[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-45607-en":3,"doc-seo-45607-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},45607,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Conditions Under Which Mean Square Ratios in Repeated Measurements Designs Have Exact F-Distributions","The study characterizes covariance-matrix structures that yield exact F-distributions for treatment and interaction variance ratios in repeated measurements analyses of variance. Under multivariate normality, equality of variances and equality of covariances with identical matrices across levels of a second factor are sufficient but not necessary. Necessary and sufficient conditions are framed via equality of variances of differences across all correlated pairs of treatment measures, equivalently expressed through the Box–Geisser–Greenhouse e=1.0 condition, with a related test proposed.","This article was downloaded by:[Michigan State University]  \nOn:15 January 2015,At:00:52  \nPublisher:Taylor &Francis  \nInforma Ltd Registered in England and Wales Registered Number:1072954 Registered  \noffice:Mortimer House,37-41 Mortimer Street,London W1T 3JH,UK  \nJournal of the American StatisticalAssociation  \nPublication details,including instructions for authors andsubscription information:http://www.tandfonline.com/loi/uasa20  \nConditions under Which Mean SquareRatios in Repeated MeasurementsDesigns Have Exact F-Distributions  \nHuynh Huynha&Leonard S.FeldtbUniversity of Saigon,USAUniversity of lowa,USAPublished online:05 Apr 2012.  \nTo cite this article:Huynh Huynh &Leonard S.Feldt(1970)Conditions under Which Mean SquareRatios in Repeated Measurements Designs Have Exact F-Distributions,Journal of the AmericanStatistical Association,65:332,1582-1589  \nTo link to this article:httn://dx doi ora/10.1080/01621459.1970.10481187  \nPLEASE SCROLL DOWN FOR ARTICLE  \nTaylor &Francis makes every effort to ensure the accuracy of all the information(the“Content”)contained in the publications on our platform.However,Taylor &Francis,our agents,and our licensors make no representations or warranties whatsoever as tothe accuracy,completeness,or suitability for any purpose of the Content.Any opinionsand views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor &Francis.The accuracy of the Contentshould not be relied upon and should be independently verified with primary sourcesof information.Taylor and Francis shall not be liable for any losses,actions,claims,proceedings,demands,costs,expenses,damages,and other liabilities whatsoever orhowsoever caused arising directly or indirectly in connection with,in relation to or arisingout of the use of the Content.  \nThis article may be used for research,teaching,and private study purposes.Anysubstantial or systematic reproduction,redistribution,reselling,loan,sub-licensing,systematic supply,or distribution in any form to anyone is expressly forbidden.Terms &Conditions of access and use canbe found at http://www.tandfonline.com/page/terms-and-conditions  \nDownloaded by [Michigan State University] at00:52 15 January 2015  \nConditions Under Which Mean Square Ratios in RepeatedMeasurements Designs Have Exact F-Distributions  \nHUYNH HUYNH and LEONARD S.FELDT*  \nInvestigation is made of the character of the covarisnce matrix which will resultin exact F-distributions for the treatments and interaction variance ratios in re-peated measurements designs.It is shown,assuming multivariate normality,thatthe matrix may exhibit a more general character than is typically implied to beessential.Equality of variances and equality of covariances,with identical matricesfor alllevels of a second treatment factor,are sufficient but not necessary condi-tions.The necessary and sufficient condition s the equality of variances of differ-ences for all pairs of treatment measures assumed to be correlated.An alternativestatement is that the Box-Geisser-Greenhouse parametere=1.0.A test is describedwhich bears on the tenability of this condition.  \n# 1.INTRODUCTION\n\nRepeated measurements analyses of variance,often referred to as random-ized block and split-plot designs [3,13,14],are extensively used in agricul-tural,industrial,and psychological research.In the single factor design thehypothesis of no treatment effect is normally tested by the mean square ratioMSRA=MSA/MSAs,where MSAs is the mean square for interaction of blocksby treatments.It is well known that whenever the measures are normallydistributed,exhibit equal variances under all treatments,and have equal co-variances between all pairs of treatments,this ratio has an exact F-distribution,central if the null hypothesis is true.That is,the mean square ratio is distrib-uted as F if the populations of measures under the k levels of treatment factorA have equal variances and equal covariances when k≥3.If n","cbCaiaW0mAayUW0I","https://ap.wps.com/l/cbCaiaW0mAayUW0I","pdf",438529,2,1,9,"English","en",105,"# Introduction\n# Conditions Under Which the Ratio MSRA in the Randomized Block Design is Distributed as F","[{\"question\":\"What is the main goal of the paper regarding repeated measurements designs?\",\"answer\":\"To determine the covariance-matrix conditions under which mean square ratios for treatment and interaction variance ratios follow exact F-distributions.\"},{\"question\":\"Why are equal variances and equal covariances not strictly necessary?\",\"answer\":\"The paper shows that these conditions are sufficient but not necessary, because exact F-distributions can occur under broader covariance structures.\"},{\"question\":\"What condition is equivalent to the Box–Geisser–Greenhouse parameter e=1.0?\",\"answer\":\"The equivalence is expressed through the necessary and sufficient requirement that the variances of differences for all correlated pairs of treatment measures are equal.\"}]",1783462474,23,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"conditions-under-which-mean-square-ratios-in-repeated-measurements-designs-have-exact-f-distributions","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/conditions-under-which-mean-square-ratios-in-repeated-measurements-designs-have-exact-f-distributions/45607/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-11","2026-07-07",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the main goal of the paper regarding repeated measurements designs?","Question",{"text":75,"@type":76},"To determine the covariance-matrix conditions under which mean square ratios for treatment and interaction variance ratios follow exact F-distributions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are equal variances and equal covariances not strictly necessary?",{"text":80,"@type":76},"The paper shows that these conditions are sufficient but not necessary, because exact F-distributions can occur under broader covariance structures.",{"name":82,"@type":73,"acceptedAnswer":83},"What condition is equivalent to the Box–Geisser–Greenhouse parameter e=1.0?",{"text":84,"@type":76},"The equivalence is expressed through the necessary and sufficient requirement that the variances of differences for all correlated pairs of treatment measures are 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