Theoretical studies on the necessary number of components in mixtures

Summary Theoretical studies on the optimal numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to phenotypic yield stability (measured by the parameter variance). For each component i, i = 1, 2,..., n, a parameter u_i with 0 u_i 1 has been introduced reflecting the different survival and yielding ability of the components. For the stochastic analysis the mean of each u_i is denoted by u ₁ and its variance by _i ² For the character total yield the phenotypic variance V can be explicitly expressed dependent on 1) the number n of components in the mixture, 2) the mean of the _i ² 3) the variance of the _i ² 4) the ratio and 5) the ratio _i ² /² where denotes the mean of the u _i and _u ² is the variance of the u _j. According to the dependence of the phenotypic stability on these factors some conclusions can be easily derived from this V-formula. Furthermore, two different approaches for a calculation of necessary or optimal numbers of components using the phenotypic variance V are discussed: A. Determination of optimal numbers in the sense that a continued increase of the number of components brings about no further significant effect according to stability. B. A reduction of b % of the number of components but nevertheless an unchanged stability can be realized by an increase of the mean of the u _i by 1% (with and _u ² assumed to be unchanged). Numerical results on n (from A) and 1 (from B) are given. Computing the coefficient of variation v for the character total yield and solving for the number n of components one obtains an explicit expression for n dependent on v and the factors 2.-5. mentioned above. In the special case of equal variances, _i ² = _o ² for each i, the number n depends on v, x = (₀/)² and y = (_u/)². Detailed numerical results for n = n (v, x, y) are given. For x 1 and y 1 one obtains n = 9, 20 and 79 for v = 0.30, 0.20 and 0.10, respectively while for x 1 and arbitrary y-values the results are n = 11, 24 and 95.This publication is an extended version of a lecture given at the 1984-EUCARPIA meeting (Section Biometrics in Plant Breeding) in Stuttgart-Hohenheim (Federal Republic of Germany)