##### Abstract

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.