教育研究的統計方法(2)--單因子變異數分析與多重比較法

變異數分析(Analysis of Variance， 簡稱ANOVA)為R.A.Fisher於1923年所創用，係一種基於F分配(F-distributions)而與實驗設計密切結合的統計方法。該法最初應用於農業試驗結果之分析，後來逐漸推廣至其他科學的研究領域，目前在心理學與教育研究方面採用此一方法以處理資料者相當普遍。在行為科學研究中最常見的統計假設為「平均數相等」，t檢定(t test)為考驗兩組平均數差異顯著性的有效方法，但若有兩組以上的平均數，固然仍可兩兩加以比較，不過所需比較的次數隨組數而遞增，且真實的虛無假設被拒斥之機率大於事先預定的顯著水準(level of significance)，此時如用變異數分析，則所有各組平均數之間的差異是否顯著，可同時一次加以考驗，更符合經濟實用的要求。

The analysis of variance is a method by which the sources of variation observed in experimental data may be segregated and analyzed. In all problems where the samples are randomly drawn from normal populations having the same variance， the analysis of variance provides an effective and powerful technique. The simplest type of analysis of variance model is the one in which observations are classified into groups on the basis of a single property. The kn subjects are randomly assigned into each of k treatments in such a way that for each treatment there is n subjects. In this article the analysis of variance for a simple randomized， or completely randomized， design is illustrated. The following steps are involved:
1. Partition the total sum of squares into two components， a within-groups and a between-groups sum of squares， using the appropriate computation formulas.
2. Divide these sums of squares by the associated number of degrees of freedom to obtain 「組內均方」and「組間均方」， the within- and between- group variance estimates.
3. Calculate the F ratio， 「組間均方」/「組內均方」， and refer this to the table F(Table A of the Appendix).
4. If the probability of obtaining the observed F value is small， say， less than .05 or .01， under the null hypothesis， reject that hypothesis.
There are a variety of statistical procedures available for multiple comparison between specific means following analysis of variance if the null hypothesis is rejected. Methods in common use， using a t statistic， the F test， and studentized range， have been developed by Dunnett(1955)， Scheffe(1953)， Tukey(1949)， Newman(1939)， Keuls(1952)， and Duncan(1955， 1957). In terms of per-comparison Type I error， multiple-comparison procedures may be ordered from low to high as follows: Scheffe， Tukey， Newman-Keuls， and Duncan. In terms of Type II error， the order of the procedure is the reverse: Duncan， Newman-Keuls， Tukdy， and Scheffe.