A Robust Single-stage Method for the One-way ANOVA with Unequal Sample Sizes and Heterogeneous Variances

樣本數不等時單階段異質性變異數分析法

在行為科學研究中我們常使用變異數分析來比較組間的平均數差異，但往往研究者會忽略了統計分析中的組間變異數同質性之假定。二階段變異數法（Two-stage analysis of variance）是利用第一階段的資料求出第二階段必須取得的額外樣本，以處理變異數不同質的問題。單階段變異數法（Singlestage sampling）則做進一步的改良，只取一次樣本會較為簡便可行。然上述的方法皆限制於組間樣本數必須一樣，故本研究應用Hill's轉換提出新的統計量，成為卡方分配，使得組間不同樣本數時單階段法仍可使用，以增加應用的可行性。本文使用電腦模擬方法檢驗發現，不論在不同的樣本數、變異數異質性或不同組數下，本方法在實徵上的第一型錯誤率和名目顯著水準非常接近，而且統計檢定力也比其他的方法來的高。本文並舉一計算範例說明新方法的使用，並提供SAS程式供資料分析者下載使用。

Considering the prevalence of heterogeneous variances and unequal sample sizes appeared in the educational and psychological research, the present study develops a modified single-stage one-way ANOVA by employing Hill's normalizing transformation. The proposed test statistic for the singlestage ANOVA is approximately a chi-square distribution, which makes the critical values are easy to obtain. A simulation and a numerical example for the single-stage ANOVA are also presented to demonstrate the properties of the new technique. It is found that the proposed test statistic is relatively closed to the nominal level under various simulated configurations of sample size, standard deviation ratio, and number of groups. Moreover, the proposed method outperforms other competitors with respect to the empirical power. A numerical example is also demonstrated to enhance the application of the new technique and a SAS program is available at http:\\www.css.ncku.edu.tw/Edu/luh/index.