{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,17]],"date-time":"2026-09-17T03:02:36Z","timestamp":1789614156380,"version":"build-2803163510"},"reference-count":21,"publisher":"Wiley","issue":"5","license":[{"start":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T00:00:00Z","timestamp":1787702400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"},{"start":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T00:00:00Z","timestamp":1787702400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/doi.wiley.com\/10.1002\/tdm_license_1.1"}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["Pharmaceutical Statistics"],"published-print":{"date-parts":[[2026,9]]},"abstract":"<jats:title>ABSTRACT<\/jats:title>\n                  <jats:p>\n                    Linear mixed models are widely used for pharmaceutical stability trending when sufficient lots are available. Expiry support is often based on whether lot\u2010specific conditional\u2010mean confidence limits remain within specification through a proposed expiry, and these limits depend on the denominator degrees\u2010of\u2010freedom (DDF) method used for\n                    <jats:italic>t<\/jats:italic>\n                    \u2010based inference. We document an operationally important boundary\u2010proximal phenomenon: when a fitted random\u2010effect variance component is close to zero, Satterthwaite DDF\u2014and, in sensitivity analyses, Kenward\u2013Roger DDF\u2014for conditional\u2010mean predictions can collapse, inflating  critical values and producing very wide and sometimes nonmonotone pointwise confidence limits on scheduled time grids. In the settings studied here, containment DDF yields stable degrees of freedom and avoids sharp discontinuities as variance components approach the boundary. In a balanced random\u2010intercept design, a closed\u2010form calculation gives the conditional prediction variance and delta\u2010method Satterthwaite DDF, showing why DDF can be extremely small near the boundary when the prediction point is near the design centroid. A worked example and simulation studies show that DDF choice can materially change pass\/fail conclusions even when the observed measurements themselves are not close to specification limits. Containment\u2010based inference with the full random\u2010effects model provides a single framework without model switching. When containment is unavailable, a 10% variance\u2010contribution reduction procedure mitigates extreme Satterthwaite behavior by simplifying only when fitted contributions at the proposed expiry are negligible. AICc step\u2010down is best treated as a sensitivity analysis because it can be too aggressive when the proposed\u2010expiry margin is small.\n                  <\/jats:p>","DOI":"10.1002\/pst.70114","type":"journal-article","created":{"date-parts":[[2026,8,26]],"date-time":"2026-08-26T08:44:10Z","timestamp":1787733850000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Degrees\u2010of\u2010Freedom Approximations for Conditional\u2010Mean Inference in Random\u2010Lot Stability Analysis"],"prefix":"10.1002","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5933-8706","authenticated-orcid":false,"given":"Andrew T.","family":"Karl","sequence":"first","affiliation":[{"name":"Karl Statistical Services LLC  Aurora Colorado USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Heath","family":"Rushing","sequence":"additional","affiliation":[{"name":"Adsurgo LLC  San Antonio Texas USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard K.","family":"Burdick","sequence":"additional","affiliation":[{"name":"Burdick Statistical Consulting, LLC  Colorado Springs Colorado USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jeff","family":"Hofer","sequence":"additional","affiliation":[{"name":"JDH CMC Stats, LLC  Avon Indiana USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2026,8,26]]},"reference":[{"key":"e_1_2_12_2_1","unstructured":"International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use \u201cICH Harmonised Tripartite Guideline: Evaluation for Stability Data Q1E \u201d(2003) International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use Guideline Q1E Current Step 4 Version https:\/\/database.ich.org\/sites\/default\/files\/Q1E_Guideline.pdf."},{"key":"e_1_2_12_3_1","unstructured":"International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use \u201cICH Harmonised Guideline: Stability Testing of Drug Substances and Drug Products Q1 \u201d(2025) International Council for Harmonisation of Technical Requirements for Pharmaceuticals for Human Use Draft guideline Q1 Step 2b Draft Version https:\/\/database.ich.org\/sites\/default\/files\/ICH_Q1EWG_Step2_Draft_Guideline_2025_0411.pdf."},{"key":"e_1_2_12_4_1","volume-title":"SAS\/STAT User's Guide: The MIXED Procedure, MODEL Statement","author":"SAS Institute Inc","year":"2026"},{"key":"e_1_2_12_5_1","volume-title":"JMP Pro 19.1: Mixed Model Report and Options","author":"JMP Statistical Discovery LLC","year":"2025"},{"key":"e_1_2_12_6_1","volume-title":"JMP Pro 19.1: Statistical Details for the Kackar\u2013Harville Correction","author":"JMP Statistical Discovery 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\u201cSimulation Code for Conditional\u2010Mean Inference in Random\u2010Lot Stability Models \u201d(2026) https:\/\/doi.org\/10.17632\/4hcmr7gfcn.4."},{"key":"e_1_2_12_21_1","doi-asserted-by":"publisher","DOI":"10.2307\/2527669"},{"key":"e_1_2_12_22_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-01689-9"}],"container-title":["Pharmaceutical 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