{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/64673"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/64673","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"Estimation of individual variations in an unreplicated two-way classification","abstract":"Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model xij = μi + βij + εij, i=1,2,...n; j=1,2,...,r, where xij = observation on the ith treatment of the jth block, μi = true mean of the ith treatment, βj = bias of the jth block, εij = random error, distributed normally with means zero and variance σ²j, and E(xij) = μi + βj. The estimator σ̂²t, for σ²t, t=1,2,3,...,r, was derived for n ≥ 2 and r = 3, by applying the principle of maximum likelihood to a set of (n-1)(r-1) transformed variables usually ascribed to error. Equations were derived for the maximum likelihood estimators for n ≥ 2 and r ≥ 3. A general quadratic form was used and when four reasonable assumptions were applied, estimators of the variances were obtained in for form of Qt = [r(r-1)∑i(xij-xi.-x.t+x..)²-∑i∑j(xij-xi.-x.j+x..)²] ÷ [(n-1)(r-1)(r-2)] where xi., x.j and x.. are the means of ith treatment, jth block and grand mean respectively. σ̂²t and Qt were shown to be identical when σ²t was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Qj is equal to the estimators proposed by Grubbs [J.A.S.A., Vol. 43 (1948)] and Ehrenberd [Biometrika, Vol 37. (1950).] It was shown that Qt/σ² = [(r-1)²x(n-1)²-x(n-1)(r-2)²]/[(n-1)(r-1)(r-2)], a linear difference of two independent central chi-square variates. The statistic Q/E was derived such that Qt/E = [(((r-1)²)/(1+(r-2)F))-1]/[(n-1)(r-1)(r-2)] with F, a central F-statistic with (n-1)(r-2) and (n-1) degrees of freedom in the numerator and denominator respectively and E =∑i∑j(xij-xi.-x.j+x..)². It was noted that this statistic may be used to test Ho: σ²t = σ²against one of Ha₁: σ²t > σ²; Ha₂: σ²t < σ² and Ha₃: σ²t ≠ σ² assuming σ²j = σ², j≠t, j=1,2,...,r. A final test was of homogeneity of variances when r = 3 and was based on - 2 ln λ = (n-1)[2 ln (n-1) + ln(Q₁Q₂+Q₁Q₃+Q₂Q₃) - 2 ln E + ln 4/3], where λ is a likelihood ratio and -2 ln λ is approximately distributed as x² with 2 degrees of freedom for large n. A more general statistic for testing homogeneity of variance for r ≥ 3 was proposed and its distribution discussed in a special case.","abstract_html":"Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model x<sub>ij</sub> = μ<sub>i</sub> + β<sub>ij</sub> + ε<sub>ij</sub>, i=1,2,...n; j=1,2,...,r, where x<sub>ij</sub> = observation on the i<sup>th</sup> treatment of the j<sup>th</sup> block, μ<sub>i</sub> = true mean of the i<sup>th</sup> treatment, β<sub>j</sub> = bias of the j<sup>th</sup> block, ε<sub>ij</sub> = random error, distributed normally with means zero and variance σ²<sub>j</sub>, and E(x<sub>ij</sub>) = μ<sub>i</sub> + β<sub>j</sub>. The estimator σ̂²<sub>t</sub>, for σ²<sub>t</sub>, t=1,2,3,...,r, was derived for n ≥ 2 and r = 3, by applying the principle of maximum likelihood to a set of (n-1)(r-1) transformed variables usually ascribed to error. Equations were derived for the maximum likelihood estimators for n ≥ 2 and r ≥ 3. A general quadratic form was used and when four reasonable assumptions were applied, estimators of the variances were obtained in for form of Q<sub>t</sub> = [r(r-1)∑<sub>i</sub>(x<sub>ij</sub>-x<sub>i.</sub>-x<sub>.t</sub>+x<sub>..</sub>)²-∑<sub>i</sub>∑<sub>j</sub>(x<sub>ij</sub>-x<sub>i.</sub>-x<sub>.j</sub>+x<sub>..</sub>)²] ÷ [(n-1)(r-1)(r-2)] where x<sub>i.</sub>, x<sub>.j</sub> and x<sub>..