Theoretical studies on the necessary number of components in mixtures

Summary Theoretical studies on the necessary numbers of components in mixtures (for example multiclonal varieties or mixtures of lines) have been performed according to the relations between the juvenile-mature correlations of mixtures and their number of components. For the juvenile-mature correlation r_E based upon the values of the single components (= component means at juvenile and mature ages) and the juvenile-mature correlation r_M based upon the means of mixtures of different components we usually will have r_M>r_E. Furthermore, r_M will increase with an increasing number of components in the mixtures. The effectiveness of an early selection will be mainly determined by the magnitude of the juvenile-mature correlation. If we have r_M>r_E an improvement of early testing can be realized by using mixtures instead of single components. But, what are the necessary numbers of components so that r_M will be sufficiently high to enable an effective early selection of mixtures? Some relations between r_E and r_M can be obtained and conclusions have been derived.The statistical approach significant difference between r_E and r_M for a given numerical value of r_M leads to estimates for the necessary number n of components dependent on r_M, , r_E and N where: N = total number of components, which are available for the composition of mixtures and = error probability. For different tree species r_E can be estimated by an appropriate formula which depends on T with T = time (in years) from planting date until the mature age.Lambeth's formula, for example, has been developed for height growth in pines. For this situation numerical calculations are performed using r_M=0.90 and =0.05. The necessary numbers n for T=5, T=10, T=20 and T=50 are: 6, 9, 10 and 12 (for N=50); 13, 17, 20 and 23 (for N=100); 26, 34, 40 and 46 (for N=200); 38, 51, 60 and 69 (for N=300); 64, 85, 100 and 114 (for N=500) and 128, 171, 199 and 228 (for N=1,000). The dependence of these necessary numbers n of components on different type I errors and different levels of r_M have been investigated numerically.