Estimation of individual variations in an unreplicated two-way classification

Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model

x_ij = μ_i + β_ij + ε_ij, i=1,2,...n; j=1,2,...,r,

where

x_ij = observation on the i^th treatment of the j^th block,

μ_i = true mean of the i^th treatment,

β_j = bias of the j^th block,

ε_ij = random error, distributed normally with means zero and variance σ²_j,

and E(x_ij) = μ_i + β_j.

The estimator σ̂²_t, for σ²_t, 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_t = [r(r-1)∑_i(x_ij-x_i.-x_.t+x_..)²-∑_i∑_j(x_ij-x_i.-x_.j+x_..)²] ÷ [(n-1)(r-1)(r-2)]

where x_i., x_.j and x_.. are the means of i^th treatment, j^th block and grand mean respectively. σ̂²_t and Q_t were shown to be identical when σ²_t was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Q_j 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_t/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 Q_t/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(x_ij-x_i.-x_.j+x_..)². It was noted that this statistic may be used to test H_o: σ²_t = σ²against one of H_a₁: σ²_t > σ²; H_a₂: σ²_t < σ² and H_a₃: σ²_t ≠ σ² assuming σ²_j = 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.