Correcting Two-Sample z and t Tests for Correlation: An Alternative to One-Sample Tests on Difference Scores
Bibliographic record
Abstract
In order to circumvent the influence of correlation in paired-samples and repeated measures experimental designs, researchers typically perform a one-sample Student t test on difference scores. That procedure entails some loss of power, because it employs N – 1 degrees of freedom instead of the 2N – 2 degrees of freedom of the independent-samples t test. In the case of non-normal distributions, researchers typically substitute the Wilcoxon signed-ranks test for the one-sample t test. The present study explored an alternate strategy, using a modified two-sample t test with a correction for correlation, analogous to the “z test for correlated samples” used at one time for paired observations. For non-normal distributions, the same modified t test was performed on rank-transformed data. Simulations disclosed that this procedure protects the Type I error rate for moderate and large sample sizes, maintains power for normal distributions and several symmetric non-normal distributions, and substantially increases power for various skewed nonnormal distributions. Statistical analysis of paired-samples or repeated-measures experimental designs typically employs the one-sample Student t test on difference scores in place of the independent-samples t test. This method, widely used in the past, entails some loss of power, because the test on differences is necessarily based on N – 1 instead of 2N − 2 degrees of freedom. In the first part of the last century, data from paired-samples was often analyzed in a different way. Many introductory textbooks in that period, focusing mainly on large-sample studies for which the z-test is appropriate, presented methods of analyzing what were called correlated samples, using a modification of the familiar two-sample z test. These
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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 | 0.067 | 0.391 |
| Meta-epidemiology (narrow) | 0.003 | 0.001 |
| Meta-epidemiology (broad) | 0.005 | 0.004 |
| Bibliometrics | 0.005 | 0.011 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.005 | 0.010 |
| Open science | 0.007 | 0.005 |
| Research integrity | 0.004 | 0.008 |
| Insufficient payload (model declined to judge) | 0.026 | 0.006 |
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".