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題 名 | 混雜設計二次效應式求法=Fitting a Quadratic Response Equatin to the Data Obtained from the Confounded Factorial Experiment |
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作 者 | 林燦隆; | 書刊名 | 中華農學會報 |
卷 期 | 56 民55.12 |
頁 次 | 頁5-17 |
關鍵詞 | 混雜設計; 二次效應; |
語 文 | 中文(Chinese) |
中文摘要 | 在本文誘導由混雜設計資料估計二次效應式未知回歸係數及進行有關統計擬說測驗所需之計算公式。此等混雜設計包括(一) 4×2×2 混雜設計,(二) 4×4 混雜設計,(三) 4×4×3 混雜設計,(四) 4×4×3 混雜設計(使用第一及第二重複),(五) 4×4×3混雜設計(使用第二及第三重複),(六) 4×3×3 混雜設計,(七) 4×3×3 混雜設計(使用第一及第二重複),(八) 4×3×3 混雜設計(使用第三及第四重複)。田間排列圖係根據文獻(2) |
英文摘要 | Some working formulas necessary for fitting a p-variable quadratic response equation y=b0 +Σbiε'?i(Xi) +Σbiiε'?i(Xi) +ΣΣbijε'?i(Xi)ε'?j(Xj) (ε'?i(Xi) being an orthogonal polynomial in Xi of the first degree, ε'?i(Xi) an orthogonal polynomal in Xi of second degree) to the data from the confounded factorial experiment and for testing statistical hypotheses are derived. The confounded designs considered are: (1) 4×2×2 confounded factorial experiment in blocks of 4 plots (2) 4×4 confounded factorial experiment in blocks of 4 plots (3) 4×4×3 confounded factorial experiment in blocks of 12 plots (4) 4×4×3 confounded factorial experiment in blocks of 12 plots using 1st and 2nd replications (5) 4×4×3 confounded factorial experiment in blocks of 12 plots using 2nd and 3rd replications (6) 4×3×3 confounded factorial experiment in blocks of 12 plots (7) 4×3×3 confounded factorial experiment in blocks of 12 plots using 1st and 2nd replications (8) 4×3×3 confounded factorial experiment in blocks of 12 plots using 3rd and 4th replications. The blocking plans in use were provided by Li (1944). In a confounded factorial experiment, if Yghk is the observed yield of the plot in the hth replication receiving kth factorial treatment (X1k1X2k2…Xpkp), we set up a mathematical model Yghk=A +Rg +Cgh +P(Xiki) +eghk where Rg (g=1, 2, …L) is the effext of gth replication, Cgh (h=1, 2, ...n) is the effect of hth incomplete block in gth replication, A is a constant, Xiki is the amount of the ith (i=1, 2, …, p) factor applied, P(Xiki) is a polynomial in orthogonal form such as P(Xiki)= ΣΣ…Σπεqii(Xiki) for 0<Σqi where ε'qi, is an orthonal polymial in Xi of the qith degree. The number of terms of P(Xiki) coincides with the number of degrees of greedom for treatment sum of squares. If we use "x" to denote that the estimator involces block sums and "w" that does not, and use "2" to denote that the regression coefficient is included in the quardratic response equation and "1" is not, we can divide the regression coefficients in P(Xiki) into four categories, i.e.,bw1, bw2, bx1 and bx2. Hence, using the matrix natation, this mathematical model can be written in the form: (y)= (G: H: W1: X1: W2: X2) (R': C': b'w1: b'x1: b'w2: b'x2)' where (y) is an (Lnm×1) coloumn vector (m being the number of plots in an incomplete block) whose elements are plot yields, G is an (Lnm×(L-1)) matrix of rank (L-1) whose elements are the coefficients of the replication effect satisfying the condition RL=-(R1+ …+RL-1), H is an (Lnm×(L-1)) matrix of rank L(n-1) whose elements are the coefficients of the block effect satisfying the condition Cgn=-ΣCgn and W1, X1, W2 and X2 are the matrices matrices correponding to bw1, bx1, bw2 and bx2 respectively. The coefficients of A are included in W2. By the mathod of least squares, this leads to a set of normal equations whose solution gives: bw2=(W'2W2) -1W'2(y) bx2= (X'X2) -1(X'2(y) -X'2H(H'KH)-1 (H'(y) -H'X1(X1X1)-1 X'1(y) -H'X2 (X2X2)-1 X'2(Y)) where K is an idenpodent matrix (I- X1(X'1X1)-1X'1-X2(X'2X2)-1X'2). The formulas used for estimating bw2 and bx2 for the designs mentioned are numbered from (2-1) to (2-8). The loss of efficientcy in estimating bx2 due to confounding is listed in Table 2. On the testing statistical null hypothesis, two kinds of hypothese are considered: (1) H0:bi=0 i belongs to w2 or x2 (2) H0:bwi= and bx1=0 simultaneously. For testing the first null hypothesis, the use of t-test is assumed. "cii" values necessary for calculating t=b/√ciiMSE are listed in Table 1. For testing the second kind of hypothesis, the use of F-test is assumed, i.e., F=Mean square for lack of fitness/ MSE. The sum of squares for lack of fitness is calculated as follows: s.s for lack of fitness= s.s. for treatment (adjusted for block effects)- ss(bw2, bx2|bw1=0, bx1=0) and ss(bw2, bx2|bw1=0, bx1=0)= b'w2 W'2(y) +ss(bx2|bw1=0) The formulas used for calculating ss(bx2|bw1=0) are derived and numbered form (6-1) to (6-8). |
本系統中英文摘要資訊取自各篇刊載內容。