Subset Selection for Exponential Populations: Determination of the Selection Constant

Given are k(≧2) exponential populations differing only in their location parameter. One wishes to choose the best one, that is the population with the largest value of the location parameter. A possible method for solving this problem is to select a subset of the k populations of size at least one which includes the best population with a required confidence P*(k^?1 ≤ P* ≤1). In this paper the required selection constant is determined for different values of k and P*. Also an approximation for the selection constant is derived. A comparison with the exact results is made.     

Selecting the Best Population,Provided It Is Better than a Control: The Unequal Variance Case

Bechhofer and Turnbull (1978) proposed two procedures to compare k normal means with a standard and the procedures guarantee that (1) with probability at least P₀* (specified), no category is selected when the best experimental category is sufficiently worst than the standard, and (2) with probability at least P₁* (specified), the best experimental category is selected when it is sufficiently better than the second best and the standard. For the case of common known variance, they studied a single-stage procedure. For the case of common unknown variance, they studied a two-stage procedure. Under the same formulation of Bechhofer and Turnbull (1978) and for the same selection goals (1) and (2) described above, Wilcox (1984a) proposed a procedure to the case of unknown and unequal variances, and supplied a table of the necessary constants to implement the procedure. This paper considers the case of unknown and unequal variances for the same formulation of Bechhofer and Turnbull, and Wilcox, but assumes that μ₀ is an unknown control. A two-stage procedure is proposed to solve the problem. A lower bound of the probability of a correct selection is derived and it takes the same form as the double integral appeared in Rinott (1978) which was used for the lower bound of the probability of a correct selection for a different selection goal.     

Simultaneous Selection of Extreme Populations: A Subset Selection Approach

The problem of selecting a “best” (largest mean, or smallest mean) population from a collection of k independent populations was formulated and solved by Bechhofer (1954). Gupta (1965) solved another important problem, that of selecting a subset of populations containing the “best” population from the original collection of populations. Since then many variations of the problem have been considered. Tong (1969) and Lewis (1980) have investigated the problem of selecting extreme populations (populations with a largest, and populations with a smallest, mean) with respect to one and two standard populations, respectively. In this paper we study the selection of extreme populations in absence of any standard population. We formulate subset-selection procedures when variances are known and equal, and also in the most general case when they are unknown and unequal. Nonexistence of a single-stage procedure is noted for this latter case (even if variances are equal). A two-stage procedure and some of its associated properties are discussed. Tables needed for application are provided, as is a worked example.     

THREE SELECTIONS ARE BETTER THAN ONE: CLINAL VARIATION OF THERMAL QTL FROM INDEPENDENT SELECTION EXPERIMENTS IN DROSOPHILA

We report the results of two independent selection experiments that have exposed distinct populations of Drosophila melanogaster to different forms of thermal selection. A recombinant population derived from Arvin California and Zimbabwe isofemale lines was exposed to laboratory natural selection at two temperatures (T_AZ: 18°C and 28°C). Microsatellite mapping identified quantitative trait loci (QTL) on the X‐chromosome between the replicate “Hot” and “Cold” populations. In a separate experiment, disruptive selection was imposed on an outbred California population for the “knockdown” temperature (T_KD) in a thermal column. Microsatellite mapping of the “High” and “Low” populations also uncovered primarily X‐linked QTL. Notably, a marker in the shaggy locus at band 3A was significantly differentiated in both experiments. Finer scale mapping of the 3A region has narrowed the QTL to the shaggy gene region, which contains several candidate genes that function in circadian rhythms. The same allele that was increased in frequency in the High T_KD populations is significantly clinal in North America and is more common at the warm end of the cline (Florida vs. Maine; however, the cline was not apparent in Australia). Together, these studies show that independent selection experiments can uncover the same target of selection and that evolution in the laboratory can recapitulate putatively adaptive clinal variation in nature.     

On Choosing Among Several Experimental Treatments and a Control

The two classical selection approaches in comparing experimental treatments with a control are combined to form an integrated approach. In this integrated approach, there is a preference zone (PZ) and an indifference zone (IZ), and the concept of a correct decision (CD) is defined differently in each of these zones. In the PZ, we are required to select the best treatment for a correct decision (CD₁) but in the IZ, we define any selected subset to be correct (CD₂) if it contains the best treatment among all the experimental treatments and the controlled treatment. We propose a single-stage procedure R to achieve the selection goals CD₁ and CD₂ simultaneously with certain probability requirements. It is shown that both the probability of a correct decision under PZ, P(CD₁ | PZ), and the probability of a correct decision under IZ, P(CD₂ | IZ), satisfy some monotonicity properties and the least favorable configuration in PZ and the worst configuration in IZ are derived by these properties. We also derive formulas for the probabilities of correct decision and provide a brief table to illustrate the magnitude of the procedure parameters and the common sample sizes needed for various probability requirements and configurations.     

