated by Alzheimer prevention trials, evaluates covariate balance, efficiency gains, coverage, power, and the effect of pool oversampling. The findings shape four practical conditions under which matched-tuple randomization is advantageous:
- The full pool is available before randomization. This is what permits global rather than sequential matching, and it is the situation in a trial-ready cohort such as TRC-PAD, AHEAD 3-45, DIAN-TU, or EPAD, where thousands of participants are characterized in advance.
- Strong prognostic covariates are measured at baseline. Matching on weak predictors buys nothing.
- The number of arms is not too large, up to about \(k = 8\), that is a \(2^3\) design, to avoid excessive degrees-of-freedom loss.
- The pool-to-sample ratio exceeds 1. Excess candidates allow the matching algorithm to discard poor matches, and the simulation shows the quality of matching, and hence the efficiency gain, improving with oversampling.
The degrees-of-freedom cost. With \(M\) tuples the effective degrees of freedom for variance estimation are \(M - 1\), against \(N - k\) under unrestricted randomization. For a \(2^3\) design with \(N = 160\), \(M = 20\) and the variance is estimated on 19 degrees of freedom rather than 152. This is the binding limitation, and it is attenuated when \(M\) exceeds roughly 30.
Computational cost. Optimal \(k\)-tuple formation by integer programming is feasible for pools up to about 1,000 and needs heuristics beyond that. Trial-ready cohort pools of 1,000 to 2,000 sit at the edge of the tractable range.
17.9 Worked example: a \(2 \times 2\) prevention trial
An Alzheimer prevention trial in a trial-ready cohort of 1,400 characterized participants. Two interventions: an anti-amyloid agent and an intensive lifestyle program. Target enrollment 400. Primary endpoint: change in a composite cognitive score over three years.
Design. \(2 \times 2\) factorial, \(k = 4\). The mechanisms are distinct, so a substantial interaction is not expected; the trial is powered for main effects and reports the interaction descriptively.
Randomization. Matched-tuple. From the pool of 1,400, form 100 tuples of four participants each, matched on a prognostic score built from age, APOE genotype, baseline composite score, and baseline amyloid PET centiloid. Pool ratio is 3.5, comfortably above 1, which allows the algorithm to discard poor matches.
Estimation. Difference-in-means across tuples for each main-effect contrast, with the conservative within-tuple variance estimator, augmented by regression adjustment on residual covariate differences.
Degrees of freedom. \(M = 100\), so 99 degrees of freedom for the variance estimate. Well above the threshold of concern.
Efficiency. Simulated relative efficiency against stratified block randomization on the same covariates, categorized: matched-tuple gives a variance reduction of roughly 15 to 20% for the main effects in the compendium’s Alzheimer-motivated scenarios, with the gain increasing in the prognostic strength of the matching variables and in the pool ratio.
What it buys, stated plainly. A 15 to 20% variance reduction is equivalent to enrolling 15 to 20% more patients. At 400 patients in a prevention trial costing tens of millions, that is a substantial saving, and it is available only because the cohort was characterized in advance.
17.10 Collaborating with an LLM on factorial designs
Prompt 1: ‘Compute the sample size for a 2x2 factorial trial.’
What to watch for. The main-effect calculation is usually correct. Models often fail to note that the interaction requires four times the sample size, and sometimes size the trial for the interaction without saying so, which quadruples the answer.
Verification. Confirm which contrast the calculation targets, and check that the main-effect variance is \(\sigma^2/N\) rather than \(4\sigma^2/N\).
Prompt 2: ‘Implement matched-tuple randomization.’
What to watch for. Code that matches into pairs and then splits, which does not give a valid \(k\)-tuple structure, or that assigns treatments without drawing uniformly from all \(k!\) permutations within tuple.
Verification. Check that every tuple contains each treatment exactly once, and simulate the allocation distribution to confirm each permutation is equally likely.
Prompt 3: ‘How should we analyze a matched-tuple randomized trial?’
What to watch for. A tendency to recommend the standard unadjusted analysis, which is valid but conservative because it ignores the matching.
Verification. The analysis should either use the tuple-based variance estimator or condition on the tuple structure, and a permutation test over the actual randomization distribution is always available as a check.
17.11 Principle in use
Use factorial designs when the mechanisms are distinct. Two answers for one trial is real, and the interaction concern is usually theoretical.
Match only when the pool exists in advance. Sequential enrollment forecloses global matching, and minimization is then the right tool.
Count the tuples, not the patients, when assessing the variance estimate. \(M - 1\) degrees of freedom is the operative number.
17.12 Exercises
Derive the variance of the main-effect and interaction contrasts in a balanced \(2 \times 2\) design and confirm the factor of four.
Simulate a \(2 \times 2\) trial with a true interaction equal to half the main effect. What is the power of the interaction test at the main-effect-powered sample size?
Implement matched-pair randomization for \(k = 2\) and compare the variance of the treatment effect to simple randomization, for covariate-outcome correlations of 0.3, 0.5, and 0.7.
Extend to \(k = 4\) tuples and reproduce the relationship between pool ratio and matching quality.
For a \(2^3\) design with \(N = 120\), compute the degrees-of-freedom cost of matched-tuple randomization and determine whether the efficiency gain from matching compensates for it, using a simulated prognostic score with \(R^2 = 0.4\).