Theoretical studies on the necessary number of components in mixtures

Summary

Theoretical studies on the necessary number of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to yielding ability. All theoretical investigations are based upon a Gram-Charlier frequency distribution of the component means with skewness γ₁ and kurtosis γ₂. The selected fraction p of the best components constitutes the mixture under consideration. The same selection differential S = S (p, γ₁, γ₂) can be realized by different parameter values of p, γ₁ and γ₂. Therefore, equal yield levels of the mixture can be achieved by different selected fractions p which implies different numbers of components in the mixture. Numerical results of S = S(p) for different values of γ₁ and γ₂ are presented and discussed. Of particular interest are the selected fractions p which lead to a maximal selection differential S. These results on S for ‘large populations’ must be reduced in the case of finite population size. For this correction term we used an approximation B = B (p, n, γ₁, γ₂) given by Burrows (1972) where n = number of selected components. For given parameter values of γ₁, γ₂ and p, the necessary number n of components can be calculated by using the condition: Burrows-correction less than a certain percentage g of S — for example with g = 0.05 or g = 0.01. For given γ₁ and γ₂, the number n leading to a maximal selection differential S can be regarded as necessary number of components (necessary = maximum gain of selection under the given conditions). Numerical results are given for γ₂ = 0 and for eight situations which are defined by linear relations γ₂ = c γ₁ between skewness and kurtosis. These cases will contain all possible numerical situations for γ₁ and γ₂, which may be relevant for practical applications. The necessary number of components turns out to be nearly independent of the numerical value of the kurtosis γ₂. The n-intervals leading to selected fractions p from 0.01 to 0.20 approximately are: 2 ≤ n ≤ 4 for g = 0.05, 6 ≤ n ≤ 20 for g = 0.01 and 11 ≤ n ≤ 40 for g = 0.005, respectively. However, percentages g less than 0.01 would be unrealistically excessive. Therefore, following the assumptions and restrictions given in this paper one may conclude that n = 20 seems to be an appropriate upper bound for the necessary number of components in mixtures.