On the Model-Based Bootstrap With Missing Data: Obtaining a <i>P</i>-Value for a Test of Exact Fit
Bibliographic record
Abstract
Evaluating the fit of a structural equation model via bootstrap requires a transformation of the data so that the null hypothesis holds exactly in the sample. For complete data, such a transformation was proposed by Beran and Srivastava (1985) Beran, R. and Srivastava, M. S. 1985. Bootstrap tests and confidence regions for functions of a covariance matrix. The Annals of Statistics, 13: 95–115. [Crossref], [Web of Science ®] , [Google Scholar] for general covariance structure models and applied to structural equation modeling by Bollen and Stine (1992) Bollen, K. A. and Stine, R. A. 1992. Bootstrapping goodness-of-fit measures in structural equation models. Sociological Methods and Research, 21: 205–229. [Crossref], [Web of Science ®] , [Google Scholar]. An extension of this transformation to missing data was presented by Enders (2002) Enders, C. K. 2002. Applying the Bollen-Stine bootstrap for goodness-of-fit measures to structural equation models with missing data. Multivariate Behavioral Research, 37: 359–377. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], but it is an approximate and not an exact solution, with the degree of approximation unknown. In this article, we provide several approaches to obtaining an exact solution. First, an explicit solution for the special case when the sample covariance matrix within each missing data pattern is invertible is given. Second, 2 iterative algorithms are described for obtaining an exact solution in the general case. We evaluate the rejection rates of the bootstrapped likelihood ratio statistic obtained via the new procedures in a Monte Carlo study. Our main finding is that model-based bootstrap with incomplete data performs quite well across a variety of distributional conditions, missing data mechanisms, and proportions of missing data. We illustrate our new procedures using empirical data on 26 cognitive ability measures in junior high students, published in Holzinger and Swineford (1939) Holzinger, K. J. and Swineford, F. 1939. A study in factor analysis: The stability of a bi-factor solution. Supplementary Educational Monographs, 48: 1–91. [Google Scholar].
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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 | 0.148 | 0.591 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.007 | 0.006 |
| Bibliometrics | 0.010 | 0.014 |
| Science and technology studies | 0.003 | 0.018 |
| Scholarly communication | 0.007 | 0.012 |
| Open science | 0.007 | 0.010 |
| Research integrity | 0.008 | 0.015 |
| Insufficient payload (model declined to judge) | 0.009 | 0.003 |
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".