</sub> are the means of i<sup>th</sup> treatment, j<sup>th</sup> block and grand mean respectively. σ̂²<sub>t</sub> and Q<sub>t</sub> were shown to be identical when σ²<sub>t</sub> was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Q<sub>j</sub> is equal to the estimators proposed by Grubbs [J.A.S.A., Vol. 43 (1948)] and Ehrenberd [Biometrika, Vol 37. (1950).] It was shown that Q<sub>t</sub>/σ² = [(r-1)²x<sub>(n-1)</sub>²-x<sub>(n-1)(r-2)</sub>²]/[(n-1)(r-1)(r-2)], a linear difference of two independent central chi-square variates. The statistic Q/E was derived such that Q<sub>t</sub>/E = [(((r-1)²)/(1+(r-2)F))-1]/[(n-1)(r-1)(r-2)] with F, a central F-statistic with (n-1)(r-2) and (n-1) degrees of freedom in the numerator and denominator respectively and E =∑<sub>i</sub>∑<sub>j</sub>(x<sub>ij</sub>-x<sub>i.</sub>-x<sub>.j</sub>+x<sub>..</sub>)². It was noted that this statistic may be used to test H<sub>o</sub>: σ²<sub>t</sub> = σ²against one of H<sub>a₁</sub>: σ²<sub>t</sub> > σ²; H<sub>a₂</sub>: σ²<sub>t</sub> < σ² and H<sub>a₃</sub>: σ²<sub>t</sub> ≠ σ² assuming σ²<sub>j</sub> = σ², j≠t, j=1,2,...,r. A final test was of homogeneity of variances when r = 3 and was based on - 2 ln λ = (n-1)[2 ln (n-1) + ln(Q₁Q₂+Q₁Q₃+Q₂Q₃) - 2 ln E + ln 4/3], where λ is a likelihood ratio and -2 ln λ is approximately distributed as x² with 2 degrees of freedom for large n. A more general statistic for testing homogeneity of variance for r ≥ 3 was proposed and its distribution discussed in a special case.","abstract_has_math":false,"creators":["Russell, Thomas Solon"],"institution":"Virginia Polytechnic Institute","degree_name":"Ph. D.","degree_level":"doctoral","degree_discipline":"Statistics","degree_department":"Statistics","school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":1956,"date_issued":"1956","date_published":"1956","updated_at":"2026-07-22T22:18:44Z","subjects":[],"languages":["en_US"],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10919/64673","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.department","label":"Department","values":["Statistics"]},{"key":"dc:creator","label":"Author","values":["Russell, Thomas Solon"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2016-02-01T15:31:38Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2016-02-01T15:31:38Z"]},{"key":"dc:date.issued","label":"Date","values":["1956"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Polytechnic Institute"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"dc:type.dcmitype","label":"Dc Type Dcmitype","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Statistics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph. D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en_US"]},{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/64673"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model xij = μi + βij + εij, i=1,2,...n; j=1,2,...,r, where xij = observation on the ith treatment of the jth block, μi = true mean of the ith treatment, βj = bias of the jth block, εij = random error, distributed normally with means zero and variance σ²j, and E(xij) = μi + βj. The estimator σ̂²t, for σ²t, t=1,2,3,...,r, was derived for n ≥ 2 and r = 3, by applying the principle of maximum likelihood to a set of (n-1)(r-1) transformed variables usually ascribed to error. Equations were derived for the maximum likelihood estimators for n ≥ 2 and r ≥ 3. A general quadratic form was used and when four reasonable assumptions were applied, estimators of the variances were obtained in for form of Qt = [r(r-1)∑i(xij-xi.-x.t+x..)²-∑i∑j(xij-xi.-x.j+x..)²] ÷ [(n-1)(r-1)(r-2)] where xi., x.j and x.. are the means of ith treatment, jth block and grand mean respectively. σ̂²t and Qt were shown to be identical when σ²t was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Qj is equal to the estimators proposed by Grubbs [J.A.S.A., Vol. 43 (1948)] and Ehrenberd [Biometrika, Vol 37. (1950).] It was shown that Qt/σ² = [(r-1)²x(n-1)²-x(n-1)(r-2)²]/[(n-1)(r-1)(r-2)], a linear difference of two independent central chi-square variates. The statistic Q/E was derived such that Qt/E = [(((r-1)²)/(1+(r-2)F))-1]/[(n-1)(r-1)(r-2)] with F, a central F-statistic with (n-1)(r-2) and (n-1) degrees of freedom in the numerator and denominator respectively and E =∑i∑j(xij-xi.