Subset Selection of an Almost Best Treatment

A generalized goal using subset selection is discussed for the location parameter case. This goal is to select a non-empty subset from a set of k (k ≥ 2) treatments that contains at least one ε-best treatment with confidence level P*. For a set of treatments an ε-best treatment is defined as a treatment with location parameter on a distance less than or equal to ε(ε ≥o) from the best treatment, where best is defined as largest value of the location parameter. The efficiency of subset selection of an ε-best treatment relative to subset selection of the best treatment is investigated and is computed for some values of k and the confidence level for the Normal case as well as for the Logistic case.     

Optimal restricted phenotypic selection

Phenotypic selection is modified by introducing upper limits on the portion (P ₁) of individuals selected from a family as well as on the portion (P ₂) of family number that are allowed to contribute. At a preset selection proportion, P and P ₁, the maximum genetic gain is obtained by finding an optimum restriction on family number (P ₂ ^* ). A numerical procedure for solving the problem of optimization is developed for infinite populations. In small populations, maximum gain and P ₂ ^* can be found by simply comparing all possible P₂. Numerical examples are demonstrated for infinite breeding populations, assuming a normally-distributed family mean and within-family deviation. Selection and its simulation were applied to the fieldtest results of two tree species. Optimum restriction on family number is very close to P/P ₁, especially when heritability is low. In the real world of tree breeding, P ₂ ^* is given, or approximated, by P/P ₁+1/ _tm where m is the initial family number. The improvement of gain and the conservation of inbreeding effective population size are easy with high heritability and could be simultaneously obtained by using intense selection with a relatively low P ₁.     

Hybrid maize breeding with doubled haploids: I. One-stage versus two-stage selection for testcross performance

Optimum allocation of resources is of fundamental importance for the efficiency of breeding programs. The objectives of our study were to (1) determine the optimum allocation for the number of lines and test locations in hybrid maize breeding with doubled haploids (DHs) regarding two optimization criteria, the selection gain ΔG _k and the probability P _k of identifying superior genotypes, (2) compare both optimization criteria including their standard deviations (SDs), and (3) investigate the influence of production costs of DHs on the optimum allocation. For different budgets, number of finally selected lines, ratios of variance components, and production costs of DHs, the optimum allocation of test resources under one- and two-stage selection for testcross performance with a given tester was determined by using Monte Carlo simulations. In one-stage selection, lines are tested in field trials in a single year. In two-stage selection, optimum allocation of resources involves evaluation of (1) a large number of lines in a small number of test locations in the first year and (2) a small number of the selected superior lines in a large number of test locations in the second year, thereby maximizing both optimization criteria. Furthermore, to have a realistic chance of identifying a superior genotype, the probability P _k of identifying superior genotypes should be greater than 75%. For budgets between 200 and 5,000 field plot equivalents, P _k > 75% was reached only for genotypes belonging to the best 5% of the population. As the optimum allocation for P _k (5%) was similar to that for ΔG _k , the choice of the optimization criterion was not crucial. The production costs of DHs had only a minor effect on the optimum number of locations and on values of the optimization criteria. C. Friedrich H. Longin and H. Friedrich Utz contributed equally to this work.     

