{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T18:33:55Z","timestamp":1770489235029,"version":"3.49.0"},"reference-count":21,"publisher":"Oxford University Press (OUP)","issue":"2","license":[{"start":{"date-parts":[[1973,1,1]],"date-time":"1973-01-01T00:00:00Z","timestamp":94694400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[1973,1,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>The set of product-type interaction contrasts, which contains the subsets of interaction residuals, tetrad contrasts, double-dichotomy contrasts and pooled-tetrad contrasts, is discussed. In the Normal case with equal and known variances, simultaneous confidence intervals for a given set of such contrasts can be constructed in several ways. Five such methods (the Scheff\u00e9 method, the Tukey method, the Dunn method, a method based on Roy's maximum-root statistic and a modification of Tukey's method) are presented and compared in terms of the widths of the resulting 95 and 99 per cent confidence intervals. The Dunn method is found to yield the shortest intervals for interaction residuals, for tetrad contrasts and for double-dichotomy contrasts. On the other hand, for the set of pooled-tetrad contrasts, and therefore also for any larger set of product-type interaction contrasts, the maximum-root method yields the shortest intervals. A table of the half-widths of the intervals given by the recommended methods is provided.<\/jats:p>","DOI":"10.1111\/j.2517-6161.1973.tb00954.x","type":"journal-article","created":{"date-parts":[[2018,12,7]],"date-time":"2018-12-07T07:00:49Z","timestamp":1544166049000},"page":"234-244","source":"Crossref","is-referenced-by-count":18,"title":["Simultaneous Confidence Intervals for Product-Type Interaction Contrasts"],"prefix":"10.1093","volume":"35","author":[{"given":"K. R.","family":"Gabriel","sequence":"first","affiliation":[{"name":"Hebrew University"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J.","family":"Putter","sequence":"additional","affiliation":[{"name":"Israel Agricultural Research Organization"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Y.","family":"Wax","sequence":"additional","affiliation":[{"name":"Hebrew University"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"286","published-online":{"date-parts":[[2018,12,5]]},"reference":[{"key":"2024090920482684500_rssb00954-cit-0001","article-title":"Analysis of variance of the factorial two-way model: A study of the structure of interactions","volume-title":"Ph.D. thesis","author":"Bradu","year":"1970"},{"key":"2024090920482684500_rssb00954-cit-0002","volume-title":"Simultaneous statistical inference on interactions in two-way analysis of variance","author":"Bradu","year":"1971"},{"key":"2024090920482684500_rssb00954-cit-0003","doi-asserted-by":"crossref","first-page":"505","DOI":"10.1093\/biomet\/55.3.505","article-title":"Percentage points of the extreme roots of a Wishart matrix","volume":"55","author":"Choudary-Hanumara","year":"1968","journal-title":"Biometrika"},{"key":"2024090920482684500_rssb00954-cit-0004","doi-asserted-by":"crossref","first-page":"149","DOI":"10.1080\/00401706.1960.10489889","article-title":"Locating outliers in factorial experiments","volume":"2","author":"Daniel","year":"1960","journal-title":"Technometrics"},{"key":"2024090920482684500_rssb00954-cit-0005","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1214\/aoms\/1177706443","article-title":"Estimation of the means of dependent variables","volume":"29","author":"Dunn","year":"1958","journal-title":"Ann. Math. Statist."},{"key":"2024090920482684500_rssb00954-cit-0006","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1080\/01621459.1961.10482090","article-title":"Multiple comparisons among means","volume":"66","author":"Dunn","year":"1961","journal-title":"J. Amer. Statist. Ass."},{"key":"2024090920482684500_rssb00954-cit-0007","doi-asserted-by":"crossref","first-page":"153","DOI":"10.2307\/2346257","article-title":"Preliminary graphical analysis of quasi-independence for two-way contingency tables","volume":"18","author":"Fienberg","year":"1969","journal-title":"Appl. Statist."