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Order-of-addition experiments raise LLM success rates on optimal design construction from about 12% to 98%.

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2026-07-11 06:08 UTC pith:D5Y33JCS

load-bearing objection Clean transfer of OofA designs and logistic PWO models to LLM prompt order sensitivity, with large confirmed gains on a concrete design-construction task.

arxiv 2607.05537 v1 pith:D5Y33JCS submitted 2026-07-06 stat.AP

Prompt engineering using order-of-addition experiments: An application to generating two-level fractional factorial designs

Duoduo Danny Ying , Alan R. Vazquez , Hongquan Xu This is my paper
classification stat.AP MSC 62K0562J12
keywords prompt engineeringorder dependencyorder-of-addition experimentspairwise ordering modellarge language modelstwo-level fractional factorial designslogistic regression
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Large language models often change how well they perform a task when the same prompt phrases are merely reordered. The authors treat the phrases of a design-of-experiments prompt as the components of an order-of-addition experiment, run a compact set of carefully chosen orderings, and fit logistic pairwise-ordering models to the resulting success rates. On the concrete task of generating 16-run two-level fractional factorial designs, this procedure identifies prompt sequences that lift success rates from 11.7% to 98.3% for one model and from 35% to 100% for another. The same framework also estimates the size and direction of each ordering effect, so the improvement is not a black-box search result but a quantified statistical finding. A reader who needs reliable LLM output for structured technical tasks therefore obtains both a practical optimization tool and a diagnostic for order sensitivity.

Core claim

Order-of-addition designs together with logistic pairwise-ordering models can quantify the effect of reordering prompt elements and can locate high-performing configurations that raise the success rate of state-of-the-art LLMs on constructing optimal two-level fractional factorial designs by tens of percentage points.

What carries the argument

The compound logistic pairwise-ordering (PWO) model: binary pseudo-factors that record whether component i precedes component j, optionally crossed with rephrasing indicators and selected two-factor interactions, linked to binomial success counts via logistic regression.

Load-bearing premise

That a logistic pairwise-ordering model fitted to an 80-run design is accurate enough to rank the remaining untested orderings among the 240 possible prompt configurations.

What would settle it

Run the five highest-ranked predicted sequences and the original base prompt for 60 independent trials each; if the observed success rates fail to match the model predictions within the reported standard errors, or if none of the new sequences substantially outperforms the base prompt, the central claim fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proposes order-of-addition (OofA) experiments as a systematic prompt-engineering method for quantifying and mitigating order dependency in LLMs. Components of a base prompt (ROLE, INSTRUCTION, STEP-BY-STEP, FORMAT, SHOT) and two rephrasings of INSTRUCTION are treated as OofA factors; binomial logistic models (full dummy-variable and pairwise-ordering/PWO, with and without two-factor interactions) are fitted to success rates of constructing minimum-aberration 16-run two-level designs. Three experiments with gpt-4.1 and gemini-2.5-flash (full OF designs for q=3; 80-run COAs for q=5) demonstrate statistically significant ordering and rephrasing effects and identify configurations that raise confirmed success rates from 11.7% to 98.3% (gpt-4.1, 9 factors) and from 35% to 100% (gemini-2.5-flash, 7 factors).

Significance. The work supplies a statistically grounded, cost-efficient alternative to ad-hoc or purely search-based prompt reordering. Strengths include the use of component-orthogonal arrays, overdispersion-corrected logistic PWO models (including a compound interaction extension new to the OofA literature), sequential deviance decompositions, and independent 60-replicate confirmation experiments whose observed rates closely match model predictions. The concrete, large gains on a nontrivial experimental-design task make the method immediately useful to statisticians and engineers who rely on LLMs for design generation, while the modeling framework is portable to other prompt-engineering settings.

minor comments (5)
  1. Section 2.2 and Tables 4–5: the initial overdispersion estimates (ϕ̂ ≈ 4.3 and 5.0) are high; a brief remark on residual diagnostics or on whether a beta-binomial alternative was considered would strengthen the modeling discussion.
  2. Section 4.2.2 / Table 5: stepwise BIC selection of the interaction model is pragmatic but opaque; listing the candidate pool size and the final term-selection path (or providing the code path) would aid reproducibility.
  3. Figure 1 and Table 1: success-rate boxplots and the full OF table are dense; a short textual summary of the dominant ordering patterns per task would improve readability.
  4. Supplementary Section S4: the brief demonstration that newer models (gpt-5.4, gemini-3-flash-preview) still exhibit order dependency is valuable; a one-sentence pointer in the main-text concluding remarks would help readers locate it.
  5. Notation: the switch between s_i (dummy variables for full orderings) and z_ij (PWO pseudo-factors) is clear once introduced, but a single sentence reminding the reader of the mapping would reduce cognitive load.

