Estimation of individual variations in an unreplicated two-way classification
Russell, Thomas Solon
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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.
- Doctoral Dissertations