Small Sample Statistics for Incomplete Nonnormal Data: Extensions of Complete Data Formulae and a Monte Carlo Comparison
Bibliographic record
Abstract
Incomplete nonnormal data are common occurrences in applied research. Although these 2 problems are often dealt with separately by methodologists, they often cooccur. Very little has been written about statistics appropriate for evaluating models with such data. This article extends several existing statistics for complete nonnormal data to incomplete data and evaluates their performance via a Monte Carlo study. The focus is on statistics that also perform well in small samples. The following statistics are defined and studied: corrected residual-based statistic, residual-based F statistic, scaled chi-square, adjusted chi-square, Bartlett-corrected scaled chi-square, and Swain-corrected scaled chi-square. Both Type I error rates and power are studied with missing completely at random nonnnormally distributed data and varying degrees of nonnormality. Sample size, model size, and number of variables containing missingness are also varied. For power comparisons, both minor and major model misspecifications are considered. Two statistics had the best Type I error control and power: the adjusted chi-square and Bartlett-corrected chi-square. These statistics are recommended to practitioners. It is concluded that model fit can be assessed reliably and with sufficient power even at the intersection of all 3 problems: incomplete data, nonnormality, and small sample size.
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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 | 0.136 | 0.507 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.004 | 0.003 |
| Bibliometrics | 0.005 | 0.006 |
| Science and technology studies | 0.001 | 0.008 |
| Scholarly communication | 0.004 | 0.014 |
| Open science | 0.005 | 0.005 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.010 | 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".