Comparing Two Independent Groups: Inferences About the Lower and Upper Tails of the Distribution of D = X − Y
Bibliographic record
Abstract
When comparing two independent groups, a way of getting a more detailed understanding of how the groups compare is to focus on multiple quantiles rather than a single measure of location. There are two distinct approaches regarding how this might be done. The first is to estimate a collection of quantiles for each group and choose an appropriate inferential method for comparing them. This approach has been studied extensively. For example, Doksum and Sievers (1976) derived a nonparametric method for computing confidence intervals for the difference between all quantiles for which the simultaneous probability coverage can be determined exactly assuming random sampling only. An analog of the Doksum–Sievers method was derive by Lombard (2005), which is based on the marginal distributions of two dependent groups. Other methods and applications are described in Wilcox (2023). Let X and Y denote two independent random variables and let D = X − Y. Note that under general conditions, the q quantile of D is not equal to the q quantile of X minus the q quantile of Y. Methods for making inferences about the median of D have been studied that have a close connection to the Wilcoxon–Mann–Whitney test. Here, four methods are suggested and studied via simulations that are aimed at making inferences about the tails of the distribution D with the goal of providing a deeper and more nuanced understanding of how groups compare.
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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 | 0.089 | 0.354 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.005 | 0.004 |
| Science and technology studies | 0.002 | 0.013 |
| Scholarly communication | 0.005 | 0.012 |
| Open science | 0.005 | 0.007 |
| Research integrity | 0.005 | 0.006 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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".