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.     

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.     

A Selection Procedure for Selecting Good Exponential Populations

Consider k independent exponential populations with location parameters μ₁,…, μ_k and a common scale parameter or standard deviation θ. Let μ_(k) be the largest of the μ's and define a population to be good if its location parameter exceeds μ_(k) –Δ₁. A selection procedure is proposed to select a subset of the k populations which includes the good populations with probability at least P*, a pre-assigned value. Simultaneous confidence intervals, that can be derived with the proposed selection procedure, are discussed. Moreover, if populations with locations below μ_(k) –δ₂, (δ₂ > δ₁) are “bad”, a selection procedure is proposed and a sample size is determined so that the probability of omitting a “good” population or selecting a “bad” population is at most 1 – P*.     

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.     

A Selection and Testing Procedure for Comparing Several Experimental Treatments with a Control

We consider a selection and testing procedure for comparing k experimental treatments with a control treatment where the treatments are assumed to be normally distributed with unknown means and a common, unknown variance. Stein‐type sampling is used in the selection phase to screen for an experimental treatment that exhibits evidence of being better than the control treatment and each of the other experimental treatments, where better is defined in terms of the largest mean. In the testing phase, the best experimental treatment is compared to the control using a hypothesis test. If no experimental treatment indicates that it is an improvement over the control during the selection phase, our procedure allows for early termination. We provide definitions of level and power appropriate for our hybrid procedure and compute procedure parameters required to implement our procedure.     

Subset Selection Procedures for Randomized Designs

Starting from the principle that there exists a randomization procedure that assigns treatments to experimental units, four subset selection rules for the problem of selecting the best treatment from a set of different treatments are proposed. Two of these are extensions of already existing subset selection procedures, which were defined for unbalanced designs, and need a separate selection constant for each individual treatment. The other two rules proposed are new and need only one selection constant for all treatments. The various procedures are compared, and illustrated by application to a plant breeding variety trial.     

Two-Stage Procedures for Comparing Treatments with a Control: Elimination at the First Stage and Estimation at the Second Stage

We consider the problem of comparing a set of p₁ test treatments with a control treatment. This is to be accomplished in two stages as follows: In the first stage, N₁ observations are allocated among the p₁ treatments and the control, and the subset selection procedure of Gupta and Sobel (1958) is employed to eliminate “inferior” treatments. In the second stage, N₂ observations are allocated among the (randomly) selected subset of p₂(≤p₁) treatments and the control, and joint confidence interval estimates of the treatment versus control differences are calculated using Dunnett's (1955) procedure. Here both N₁ and N₂ are assumed to be fixed in advance, and the so-called square root rule is used to allocate observations among the treatments and the control in each stage. Dunnett's procedure is applied using two different types of estimates of the treatment versus control mean differences: The unpooled estimates are based on only the data obtained in the second stage, while the pooled estimates are based on the data obtained in both stages. The procedure based on unpooled estimates uses the critical point from a p₂-variate Student t-distribution, while that based on pooled estimates uses the critical point from a p₁-variate Student t-distribution. The two procedures and a composite of the two are compared via Monte Carlo simulation. It is shown that the expected value of p₂ determines which procedure yields shorter confidence intervals on the average. Extensions of the procedures to the case of unequal sample sizes are given. Applicability of the proposed two-stage procedures to a drug screening problem is discussed.     

Construction and Analysis of an Augmented Lattice Square Design

Augmented designs are useful for screening experiments involving large numbers of new and untried treatments. Since resolvable row‐column designs are useful for controlling extraneous variation, it is desirable to use such designs for the check or standard treatments to construct augmented lattice square experiment designs. A simple procedure is described for constructing such designs using c = 2k and c = 3k check treatments and n = rk(k ‐— 2) and n = rk(k — 3) new treatments, respectively, r being the number of complete blocks. A trend analysis for these designs, which allows for solutions of fixed effects, is presented. The random effects case is also discussed. A SAS computer code and the output from this code illustrated with a small numerical example are available from the author.     

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.     

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.     

