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題名 | 複因子試驗資料之變方成分分析與顯著性測驗=On the Variance Component Analysis and Test of Significance in Factorial Experimental Results |
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作者姓名(中文) | 魏應澤; | 書刊名 | 中華農學會報 |
卷期 | 30 民49.06 |
頁次 | 頁31-62 |
關鍵詞 | 複因子試驗; 變方成分; 顯著性試驗; |
語文 | 中文(Chinese) |
中文摘要 | 應用變方分析於科學研究之重要,已?人所周知,但為使變方分析臻於至善之境,必先依據參與試驗研究中之因子的性質,確立適當的前提,而後進行變方分析,始能顯現變方分析之卓著功效獲得正確可靠之結論。 一般試驗研究資料中,各種因子在取樣的觀點上,可分為三種類型:第一為固定型,?參試因子各變級係試驗者所特定的,並非由該參試因子所有變級之族?中逢機取得,但視該特定變級為一有限族?;第二為逢機型,?某參試因子各變級,為該因子所有變級之族?中的逢機樣品;第三為混合型,?固定型與逢機型共同存在於同一資料中者,根據此三種不同類型之前提,作者分析二因子及三因子參試之各種設計複因子試驗資料的均方期望值,得結果見表1,2,3,4,5,7及8。由此等表中可見參試因子變級之取樣法不同(逢機或特定)所得均方期望值各異,因此各種變異原因之均方顯著性測驗方法亦應隨之而改變;在二因子參試之複因子試驗中,僅用SNEDECOR 氏之F值測驗?妥,但在三因子或以上參試之複因子試驗中,則有者需用COCHRAN 試所倡議之近似F值測驗,詳情請參看公式(II.7), (II.12), (II.17), (II.28), (II.48), (II.57)及表6。 關於變方間直線關係F' 值測驗法,本文舉出COCHRAN氏F' 測驗及互換法F' 測驗兩種,其中以COCHRAN氏F' 測驗較佳,尤以當各均方之自由度低小時為然,蓋COCHRAN氏F' 分布較近於SNEDECOR 氏分布故也。此種測驗法雖未臻至善之境,但在沒有更精密方法研究出來以前,只好採用之。 在混合型場合下所求得均方期望值內容,作者發現SNEDECOR (7.) KEMPTHORNE(5.), OSTLE(6.)等諸氏所求得結果略異,例如二因子參試之複因子試驗中(II.14)氏εMSA=σ2δ+ na/(a-1).σ2αβ+ nb/(a-1).Σσ2j,但諸氏求得為εMSA=σ2ε+nσ2αβ+ nb/(a-1).Σσ2j,兩式比較之餘僅關於σ2αβ之係數前者多乘 a/(a-1)值耳。此 a/(a-1)值可認為係一種有限族?改正項(Finite population correction),蓋固定型因子變級自成一有限族?,其與逢機型因子變級(無限族?)之交感變方應有此改正項之存在。當該固定型因子變級數漸增大時,此 a/(a-1)值則趨近於1,而與諸氏之結果相一致。其他(II.16), (II.41), (II.44), (II.45), (II.47), (II.50), (II.51), (II.53),(II.54), (II.55)及(II.56)等式皆有類似情形。 |
英文摘要 | 1. Before we interpret analysis of variance applied to a set of experimental or survey data, certain assumptions must be made. If we desire our data to be capable of rigorous analysis and our conclusions accompained with reliable statements, such assumptions must be of a mathematical nature. They shall be considered below from the point of view of both nature of materials and sampling methods. According to EISENHART, there are two very different but useful assumptions approaches to problems involving statistical inferences, i.e. fixed modele and random model, and then proceed to particular cases where these assumptions will play an important role. Now let us consider a two-factor factorial experiment in a completely randomized design. Assuming that the observed values Xijk are given additively, the mathematical model which is customarily assumed for situations involving two factors in a completely randomized design is Xijk=μ+αj+βk+(αβ)jk+εijk, (i=1, 2,…, n, j=1, 2 ,…, a, k=1 , 2 ,…, b) where Xijk is normally distributed about μ as true mean and its distribution is a joint distribution of αj, βk, (αβ)jk andεijk.(μ= population mean, αj= effect of the j-th level of factor A, βk= effect of the k-th level of factor B, (αβ)jk= effect of the interaction of the j-th level of factor A with the k-th level of factor B, εijk= experimental error). A more general case, known as a mixed model, involving both random and fixed elements, that is, a mixture of Models I and II. The expected mean squares under mixed models are given in Tables 3, 7 and 8. 2. The differences among fixed, random and mixed models in the resulting analysis of variance are in the expected mean squares, ane these changes are, of course, reflected in the test procedures. The appropriate tests of hypothesis are clearly indicated by the expected mean squares. The peculiar formulas for tests of hypothesis are given in equations (II-7), (II-12) and (II-17) for a two-factor factorial experiment; and in Table 6, equations (II-28), (II-48) and (II-57) for a three-factor factorial experiment under their respective models. We have considered two criteria for approximate F-test, namely Cochran's F'-criterion and alternative criterion. For instance, we would test the hypothesis σ2α=0 by comparing Cochran's F'-criterion (MSA+MSABC)/(MSAB+MSAC) with F distribution with df1 and df2 degrees of freedom, where df1=(MSA+MSABC)2/ [(MSA)2/(a-1)]+[(MSABC)2/(a-1)(b-1)(c-1)], df2=(MSAB+MSAC)2/[(MSAB)2/(a-1)(b-1)]+[(MSAC)2/(a-1)(c-1)]. Alternatively, we may compare MSA/(MSAB+MSAC-MSABC) with F distribution with df1 and df2 degrees of freedom, where df1=(a-1), df2=(MSAB+MSAC-MSABC)2/[(MSAB)2/(a-1)(b-1)]+[(MSAC)2/(a-1)(c-1)]+[(MSABC)2/(a-1)(b-1)(c-1)]. It is not known how reliable these tests are. It might be expected that the test based on Cochran's F'-criterion would be better, especially when all degrees of freedom of mean squares used in test are small. In samples of practical size, Cochran's F'-value may be distributed more like F than the alternative criterion (2). At best these tests are still very crude. However, Cochran's F'-criterion is recommended for tests of hypothesis until a more exact test may be found. 3. Let us compare the expected mean squares under mixed models in this paper with that of SNEDECOR (7), KEMPTHORNE (5), OSTLE(6), etc.. The only difference is in the coefficent of certain components of variance in an expected mean square. For instance, in the former the expected mean square of factor A for a two-factor factorial experiment is εMSA=σ2δ+na/(a-1).σ2αβ+nb/(a-1).Σσ2j as shown in equation (II-14), while in the latter is σ2δ+nσ2αβ+nb/(a-1).Σσ2j. The only difference in these results is that the former is introduced with the extra factor, a/(a-1) in the coefficient of interaction component of variance. The factor a/(a-1) may be defined as the finite population correction. In this example, the levels of factor A is concerned with a finite population, while the levels of factor B with an infinite population. Consequently, the interaction component of variance has the finite population correction, a/(a-1). When th number of levels of factor A become large, the value a/(a-1) approaches to unity and the equation (II-14) is equal to σ2δ+nσ2αβ+nb/(a-1).Σσ2j. We can also find the finite population corrections in equations (II-16), (II-41), (II-44), (II-45), (II-47), (II-50), (II-51), (II-53), (II-54), (II-55) and (II-56). |
本系統之摘要資訊系依該期刊論文摘要之資訊為主。