Estimation of individual variations in an unreplicated two-way classification
| dc.contributor.author | Russell, Thomas Solon | en |
| dc.contributor.department | Statistics | en |
| dc.date.accessioned | 2016-02-01T15:31:38Z | en |
| dc.date.available | 2016-02-01T15:31:38Z | en |
| dc.date.issued | 1956 | en |
| dc.description.abstract | 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. | en |
| dc.description.degree | Ph. D. | en |
| dc.format.extent | 111 leaves | en |
| dc.format.mimetype | application/pdf | en |
| dc.identifier.uri | http://hdl.handle.net/10919/64673 | en |
| dc.language.iso | en_US | en |
| dc.publisher | Virginia Polytechnic Institute | en |
| dc.relation.isformatof | OCLC# 20470577 | en |
| dc.rights | In Copyright | en |
| dc.rights.uri | http://rightsstatements.org/vocab/InC/1.0/ | en |
| dc.subject.lcc | LD5655.V856 1956.R877 | en |
| dc.subject.lcsh | Error analysis (Mathematics) | en |
| dc.subject.lcsh | Error functions -- Research | en |
| dc.title | Estimation of individual variations in an unreplicated two-way classification | en |
| dc.type | Dissertation | en |
| dc.type.dcmitype | Text | en |
| thesis.degree.discipline | Statistics | en |
| thesis.degree.grantor | Virginia Polytechnic Institute | en |
| thesis.degree.level | doctoral | en |
| thesis.degree.name | Ph. D. | en |
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