Pairwise Comparisons Using Trimmed Means or M-Estimators when Working with Dependent Groups

For J dependent groups, let θ_j, j = 1, …, J, be some measure of location associated with the jth group. A common goal is computing confidence intervals for the pairwise differences, θ_j — θ_k, j < k, such that the simultaneous probability coverage is 1 — α. If means are used, it is well known that slight departures from normality (as measured by the Kolmogorov distance) toward a heavy-tailed distribution can substantially inflate the standard error of the sample mean, which in turn can result in relatively low power. Also, when distributions differ in shape, or when sampling from skewed distributions with relatively light tails, practical problems arise when the goal is to obtain confidence intervals with simultaneous probability coverage reasonably close to the nominal level. Extant theoretical and simulation results suggest replacing means with trimmed means. The Tukey-McLaughlin method is easily adapted to the problem at hand via the Bonferroni inequality, but this paper illustrates that practical concerns remain. Here, the main result is that the percentile t bootstrap method, used in conjunction with trimmed means, gives improved probability coverage and substantially better power. A method based on a one-step M-estimator is also considered but found to be less satisfactory.     

Cadmium Free Cu2ZnSnS4 Solar Cells with 9.7% Efficiency

Cu₂ZnSnS₄(CZTS) thin‐film solar cell absorbers with different bandgaps can be produced by parameter variation during thermal treatments. Here, the effects of varied annealing time in a sulfur atmosphere and an ordering treatment of the absorber are compared. Chemical changes in the surface due to ordering are examined, and a downshift of the valence band edge is observed. With the goal to obtain different band alignments, these CZTS absorbers are combined with Zn_1?xSn_xO_y (ZTO) or CdS buffer layers to produce complete devices. A high open circuit voltage of 809 mV is obtained for an ordered CZTS absorber with CdS buffer layer, while a 9.7% device is obtained utilizing a Cd free ZTO buffer layer. The best performing devices are produced with a very rapid 1 min sulfurization, resulting in very small grains.     

Optimal control of a vectored plant disease model for a crop with continuous replanting

ABSTRACT

Vector-transmitted diseases of plants have had devastating effects on agricultural production worldwide, resulting in drastic reductions in yield for crops such as cotton, soybean, tomato, and cassava. Plant-vector-virus models with continuous replanting are investigated in terms of the effects of selection of cuttings, roguing, and insecticide use on disease prevalence in plants. Previous models are extended to include two replanting strategies: frequencyreplanting and abundance-replanting. In frequency-replanting, replanting of infected cuttings depends on the selection frequency parameter ε, whereas in abundance-replanting, replanting depends on plant abundance via a selection rate parameter also denoted as ε. The two models are analysed and new thresholds for disease elimination are defined for each model. Parameter values for cassava, whiteflies, and African cassava mosaic virus serve as a case study. A numerical sensitivity analysis illustrates how the equilibrium densities of healthy and infected plants vary with parameter values. Optimal control theory is used to investigate the effects of roguing and insecticide use with a goal of maximizing the healthy plants that are harvested. Differences in the control strategies in the two models are seen for large values of ε. Also, the combined strategy of roguing and insecticide use performs better than a single control.     

Mining statistically-solid k-mers for accurate NGS error correction

Background

NGS data contains many machine-induced errors. The most advanced methods for the error correction heavily depend on the selection of solid k-mers. A solid k-mer is a k-mer frequently occurring in NGS reads. The other k-mers are called weak k-mers. A solid k-mer does not likely contain errors, while a weak k-mer most likely contains errors. An intensively investigated problem is to find a good frequency cutoff f₀ to balance the numbers of solid and weak k-mers. Once the cutoff is determined, a more challenging but less-studied problem is to: (i) remove a small subset of solid k-mers that are likely to contain errors, and (ii) add a small subset of weak k-mers, that are likely to contain no errors, into the remaining set of solid k-mers. Identification of these two subsets of k-mers can improve the correction performance.

Results

We propose to use a Gamma distribution to model the frequencies of erroneous k-mers and a mixture of Gaussian distributions to model correct k-mers, and combine them to determine f₀. To identify the two special subsets of k-mers, we use the z-score of k-mers which measures the number of standard deviations a k-mer’s frequency is from the mean. Then these statistically-solid k-mers are used to construct a Bloom filter for error correction. Our method is markedly superior to the state-of-art methods, tested on both real and synthetic NGS data sets.

Conclusion

The z-score is adequate to distinguish solid k-mers from weak k-mers, particularly useful for pinpointing out solid k-mers having very low frequency. Applying z-score on k-mer can markedly improve the error correction accuracy.

