{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T16:10:04Z","timestamp":1788883804214,"version":"build-2803163510"},"reference-count":29,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T00:00:00Z","timestamp":1788220800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Psychol."],"abstract":"\n Introduction<\/jats:title>\n \n Nonlinear transformations of pretest and posttest scores are widely used in educational and psychological measurement to estimate group-level change, yet the statistical behavior of estimators derived from such transformations under measurement error remains poorly understood. We examine this problem in the context of normalized gains (ngains), a ratio-based transformation used to estimate group-level \u201clearning rates\u201d in pretest-posttest designs. Two standard estimation methods \u2014 the\n average ngain of the group<\/jats:italic>\n (\n \n \n \n \n n<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n \u00af<\/mml:mo>\n <\/mml:mover>\n <\/mml:math>\n <\/jats:inline-formula>\n ) and the\n ngain of the average learner<\/jats:italic>\n (\n \n \n \n \n n<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n <\/mml:math>\n <\/jats:inline-formula>\n ) \u2014 routinely produce different results. A prior study established a mathematical relationship between this discrepancy and the pretest-ngain correlation, interpreting it as a characterization of the learning process. The pretest-ngain correlation has itself sparked debate: researchers have argued it indicates that ngains favor high-pretest populations, undermining their validity as a measure of student growth.\n <\/jats:p>\n <\/jats:sec>\n \n Methods<\/jats:title>\n Using Classical Test Theory along with a rencently proposed statistical framework to analize ngains, we show that measurement error is one common cause behind both phenomena.<\/jats:p>\n <\/jats:sec>\n \n Results<\/jats:title>\n \n When measurement errors are absent, both\n \n \n \n \n n<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n \u00af<\/mml:mo>\n <\/mml:mover>\n <\/mml:math>\n <\/jats:inline-formula>\n and\n \n \n \n \n n<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n <\/mml:math>\n <\/jats:inline-formula>\n are unbiased and any discrepancy between them reflects only sampling variation. When measurement errors are present,\n \n \n \n \n n<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n \u00af<\/mml:mo>\n <\/mml:mover>\n <\/mml:math>\n <\/jats:inline-formula>\n acquires a systematic negative bias \u2014 consistently underestimating the true learning rate \u2014 while\n \n \n \n \n n<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n ^<\/mml:mo>\n <\/mml:mover>\n <\/mml:math>\n <\/jats:inline-formula>\n remains asymptotically unbiased. We further prove that measurement errors induce a spurious negative correlation between pretest scores and ngains, even when prior knowledge and learning capacity are statistically independent.\n <\/jats:p>\n <\/jats:sec>\n \n Discussion<\/jats:title>\n Such correlations may reflect insufficient instrument reliability rather than any inherent flaw in the transformation. These findings generalize beyond ngains: any nonlinear derived score computed from fallible instruments is susceptible to the same bias structure, and the analytical approach developed here offers a methodological template applicable to other ratio-based metrics in educational and psychological measurement. For applied researchers, we recommend computing both estimators and treating a large discrepancy as a warning sign, reporting instrument reliability alongside ngain estimates, and interpreting pretest-ngain correlations conditionally on reliability.<\/jats:p>\n <\/jats:sec>","DOI":"10.3389\/fpsyg.2026.1901493","type":"journal-article","created":{"date-parts":[[2026,9,1]],"date-time":"2026-09-01T05:34:16Z","timestamp":1788240856000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Normal gains: estimators of learning rates in pretest-posttest settings"],"prefix":"10.3389","volume":"17","author":[{"given":"Jairo A.","family":"Navarrete-Ulloa","sequence":"first","affiliation":[{"name":"Institute of Education Sciences, Universidad de O'Higgins","place":["Rancagua, Chile"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Valentina","family":"Giaconi","sequence":"additional","affiliation":[{"name":"Institute of Education Sciences, Universidad de O'Higgins","place":["Rancagua, Chile"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gonzalo","family":"Contador","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica, Universidad Tecnica Federico Santa Maria","place":["Valpara\u00edso, Chile"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1965","published-online":{"date-parts":[[2026,9,1]]},"reference":[{"key":"B1","doi-asserted-by":"publisher","first-page":"917","DOI":"10.1119\/1.2213632","article-title":"Theoretical comparisons of average normalized gain calculations","volume":"74","author":"Bao","year":"2006","journal-title":"Am. 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