연구교신 : 불균형 자료 분석과 가설 검정에 관한 연구

In the present study two sets of unbalanced two-way cross- classification data with and without empty cell(s) were used to evaluate empirically the various sums of squares in the analysis of variance table. Searle(1977) and Searle et.al.(1981) developed a method of computing R(α|μ, β) and R(β|μ, α) by the use of partitioned matrix of X`X for the model of no interaction, interchanging the columns of X in order of α, μ, β and accordingly the elements in b. An alternative way of computing R(α|μ, β), R(β|μ, α) and R(γ|μ, α, β) without interchanging the columns of X has been found by means of(X`X)^- derived, using W_2 = Z`_2Z_2-Z`_2Z_1(Z`_1Z_1)^-Z`_1Z_2. It is true that R(α|μ,β,γ)_∑ = SSA_W and R(β|μ,α,γ)_∑. = SSB_W where SSA_W and SSB_W are sums of squares for the factors A and B in the weighted squares of means analysis and R(γ|μ,α,β) = R(γ|μ,α,β)_∑ for the data without empty cell, but not for the data with empty cell(s). It is also noticed that for the data with empty cells under W - restrictions R(α|μ,β,γ)_W=R(μ,α,β,γ)_W-R(μ,α,β,γ)_W=R(α|μ) and R(β|μ,α,γ)_W=R(μ,α,β,γ)_W-R(μ,α,β,γ)_W=R(β|μ) but R(γ|μ,α,β)_W = R(μ,α,β,γ)_W-R(μ,α,β,γ)_W≠R(γ|μ,α,β). The hypotheses H_0 : K`b = 0 commonly tested were examined in the relation with the corresponding sums of squares for R(α|μ), R(β|μ), R(α|μ,β), R(β|μ,α), R(α|μ,β,γ), R(β|μ,α,γ), and R(γ|μ,α,β) under the restrictions.