     

Model Construction and Model Examination on the Basis of Simulated Selection

By reason nonlinear relations founded between selection differential and realised selection response we have been made investigations about variants of the genetic-statistical model, which include this nonlinearity. The variations of the model would not only referred to the postulate pattern of the connection between phenotype, genotype and environment but also enclosed the postulate assumption about the distribution of the variates. In an investigated special case the linear model equation P = G ± e was held, however the distributions of P and G were defined over a limited range in one direction. For P we have defined a modified normal distribution and the distribution of the random vector (G, e) non normal regarded with cov (G, e) ≠ 0, By means of a solution set of an integral equation a density function of the random vector (P, G) has been received, in which the expectation of the selection response of the usual genetic-statistical model approximate is included as a special case. The genetical parameters has been derived, which result from changed model. However their representation was only possible partially as an integral function. A subsequent paper informs of the examination this mode! variants, which depend on a parameter of the nonlinearity c.     

Intrinsic viscosities of cyclic and linear lamda DNA

The ratio of the intrinsic viscosities of the linear and circular forms of λ DNA, [η]L /[η]c, has been measured as a function of ionic strength in the range [Na⁺] = 0.6. M–0.03MCorrections were made for the presence of uncyclizable linear contaminant in circular preparations. By combining data in the literature on the ionic strength dependence of linear DNA of various molecular weights with that obtained here, it was possible to determine the expansion parameter εL as a function of [Na⁺]. εL is defined by the relation 〈L²〉 = b²N^1+εL, where 〈L¹〉 is the mean-square end-to-end distance of a chain of N segments of length b. The empirical relation εL = 0.05 ? 0.11 log [Na⁺] for native NaDNA at 25°C is found. When εL = 0, [η]L /[η]c extrapolates to 1.6, in good agreement with the theoretical prediction of 1.55. As εL increases, [η]L /[η]c increases, in agreement with a theory of Bloomfield and Zimm.     

On Asymptotically Distribution-Free Confidence Bounds for P (X1≥X2) Based on Samples not Necessarily Independent

If X₁, X₂, denote the random variables of measurement under two treatments then the probability P (X₁ ≥ X₂) is a quantity of great practical interest, especially if we consider both to be measured for the same unity. In this case the random variables cannot be assumed to be independent any longer. The following paper describes a procedure to compute approximate confidence bounds for P (X₁ ≥ X₂) where correlations between X₁, X₂ are admitted as well as between replications of the X_j. There is some relation to the FRIEDMAN-statistic with or without repeated measurements and as a special case to the sign-test. Application may be extended to ordinal data.     

A discrete branching process model for the spread of HIV via steady sexual partnerships

The transmission of HIV in a monogamous heterosexual population structured by the ordinal number of the current partnership is considered. The sexual carreer of a man (woman) is thought to be a succession of k(m) partnerships, and a multitype Galton-Watson process is defined, in which the objects are infections and the types are related to the ordinal number of the partnership during which a person has acquired the infection. Contrary to multitype models in which the types are not age-related in some sense, this process contains at least two singular types, namely infections acquired in the last partnership of a man or a woman. The criticality parameter of this branching process is the epidemic threshold parameter R₀. In the case k=m an epidemic is impossible, however large k may be, if the difference between the ordinal numbers of the partners in a pair is never > 1. When the frequency of pairs in which this difference is 2 increases, then R₀ increases. This is demonstrated for the cases k=m=3 and k=4,m=3. The formulae obtained show also the joint influence of the mixing pattern and of variable infectivity. The result for the case of uniform mixing implies that a formula of May and Anderson (1987) is an approximation for k and m large.Mathematics Subject Classification (2000): 92D (primary), 60J (secondary)     

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.     

Adaptive enrichment designs with a continuous biomarker

A popular design for clinical trials assessing targeted therapies is the two-stage adaptive enrichment design with recruitment in stage 2 limited to a biomarker-defined subgroup chosen based on data from stage 1. The data-dependent selection leads to statistical challenges if data from both stages are used to draw inference on treatment effects in the selected subgroup. If subgroups considered are nested, as when defined by a continuous biomarker, treatment effect estimates in different subgroups follow the same distribution as estimates in a group-sequential trial. This result is used to obtain tests controlling the familywise type I error rate (FWER) for six simple subgroup selection rules, one of which also controls the FWER for any selection rule. Two approaches are proposed: one based on multivariate normal distributions suitable if the number of possible subgroups, k, is small, and one based on Brownian motion approximations suitable for large k. The methods, applicable in the wide range of settings with asymptotically normal test statistics, are illustrated using survival data from a breast cancer trial.