Releve ranking based on a sum of squares criterion

Summary When a very large number of phytosociological types have to be compared, a reduction of the number of relevés is desirable. In this paper a method of relevé selection from given phytosociological tables is suggested. The method is based on a sum of squares criterion. The advantage, in comparison with other selection procedures, is that this method provides a means on the basis of which the efficiency of a relevé selection can be objectively measured.Contribution from the Working Group for Data-Processing in Phytosociology, International Society for Vegetation Science.The work was completed at the Department of Plant Sciences of the University of Western Ontario, London, Canada. We wish to thank Prof. L. Orlóci for the hospitality and the helpful discussions. The work was supported by Italian C.N.R., within the project Promozione qualità dell'ambiente subproject Metodologie matematiche e basi di dati.     

Sample size calculations based on ranking and selection in microarray experiments

Summary .   We develop formulae to calculate sample sizes for ranking and selection of differentially expressed genes among different clinical subtypes or prognostic classes of disease in genome-wide screening studies with microarrays. The formulae aim to control the probability that a selected subset of genes with fixed size contains enough truly top-ranking informative genes, which can be assessed on the basis of the distribution of ordered statistics from independent genes. We provide strategies for conservative designs to cope with issues of unknown number of informative genes and unknown correlation structure across genes. Application of the formulae to a clinical study for multiple myeloma is given.     

On the probability of correct selection for large k populations, with application to microarray data

One frontier of modern statistical research is the problems arising from data sets with extremely large k (>1000) populations, e.g. microarray and neuroimaging data. For many such problems the focus shifts from testing for significance to selecting, filtering, or screening. Classical Ranking and Selection Methodology (RSM) studied the probability of correct selection (PCS). PCS is the probability that the "best" (t = 1) of k populations is truly selected, according to some specified criteria of best. This paper extends and adapts two selection goals from the RSM literature that are suitable for large k problems (d-best and G-best selection). It is then shown how estimation of PCS for selecting multiple (t > 1) populations with d-best and G-best selection can be implemented to provide a useful measure of the quality of a given selection. A simulation study and the application of the proposed method to a benchmark microarray data set show it is an effective and versatile tool for assessing the probability that a particular gene selection or gene filtering step truly obtains the best genes. Moreover, the proposed method is fully general and may be applied to any such extremely large k problem.     

Effective size of populations under partial inbreeding and selection on a set of linked additive genes

The prediction theory of effective population size (Ne) is extended to cover selection on a set of linked additive genes and partial inbreeding (partial selfing or partial full-sib mating). Ne under selection is generally expressed as a function of the cumulative change in frequency of a neutral gene due to the random association between the neutral and selected genes generated by finite sampling. In this study, the association under partial selfing was classified into two types, the association between the neutral and selected genes on the same gamete, and the association between the neutral and selected genes each on the different gametes in the same parent. For partial full-sib mating, an additional association, i.e., the association between the neutral and selected genes each in the different parents in the same family, was included in the model. According to this classification of the association, the coefficient accounting for the cumulative change in frequency of the neutral gene was partitioned into two or three components. A method for computing the partitioned coefficients was obtained from the transition matrix approach, in which the joint effect of linkage, selection and partial inbreeding was taken into account. To assess the joint effects of linkage, selection and partial inbreeding on Ne, numerical computations with the obtained expressions were carried out. The effect of linkage on Ne was generally small, except for an extremely small genome size, while the partial inbreeding resulted in a drastic reduction in Ne. For a given genome size, Ne was essentially independent of the length and number of chromosomes. Some of these results were verified by stochastic simulations.     

A model for population regulation with density- and frequency-dependent selection

The life-cycle of a species with separate generations is divided into a reproduction phase and a growing-up phase. In the reproduction phase we assume random mating and selection due to genotype differences in fecundity of the parents and viability of the offspring. During the growing-up phase we assume a (deterministic) death process in continuous time with death rates for the genotypes which increase linearly with the genotype population sizes.In the absence of genotype differences the model gives logistic population regulation. With genotype differences the model generalizes the usual separate generations selection patterns. In addition to these we exhibit cases with three polymorphic equilibria or with a stable cycle.     

Size of breeding populations required for selection programs

The minimum population size required for selection in order to reduce the effect of genetic drift to a particular level has been considered. The model of Nicholas was extended to include the measurement-error variance in the response variance. Situations where the sex ratios among scored and breeding individuals are unequal are also considered. When the duration of a selection experiment is relatively long, Nicholas' approximation (i.e., assuming that measurement error is negligible relative to drift) is useful in determining the minimum effective population size required. However, the measurement-error variance becomes an important source of variation in short-term ( 5 generations) selection experiments, and should not be ignored.     