-x.j+x..)². It was noted that this statistic may be used to test Ho: σ²t = σ²against one of Ha₁: σ²t > σ²; Ha₂: σ²t < σ² and Ha₃: σ²t ≠ σ² assuming σ²j = σ², j≠t, j=1,2,...,r. A final test was of homogeneity of variances when r = 3 and was based on - 2 ln λ = (n-1)[2 ln (n-1) + ln(Q₁Q₂+Q₁Q₃+Q₂Q₃) - 2 ln E + ln 4/3], where λ is a likelihood ratio and -2 ln λ is approximately distributed as x² with 2 degrees of freedom for large n. A more general statistic for testing homogeneity of variance for r ≥ 3 was proposed and its distribution discussed in a special case."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph. D."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Estimation of individual variations in an unreplicated two-way classification"]}]}],"canonical_facts":{"dc:contributor.department":["Statistics"],"dc:creator":["Russell, Thomas Solon"],"dc:date.accessioned":["2016-02-01T15:31:38Z"],"dc:date.available":["2016-02-01T15:31:38Z"],"dc:date.issued":["1956"],"dc:description.abstract":["Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model xij = μi + βij + εij, i=1,2,...n; j=1,2,...,r, where xij = observation on the ith treatment of the jth block, μi = true mean of the ith treatment, βj = bias of the jth block, εij = random error, distributed normally with means zero and variance σ²j, and E(xij) = μi + βj. The estimator σ̂²t, for σ²t, t=1,2,3,...,r, was derived for n ≥ 2 and r = 3, by applying the principle of maximum likelihood to a set of (n-1)(r-1) transformed variables usually ascribed to error. Equations were derived for the maximum likelihood estimators for n ≥ 2 and r ≥ 3. A general quadratic form was used and when four reasonable assumptions were applied, estimators of the variances were obtained in for form of Qt = [r(r-1)∑i(xij-xi.-x.t+x..)²-∑i∑j(xij-xi.-x.j+x..)²] ÷ [(n-1)(r-1)(r-2)] where xi., x.j and x.. are the means of ith treatment, jth block and grand mean respectively. σ̂²t and Qt were shown to be identical when σ²t was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Qj is equal to the estimators proposed by Grubbs [J.A.S.A., Vol. 43 (1948)] and Ehrenberd [Biometrika, Vol 37. (1950).] It was shown that Qt/σ² = [(r-1)²x(n-1)²-x(n-1)(r-2)²]/[(n-1)(r-1)(r-2)], a linear difference of two independent central chi-square variates. The statistic Q/E was derived such that Qt/E = [(((r-1)²)/(1+(r-2)F))-1]/[(n-1)(r-1)(r-2)] with F, a central F-statistic with (n-1)(r-2) and (n-1) degrees of freedom in the numerator and denominator respectively and E =∑i∑j(xij-xi.-x.j+x..)². It was noted that this statistic may be used to test Ho: σ²t = σ²against one of Ha₁: σ²t > σ²; Ha₂: σ²t < σ² and Ha₃: σ²t ≠ σ² assuming σ²j = σ², j≠t, j=1,2,...,r. A final test was of homogeneity of variances when r = 3 and was based on - 2 ln λ = (n-1)[2 ln (n-1) + ln(Q₁Q₂+Q₁Q₃+Q₂Q₃) - 2 ln E + ln 4/3], where λ is a likelihood ratio and -2 ln λ is approximately distributed as x² with 2 degrees of freedom for large n. A more general statistic for testing homogeneity of variance for r ≥ 3 was proposed and its distribution discussed in a special case."],"dc:description.degree":["Ph. D."],"dc:format.mimetype":["application/pdf"],"dc:identifier.uri":["http://hdl.handle.net/10919/64673"],"dc:language.iso":["en_US"],"dc:publisher":["Virginia Polytechnic Institute"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["Estimation of individual variations in an unreplicated two-way classification"],"dc:type":["Dissertation"],"dc:type.dcmitype":["Text"],"thesis:degree_discipline":["Statistics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Ph. D."],"thesis:institution_name":["Virginia Polytechnic Institute"]},"updated_at":"2026-07-22T22:18:44Z"}