SMALL POPULATION GENETIC VARIABILITY AT LOCI UNDER STABILIZING SELECTION

Genetic variability at a locus under stabilizing selection in a finite population is investigated using analytic methods and computer simulations. Three measures are examined: the number of alleles k, heterozygosity H, and additive genetic variance Vg. A nearly-neutral theory results. The composite parameter S = NV_M/V_s (where N is the population size, V_M the variance of new mutant allelic effects and V_s the weakness of stabilizing selection) figures prominently in the results. The equilibrium heterozygosity is similar to that of strictly neutral theory, H = 4N_μc/ (1 + 4N_μc), except that μ_c = where c is about 0.5. Simulations corroborate except for very low N. Genetic variability attains similar equilibrium values at both a “lone” locus and at an “embedded” locus. This agrees with my earlier work concerning molecular clock rates. These results modify the neutralist interpretation of data concerning genetic variability and genetic distances between populations. Low H values are proportional not to N but to . This may explain the narrow observed range of H among species. Heterozygosities need not be highly correlated to genetic variances. Genetic variances are not highly dependent on population size except in very small populations which are difficult to sample without bias because the smallest populations go extinct the fastest. Nearly neutral evolution will not be easily distinguished from strictly neutral theory under the Hudson-Kreitman-Aguade inter-/intraspecific variation ratio test, since a similar effective mutation rate holds for genetic distances and D = 2μ_ct, where . As with strictly neutral theory, comparisons across loci should show D and H to be positively correlated because of the shared μ_c. But unlike neutral theory, for a given locus, comparisons across species should show D and H to be negatively correlated. There is no obvious threshold of population size below which genetic variability inevitably declines. Extinction depends on both genetic variation and natural selection. Neither theory nor observation presently indicates the measure of genetic variability (k, H, V_G or other) that best indicates vulnerability of a small population to extinction.     

Least Favorable Parameter Configurations for a Step‐down Subset Selection Procedure

Given a k‐dimensional vector X , k≥2, the range‐type statistic with J⊆I≡{1, …, k}, plays an important role in stepwise subset selection as well as in testing whether a<?tlb> prespecified subset of k populations exclusively consists of good ones. Although in previous papers least favorable parameter configurations (LFC's) for this statistic, which are worth knowing for the calculations of critical values, have been already shown to be from a small finite subset of the parameter space, further reduction has been conjectured. Under the assumption of a log‐concave and symmetric Lebesgue density with shift parameter, it is proved that in many cases the LFC can be uniquely given or, at least, found among only a few candidates. The resulting step‐down selection procedure will be illustrated for data from a balanced incomplete block design.     

Selection on the morphology–physiology‐performance nexus: Lessons from freshwater stickleback morphs

Conspecifics inhabiting divergent environments frequently differ in morphology, physiology, and performance, but the interrelationships amongst traits and with Darwinian fitness remains poorly understood. We investigated population differentiation in morphology, metabolic rate, and swimming performance in three‐spined sticklebacks (Gasterosteus aculeatus L.), contrasting a marine/ancestral population with two distinct freshwater morphotypes derived from it: the “typical” low‐plated morph, and a unique “small‐plated” morph. We test the hypothesis that similar to plate loss in other freshwater populations, reduction in lateral plate size also evolved in response to selection. Additionally, we test how morphology, physiology, and performance have evolved in concert as a response to differences in selection between marine and freshwater environments. We raised pure‐bred second‐generation fish originating from three populations and quantified their lateral plate coverage, burst‐ and critical swimming speeds, as well as standard and active metabolic rates. Using a multivariate Q_ST‐F_ST framework, we detected signals of directional selection on metabolic physiology and lateral plate coverage, notably demonstrating that selection is responsible for the reduction in lateral plate coverage in a small‐plated stickleback population. We also uncovered signals of multivariate selection amongst all bivariate trait combinations except the two metrics of swimming performance. Divergence between the freshwater and marine populations exceeded neutral expectation in morphology and in most physiological and performance traits, indicating that adaptation to freshwater habitats has occurred, but through different combinations of traits in different populations. These results highlight both the complex interplay between morphology, physiology and performance in local adaptation, and a framework for their investigation.     

Testing Equivalence Simultaneously for Location and Dispersion of two Normally Distributed Populations

In clinical trials with an active control usually therapeutical equivalence of a new treatment is investigated by looking at a location parameter of the distributions of the primary efficacy variable. But even if the location parameters are close to each other existing differences in variability may be connected with different risks for under or over treatment in an individual patient. Assuming normally distributed responses a multiple test procedure applying two shifted one-sided t-tests for the mean and accordingly two one-sided F-tests for the variances is proposed. Equivalence in location and variability is established if all four tests lead to a rejection at the (one-sided) level α. A conservative procedure “correcting” the t-tests for heteroscedasticity is derived. The choice of a design in terms of the global level α, the global power, the relevant deviations in the population means and variances, as well as the sample size is outlined. Numerical calculations of the actual level and power for the proposed designs show, that for balanced sample sizes the classical uncorrected one-sided t-tests can be used safely without exaggerating the global type I error probability. Finally an example is given.     