},{"key":"2024090920482684500_rssb00954-cit-0008","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/BF02289676","article-title":"A statistical model which combines features of factor-analytic and analysis-of-variance techniques","volume":"33","author":"Gollob","year":"1968","journal-title":"Psychometrika"},{"issue":"5","key":"2024090920482684500_rssb00954-cit-0009","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1080\/00031305.1970.10477224","article-title":"Multiple comparison procedures for interactions","volume":"24","author":"Harter","year":"1970","journal-title":"Amer. Statistician"},{"key":"2024090920482684500_rssb00954-cit-0010","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1111\/j.1467-842X.1959.tb00275.x","article-title":"Basing decisions on an analysis of variance","volume":"1","author":"Konijn","year":"1959","journal-title":"Austral. J. Statist."},{"key":"2024090920482684500_rssb00954-cit-0011","doi-asserted-by":"crossref","first-page":"309","DOI":"10.6028\/jres.073B.031","article-title":"The partitioning of interaction in analysis of variance","volume":"73","author":"Mandel","year":"1969","journal-title":"J. Res. Nat. Bur. Stand., B"},{"key":"2024090920482684500_rssb00954-cit-0012","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1080\/00401706.1971.10488751","article-title":"A new analysis-of-variance model for non-additive data","volume":"13","author":"Mandel","year":"1971","journal-title":"Technometrics"},{"key":"2024090920482684500_rssb00954-cit-0013","volume-title":"Handbook of Statistical Tables.","author":"Owen","year":"1962"},{"key":"2024090920482684500_rssb00954-cit-0014","volume-title":"Some Aspects of Multivariate Analysis.","author":"Roy","year":"1957"},{"key":"2024090920482684500_rssb00954-cit-0015","first-page":"87","article-title":"A method of judging all contrasts in the analysis of variance","volume":"40","author":"Scheff\u00e9","year":"1953","journal-title":"Biometrika"},{"key":"2024090920482684500_rssb00954-cit-0016","volume-title":"The Analysis of Variance.","author":"Scheff\u00e9","year":"1959"},{"key":"2024090920482684500_rssb00954-cit-0017","first-page":"245","article-title":"Simultaneous interval estimates of linear functions of parameters","volume":"38","author":"Scheff\u00e9","year":"1959","journal-title":"Bull. Int. Statist. Inst."},{"key":"2024090920482684500_rssb00954-cit-0018","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1214\/aoms\/1177697184","article-title":"Multiple testing versus multiple estimation; improper confidence sets; estimation of directions and ratios","volume":"41","author":"Scheff\u00e9","year":"1970","journal-title":"Ann. Math. Statist."},{"key":"2024090920482684500_rssb00954-cit-0019","first-page":"290","article-title":"A generalization of the T-method of multiple comparisons for interactions","volume":"64","author":"Sen","year":"1969","journal-title":"J. Amer. Statist. Ass."},{"key":"2024090920482684500_rssb00954-cit-0020","first-page":"626","article-title":"Rectangular confidence regions for the means of multivariate normal distributions","volume":"62","author":"","year":"1967","journal-title":"J. Amer. Statist. Ass."},{"key":"2024090920482684500_rssb00954-cit-0021","volume-title":"The problem of multiple comparisons","author":"Tukey","year":"1953"}],"container-title":["Journal of the Royal Statistical Society Series B: Statistical Methodology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.wiley.com\/onlinelibrary\/tdm\/v1\/articles\/10.1111%2Fj.2517-6161.1973.tb00954.x","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/35\/2\/234\/49096696\/jrsssb_35_2_234.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/academic.oup.com\/jrsssb\/article-pdf\/35\/2\/234\/49096696\/jrsssb_35_2_234.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,9,9]],"date-time":"2024-09-09T21:19:40Z","timestamp":1725916780000},"score":1,"resource":{"primary":{"URL":"https:\/\/academic.oup.com\/jrsssb\/article\/35\/2\/234\/7027420"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1973,1,1]]},"references-count":21,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2018,12,5]]},"published-print":{"date-parts":[[1973,1,1]]}},"URL":"https:\/\/doi.org\/10.1111\/j.2517-6161.1973.tb00954.x","relation":{},"ISSN":["1369-7412","1467-9868"],"issn-type":[{"value":"1369-7412","type":"print"},{"value":"1467-9868","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[1973,1]]},"published":{"date-parts":[[1973,1,1]]}}}