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. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/64673 |
| Date | January 1956 |
| Creators | Russell, Thomas Solon |
| Contributors | Statistics |
| Publisher | Virginia Polytechnic Institute |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | 111 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20470577 |
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