Title:
[Not Available].

Abstract:
To quantitatively investigate the competitive effects at the borders of neighbouring plots (using field experiments arranged in plots) an extensive series of field experiments, where the treatments were certain crosses of spruce, had been analysed by two different methods: 1) An analysis with almost complete elimination of the competitive effects at the borders of the plots; and 2) an analysis without explicit consideration and elimination of these competition-border-effects (see: Hühn 1974).These studies on estimating competitive effects in plant stands quantitatively finally resulted in relations (inequalities) between the phenotypic total variances of these two methods of analysis, with (V*) and without (V) eliminating the competition-border-effects: V* [Symbol: see text] V. Furthermore, relations between different variance components were obtained from the analysis of variance computations of these two methods. The main purpose of the present paper is to draw some interesting theoretical conclusions, using only the inequality V* [Symbol: see text] V, about 1) the covariance between treatment(genetic)-effects and competitive effects and 2) about the coefficients of determination and correlation-coefficients between the phenotypic values and the treatment-effects and between the phenotypic values and the competitive effects. To describe these relations quantitatively the ratio f of the competitional variance to the variance of the treatment-effects is especially suitable.In the special case V*=V, which, however, has exceptional practical relevance, one obtains - for a large part of the possible interval (0[Symbol: see text]f[Symbol: see text]2) - for the ratio H of the variance of the treatment-effects to the total phenotypic variance (under certain assumptions this is the broad sense heritability), an interesting interpretation as coefficient of multiple determination R' (phenotypic value dependent on treatment-effects and competitive-effects). For 0[Symbol: see text]f[Symbol: see text]2 this R' is explicitly totally independent of the magnitude of the competitional variance.In the main part of this publication, applications for breeding (positive mass selection) are discussed for the case V*=V, which is of special practical relevance. Starting with a simple formula for the correlation-coefficient between the phenotypic values and the treatment effects, quantitative conditions for a possible application of positive mass selection in breeding are derived; with this the demand of a certain minimal value c of the corresponding coefficient of determination, which must be exceeded, is used as a condition for selection.In the formulation of these conditions one obtains results about: 1) minimal values for H (dependent on c and on the number N of individuals, which must be selected); 2) derivation of f-intervals, where positive mass selection should not be applied (dependent on N, H and c, where an interesting special case arises for very large numbers of individuals (N → ∞)); 3) the computation of the necessary minimal number of individuals which must be selected (dependent on f, H and c), where the case: "correlation between phenotypic values and treatment-effects significantly different from zero", which is of special interest from the point of view of practical breeding, is discussed in detail.