Using normal probability plots to determine parameters for higher‐level factorial experiments with orthogonal and orthonormal bases
Bibliographic record
Abstract
ABSTRACT In chemical engineering applications such as optimizing plant operations and product quality, factorial experiments are often conducted to obtain empirical models for systems. While the mathematical underpinning for two‐level factorial designs is well understood, a methodology for analyzing higher‐level experiments is not readily available. Often an orthogonal or orthonormal basis is selected for a factorial design matrix. In factorial design, an orthonormal basis is defined as an orthogonal matrix where the Euclidean two‐norms of the column vectors are equal. This investigation examines, for full factorial design, the selection of parameters using normal probability plots and the effect that the design basis has on parameter determination. When using normal probability plots to determine parameter significance, the traditional orthogonal basis for higher‐level experiments may result in erroneous conclusions. A Monte‐Carlo experiment was developed to simulate 3‐level and mixed‐level factorial experiments with different types of measurement error. The basis chosen is shown to affect the shape of probability plots, and measurement errors from a given normal distribution are shown to result in a constant standard deviation for all parameter estimates only when using an orthonormal basis.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.027 | 0.078 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".