The conflict between natural and artificial selection in finite populations

Summary A single locus model of the interaction between natural selection and artificial selection for a quantitative character in a finite population, assuming heterozygote superiority in natural fitness but additive action on the character, has been studied using transition probability matrices.If natural selection is strong enough to create a selection plateau in which genetic variance declines relatively slowly, then the total response to artificial selection prior to the plateau will be much less than that expected in the absence of natural selection, and the half-life of response will be shorter. Such a plateau is likely to have a large proportion, if not all, of the original genetic variance still present. In selection programmes using laboratory animals, it seems likely that the homozygote favoured by artificial selection must be very unfit before such a plateau will occur. A significant decrease in population fitness as a result of artificial selection does not necessarily imply that the metric character is an important adaptive character.These implications of this model of natural selection are very similar to those derived by James (1962) for the optimum model of natural selection. In fact, there seems to be no aspect of the observable response to artificial selection that would enable anyone to distinguish between these two models of natural selection.     

Calculating sample size for studies with expected all-or-none nonadherence and selection bias

Summary .  We develop sample size formulas for studies aiming to test mean differences between a treatment and control group when all-or-none nonadherence (noncompliance) and selection bias are expected. Recent work by Fay, Halloran, and Follmann (2007, Biometrics 63, 465–474) addressed the increased variances within groups defined by treatment assignment when nonadherence occurs, compared to the scenario of full adherence, under the assumption of no selection bias. In this article, we extend the authors' approach to allow selection bias in the form of systematic differences in means and variances among latent adherence subgroups. We illustrate the approach by performing sample size calculations to plan clinical trials with and without pilot adherence data. Sample size formulas and tests for normally distributed outcomes are also developed in a Web Appendix that account for uncertainty of estimates from external or internal pilot data.     

A numerical method for calculating moments of coalescent times in finite populations with selection

A numerical method is developed for solving a nonstandard singular system of second-order differential equations arising from a problem in population genetics concerning the coalescent process for a sample from a population undergoing selection. The nonstandard feature of the system is that there are terms in the equations that approach infinity as one approaches the boundary. The numerical recipe is patterned after the LU decomposition for tridiagonal matrices. Although there is no analytic proof that this method leads to the correct solution, various examples are presented that suggest that the method works. This method allows one to calculate the expected number of segregating sites in a random sample of n genes from a population whose evolution is described by a model which is not selectively neutral.     

Early generation selection for agronomic characters in a potato breeding programme

Summary A random sample of seedlings representing high, medium and poor vigour was studied for tuber colour, tuber shape, eye depth, tuber cracking, tuber yield per plant, average tuber weight and number of tubers per plant in four successive generations (F₁, F₁, F₁C₂, and F₁C₃). Based on the performance of vigour groups in various generations and inter-generation correlation coefficients, we propose a procedure for the elimination of unproductive genotypes early in the breeding programme. The data indicates that seedlings of poor vigour can be discarded at the seedling stage prior to transplantation in the field. The rejection of clones on the basis of tuber colour, tuber shape, eye depth and tuber cracking can also be initiated at the seedling stage. For tuber yield and average tuber weight a negative selection (rejection of poor phenotypes) is suggested from the first clonal generation and for number of tubers, from second clonal generation, until statistically sound replicated trials can be conducted for carrying out positive selection.     

Efficiency of including first-generation information in second-generation ranking and selection: results of computer simulation

Using computer simulation, we evaluated the impact of using first-generation information to increase selection efficiency in a second-generation breeding program. Selection efficiency was compared in terms of increase in rank correlation between estimated and true breeding values (i.e., ranking accuracy), reduction in coefficient of variation of correlation coefficients (i.e., ranking reliability), and increase in realized gain, with best linear unbiased prediction (BLUP). The test populations were generated with varying parameters: selection strategy (forward vs backward selection of parents); number of parents (24∼96); number of crosses per parent (1∼8); heritability (0.05∼0.35); ratio of dominance to additive variance (0∼3); ratio of additive-by-site to additive variance (0∼3); and ratio of dominance-by-site to additive variance (0∼3). The two selection strategies gave distinct results. When parents of the second-generation crosses had been selected via backward selection, adding first-generation information markedly increased selection efficiency. Conversely, when parents had been selected via forward selection, first-generation information provided little increase in efficiency. The amount of increase depended more on heritabilities in both generations and less on dominance and genotype–by–environment effects. Including first-generation information helped more when there were many parents and few crosses per parent in the second generation. Only in the case of extremely low first-generation heritabilities was there no benefit to adding first-generation information in terms of improved ranking reliability and accuracy.