An Integrated Formulation for Comparing Treatments with Binary Response with a Control

The problem of comparing k(≧2) bernoulli rates of success with a control is considered. An one-stage decision procedure is proposed for either (1) choosing the best among several experimental treatments and the control treatment when the best is significantly superior or (2) selecting a random size subset that contains the best experimental treatment if it is better than the control when the difference between the best and the remaining treatments is not significant. We integrate two traditional formulations, namely, the indifference (IZ) approach and the subset selection (SS) approach, by seperating the parameter space into two disjoint sets, the preference zone (PZ) and the indifference zone (IZ). In the PZ we insist on selecting the best experimental treatment for a correct selection (CS₁) but in the IZ we define any selected subset to be correct (CS₂) if it contains the best experimental treatment which is also better than the control. We propose a procedure R to guarantee lower bounds P₁* for P(CS₁≧PZ) and P₂* for P(CS₂≧IZ) simultaneously. A brief table on the common sample size and the procedure parameters is presented to illustrate the procedure R.     

GENETIC HITCHHIKING AND THE DYNAMIC BUILDUP OF GENOMIC DIVERGENCE DURING SPECIATION WITH GENE FLOW

A major issue in evolutionary biology is explaining patterns of differentiation observed in population genomic data, as divergence can be due to both direct selection on a locus and genetic hitchhiking. “Divergence hitchhiking” (DH) theory postulates that divergent selection on a locus reduces gene flow at physically linked sites, facilitating the formation of localized clusters of tightly linked, diverged loci. “Genome hitchhiking” (GH) theory emphasizes genome‐wide effects of divergent selection. Past theoretical investigations of DH and GH focused on static snapshots of divergence. Here, we used simulations assessing a variety of strengths of selection, migration rates, population sizes, and mutation rates to investigate the relative importance of direct selection, GH, and DH in facilitating the dynamic buildup of genomic divergence as speciation proceeds through time. When divergently selected mutations were limiting, GH promoted divergence, but DH had little measurable effect. When populations were small and divergently selected mutations were common, DH enhanced the accumulation of weakly selected mutations, but this contributed little to reproductive isolation. In general, GH promoted reproductive isolation by reducing effective migration rates below that due to direct selection alone, and was important for genome‐wide “congealing” or “coupling” of differentiation (F_ST) across loci as speciation progressed.     

SINGLE‐GENE SPECIATION WITH PLEIOTROPY: EFFECTS OF ALLELE DOMINANCE,POPULATION SIZE,AND DELAYED INHERITANCE

Single‐gene speciation is considered to be unlikely, but an excellent example is found in land snails, in which a gene for left‐right reversal has given rise to new species multiple times. This reversal might be facilitated by their small population sizes and maternal effect (i.e., “delayed inheritance,” in which an individual's phenotype is determined by the genotype of its mother). Recent evidence suggests that a pleiotropic effect of the speciation gene on antipredator survival may also promote speciation. Here we theoretically demonstrate that, without a pleiotropic effect, in small populations the fixation probability of a recessive mutant is higher than a dominant mutant, but they are identical for large populations and sufficiently weak selection. With a pleiotropic effect that increases mutant viability, a dominant mutant has a higher fixation probability if the strength of viability selection is sufficiently greater than that of reproductive incompatibility, whereas a recessive mutant has a higher fixation probability otherwise. Delayed inheritance increases the fixation probability of a mutant if viability selection is sufficiently weaker than reproductive incompatibility. Our results clarify the conflicting effects of viability selection and positive frequency‐dependent selection due to reproductive incompatibility and provide a new perspective to single‐gene speciation theory.     

Vergleich der Kot- und Harn-N-Trennung in 2 verschiedenen Ratten-Stoffwechselkäfigsystemen

The energy availability from the feedstuffs and the energy requirement of animals are characterized by the metabolizable energy. The metabolizable energy of the feed (feedstuff, ration) is interpreted as potential energy.

For comparative analyses within and between animal species with regard to the energetic feed value of single feedstuffs the parameter “relative fat retention effect” is introduced.

The metabolizable energy of a ration is estimated in consideration of live weight and protein production (protein in body gain and egg). The energy requirement is calculated factorial ‐ as a rule by the partial utilization of metabolizable energy for partial performances (k_f, k _p, k _o) in connection with the level and kind of partial performance; k _m is not used in the system.

The proposed system of energetic feed evaluation is available as “computer variant” as well as “manual variant”. The computer programme is fitted first of all to application in research and teaching. (The complete documentation of the proposal for energetic feed evaluation for fowls including a diskette for using of the computer programme for scientific application can be claimed in limited extend from the author.)     

