High-resolution Monte Carlo study of the order-parameter distribution of the three-dimensional Ising model.

We apply extensive Monte Carlo simulations to study the probability distribution P(m) of the order parameter m for the simple cubic Ising model with periodic boundary condition at the transition point. Sampling is performed with the Wolff cluster flipping algorithm, and histogram reweighting together with finite-size scaling analyses are then used to extract a precise functional form for the probability distribution of the magnetization, P(m), in the thermodynamic limit. This form should serve as a benchmark for other models in the three-dimensional Ising universality class.

Pushing the limits of Monte Carlo simulations for the three-dimensional Ising model.

While the three-dimensional Ising model has defied analytic solution, various numerical methods like Monte Carlo, Monte Carlo renormalization group, and series expansion have provided precise information about the phase transition. Using Monte Carlo simulation that employs the Wolff cluster flipping algorithm with both 32-bit and 53-bit random number generators and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising Model, with lattice sizes ranging from 16^{3} to 1024^{3}. By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, e.g., logarithmic derivatives of magnetization and derivatives of magnetization cumulants, we have obtained the critical inverse temperature K_{c}=0.221654626(5) and the critical exponent of the correlation length ν=0.629912(86) with precision that exceeds all previous Monte Carlo estimates.

Exact sample sizes needed to detect dependence in 2 x 3 tables.

Many medical and biological studies entail classifying a number of observations according to two factors, where one has two and the other three possible categories. This is the case of, for example, genetic association studies of complex traits with single-nucleotide polymorphisms (SNPs), where the a priori statistical planning, analysis, and interpretation of results are of critical importance. Here, we present methodology to determine the minimum sample size required to detect dependence in 2 x 3 tables based on Fisher's exact test, assuming that neither of the two margins is fixed and only the grand total N is known in advance. We provide the numerical tools necessary to determine these sample sizes for desired power, significance level, and effect size, where only the computational time can be a limitation for extreme parameter values. These programs can be accessed at . This solution of the sample size problem for an exact test will permit experimentalists to plan efficient sampling designs, determine the extent of statistical support for their hypotheses, and gain insight into the repeatability of their results. We apply this solution to the sample size problem to three empirical studies, and discuss the results with specified power and nominal significance levels.

Detection of deleterious genotypes in multigenerational studies. III. Estimation of selection components in highly selfing populations.

New paradigms in genetics have increased the chance of finding genes that appear redundant but in fact may have been preserved due to a small level of positive selection potential acting during each generation. Monitoring changes in genotypic frequencies within and between generations allows the dissection of the fertility, viability and meiotic drive selection components acting on such genes in natural and experimental populations. Here, a formal maximum likelihood procedure is developed to identify and estimate these selection components in highly selfing populations by fitting the time-dependent solutions for genotypic frequencies to observed multigenerational counts. With adult census alone, we can not simultaneously estimate all three selection components considered. In such cases, we instead consider a hierarchy of 11 models with either fewer selection components, complete dominance, or multiplicative meiotic drive with a single parameter. We identify the best-fitting of these models by applying likelihood ratio tests to nested models and Akaike's Information Criterion (AIC) and the Bayesian Information Criterion (BIC) to non-nested models. With seed census, fertility and viability selection are not distinguishable and thus can only be estimated jointly. A combination of joint seed and adult census data allows us to estimate all three selection components simultaneously. Simulated data validate the estimation procedure and provide some practical guidelines for experimental design. An application to Arabidopsis data establishes that viability selection is the major selective force acting on the ACT2 actin gene in laboratory-grown Arabidopsis populations.

Cluster hybrid Monte Carlo simulation algorithms.

We show that addition of Metropolis single spin flips to the Wolff cluster-flipping Monte Carlo procedure leads to a dramatic increase in performance for the spin-1/2 Ising model. We also show that adding Wolff cluster flipping to the Metropolis or heat bath algorithms in systems where just cluster flipping is not immediately obvious (such as the spin-3/2 Ising model) can substantially reduce the statistical errors of the simulations. A further advantage of these methods is that systematic errors introduced by the use of imperfect random-number generation may be largely healed by hybridizing single spin flips with cluster flipping.