Circularity Check

0 steps flagged

No significant circularity: empirical LLM success rates and held-out confirmation runs stand independently of the fitted PWO models.

full rationale

The paper's derivation chain is empirical and self-contained. Order-of-addition designs (full OF or 80-run COA via Stokes-Xu algorithm) generate fresh prompt configurations that are submitted to gpt-4.1 or gemini-2.5-flash; success rates are measured by independent binomial counts of whether the returned design meets the minimum-aberration criterion. Logistic and compound logistic PWO models (main effects plus BIC-selected interactions) are then fitted to these observed rates solely for screening and ranking; the headline gains (11.67 % o 98.3 % for the 16-run 9-factor task; 35 % o 100 % for the 7-factor task) rest on separate confirmation experiments of 60 new LLM calls on the model-selected sequences (Tables 6 and S7). Those observed rates match the predictions but are not forced by them. No equation equates a claimed prediction to a fitted constant by construction, no uniqueness theorem is imported from overlapping authors, and the base-prompt elements taken from Vazquez et al. (2026) supply only the application setting, not the ordering-effect results. The modeling assumption that a low-order PWO interaction model extrapolates well is therefore non-load-bearing for the strongest claim.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 2 invented entities

The central empirical claim rests on standard binomial modeling of LLM success counts, the adequacy of pairwise-ordering pseudo-factors for capturing order effects, and a handful of controllable experimental settings (temperature, replicate count, design size). No deep free parameters are fitted to force the result; the invented modeling extension is tested against confirmation data.

free parameters (4)
  • temperature
    Fixed at 0 to reduce output stochasticity and test ‘basic knowledge’; choice affects absolute success rates and reproducibility across LLM versions.
  • thinkingBudget (gemini)
    Set to 0 to disable reasoning; later checks with low thinking still show order effects, but the main results depend on this setting.
  • number of replicates Nr
    20–60 independent LLM calls per test; chosen for cost and precision, directly enters binomial variance and overdispersion estimates.
  • OF design size N=80
    Fractional size selected via Stokes–Xu algorithm; determines which orderings are observed and therefore which interactions can be estimated.
axioms (4)
  • domain assumption Number of successful design constructions in Nr independent LLM calls follows a binomial distribution with success probability p that depends on prompt ordering and rephrasing.
    Stated in Section 2.1; underpins all logistic models and overdispersion corrections.
  • domain assumption Pairwise-ordering pseudo-factors z_ij (and selected two-factor interactions) are sufficient to capture the dominant order effects on p.
    Core modeling choice of Section 2.2; if violated, predictions for untested sequences fail.
  • domain assumption Component-orthogonal arrays produced by the Stokes–Xu algorithm remain efficient for the logistic PWO model even though the original algorithm targeted continuous responses.
    Invoked in Section 2.3 and used for the 80-run designs; efficiency is assumed rather than re-proved for the binomial case.
  • ad hoc to paper The five chosen prompt elements (ROLE, INSTRUCTION, STEP-BY-STEP, FORMAT, SHOT) and the single rephrasing of INSTRUCTION exhaust the relevant degrees of freedom for the studied tasks.
    Defines the experimental factors; other phrasings or additional elements could alter conclusions.
invented entities (2)
  • compound logistic pairwise-ordering (PWO) model (and its interaction extension) no independent evidence
    purpose: Link binomial success probability to both component orderings and two-level rephrasings of prompt elements.
    Explicitly introduced in Section 2.2 as new to the OofA literature; fitted and validated on the design-construction data.
  • base Prompt 1 (ROLE + INSTRUCTION + STEP-BY-STEP + FORMAT + SHOT) for two-level fractional factorial generation no independent evidence
    purpose: Provide a fixed, literature-motivated scaffold whose element order can be optimized.
    Constructed in Section 3.2; SHOT element is noted as new for this task.

pith-pipeline@v1.1.0-grok45 · 30909 in / 2765 out tokens · 33856 ms · 2026-07-11T06:08:58.094916+00:00 · methodology

0 comments
read the original abstract

Large language models (LLMs) are becoming ubiquitous in engineering and science because they can turn prompts into data analysis code, experimental designs, formulations of optimization problems, among other applications. However, many LLMs suffer from a phenomenon called order dependency, in which the order of phrases in the prompt affects their performance on a given task. To overcome this issue, we introduce a systematic method that uses order-of-addition experiments to quantify the ordering effect of elements in a prompt and identify their best positions. We demonstrate our methodology by constructing two-level fractional factorial designs using state-of-the-art LLMs. We show that order-of-addition experiments can elucidate order dependency in these LLMs, and can help us to identify a high-quality prompt configuration for the task.

Figures

Figures reproduced from arXiv: 2607.05537 by Alan R. Vazquez, Duoduo Danny Ying, Hongquan Xu.

Figure 1
Figure 1. Figure 1: Success rates of gpt-4.1 for constructing 16-run optimal designs with seven to 12 factors across sequences. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Success rates of gpt-4.1 for constructing 16-run optimal designs with seven to 12 factors across versions. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Success rates of gpt-4.1 for constructing the 16-run 9-factor optimal design across two versions of INSTRUCTION. To test the significance of w2, we fitted a logistic PWO model to the data in [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Success rates of gemini-2.5-flash for constructing 16-run 7-factor optimal de￾signs across two versions of INSTRUCTION. To continue with our statistical analysis, we fit the logistic PWO model in Equation (5) in the main text to the data in Table S4. Table S5 shows the estimated coefficients of this model. Consistent with the findings for gpt-4.1, the ordering of the phrases specifying the number of factor… view at source ↗

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Reference graph

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