Bibliographic record
Abstract
When two groups are compared in a pre-post study, two different conclusions can be drawn between the two-sample t-test and the analysis of covariance (ANCOVA). It is known as Lord's Paradox, and it occurs because the parameter in the two-sample t-test and the parameter of interest in the ANCOVA model are not the same quantity. The difference between the two parameters can be explained by the covariance of linearly combined random variables which is an important topic in introductory statistical theory courses. Lord's paradox is frequently observed in practice, and it is very important for students (future researchers) to have clear understanding of the paradox. The objective of this article is to explain Lord's Paradox using the covariance of linearly combined random variables. The paradox is explained using three scenarios in the context of educational research. The first scenario is when the average baseline (pre-score) is greater in the treatment group than the control group, the second scenario is when the average baseline is lower in the treatment group than the control group, and the third scenario is when the average baseline is same between the two groups by randomization. This article is written at the level of introductory statistical theory courses for undergraduate and graduate statistics students to help understanding the difference between the parameter of interest in the two-sample t-test and the parameter of interest in the ANCOVA model.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.017 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".