SEXUAL SELECTION CAN INCREASE THE EFFECT OF RANDOM GENETIC DRIFT—A QUANTITATIVE GENETIC MODEL OF POLYMORPHISM IN OOPHAGA PUMILIO,THE STRAWBERRY POISON‐DART FROG

The variation in color pattern between populations of the poison‐dart frog Oophaga pumilio across the Bocas del Toro archipelago in Panama is suggested to be due to sexual selection, as two other nonsexually selecting Dendrobatid species found in the same habitat and range do not exhibit this variation. We theoretically test this assertion using a quantitative genetic sexual selection model incorporating aposematic coloration and random drift. We find that sexual selection could cause the observed variation via a novel process we call “coupled drift.” Within our model, for certain parameter values, sexual selection forces frog color to closely follow the evolution of female preference. Any between‐population variation in preference due to genetic drift is passed on to color. If female preference in O. pumilio is strongly affected by drift, whereas color in the nonsexually selecting Dendrobatid species is not, coupled drift will cause increased between‐population phenotypic variation. However, with different parameter values, coupled drift will result in between‐population variation in color being suppressed compared to its neutral value, or in little or no effect. We suggest that coupled drift is a novel theoretical process that could have a role linking sexual selection with speciation both in O. pumilio, and perhaps more generally.     

GENETIC AND MORPHOMETRIC DIVERGENCE IN ANCESTRAL EUROPEAN AND DESCENDENT NEW ZEALAND POPULATIONS OF CHAFFINCHES (FRINGILLA COELEBS)

Descendent populations of chaffinches (Fringilla coelebs) introduced to New Zealand about 120 years ago were compared with “ancestral” populations in northern Europe and with those in a broader region of Europe (including Iberia) using protein electrophoresis at 42 loci and 12 skeletal measurements. The New Zealand populations exhibit very small scale differentiation in genetics (F_st = 0.040) and morphometrics, and the haphazard pattern of among-population variation does not align with environmental variation nor is it predicted by the geographic proximity of populations. Thus random drift is implicated in the differentiation among the descendent populations. The New Zealand chaffinches have diverged only slightly in morphometrics from an extant population in southern England, and constant heritability rate tests suggest that random drift alone could account for this small shift. In sharp contrast, the European populations are subdivided genetically (F_st = 0.222) and morphometrically, and this subdivision coincides with the Pyrenees mountains between Iberia and northern Europe which act as a barrier to gene flow between these regions. Iberian populations have smaller skulls and longer wings on average than northern European populations and are characterized by high frequencies of alternative common alleles at Ada and Np. Within both the Iberian and northern European regions, however, populations are effectively panmictic in protein-encoding genes, indicating that homogenizing gene flow is apparently extensive enough to prevent among-population differentiation in allozymes by drift. Variation in body size as represented by PC I is related to environmental productivity across Europe, unlike in New Zealand. These observations jointly suggest that longer term adaptive differentiation via selection for optimal body size has evolved in Europe. Because multilocus evolution is expected to proceed slowly in populations subject to the opposing forces of selection and homogenizing gene flow, I argue that local adaptation within “ancestral” populations in northern Europe may still be evolving.     

The effect of trait type and strength of selection on heritability and evolvability in an island bird population

The heritability (h²) of fitness traits is often low. Although this has been attributed to directional selection having eroded genetic variation in direct proportion to the strength of selection, heritability does not necessarily reflect a trait's additive genetic variance and evolutionary potential (“evolvability”). Recent studies suggest that the low h² of fitness traits in wild populations is caused not by a paucity of additive genetic variance (V_A) but by greater environmental or nonadditive genetic variance (V_R). We examined the relationship between h² and variance‐standardized selection intensities (i or β_σ), and between evolvability (I_A:V_A divided by squared phenotypic trait mean) and mean‐standardized selection gradients (β_μ). Using 24 years of data from an island population of Savannah sparrows, we show that, across diverse traits, h² declines with the strength of selection, whereas I_A and I_R (V_R divided by squared trait mean) are independent of the strength of selection. Within trait types (morphological, reproductive, life‐history), h², I_A, and I_R are all independent of the strength of selection. This indicates that certain traits have low heritability because of increased residual variance due to the age at which they are expressed or the multiple factors influencing their expression, rather than their association with fitness.