LIAN 3.0: detecting linkage disequilibrium in multilocus data. Linkage Analysis

SUMMARY: LIAN is a program to test the null hypothesis of linkage equilibrium for multilocus data. LIAN incorporates both a Monte Carlo method as well as a novel algebraic method to carry out the hypothesis test. The program further returns the genetic diversity of the sample and the pairwise distances between its members.     

A transmission/disequilibrium test approach to screen for quantitative trait loci in two selected lines of large white pigs

Pedigree and marker data from a multiple-generation pig selection experiment have been analysed to screen for loci affecting quantitative traits (QTL). Pigs from a base population were selected either for low backfat thickness at fixed live weight (L-line) or high live weight at fixed age (F-line). Selection was based on single-trait own performance and DNA was available on selected individuals only. Genotypes for three marker loci with known positions on chromosome 4 were available. The transmission/disequilibrium test (TDT) was originally described in human genetics to test for linkage between a genetic marker and a disease-susceptibility locus, in the presence of association. Here, we adapt the TDT to test for linkage between a marker and QTL favoured by selection, and for linkage disequilibrium between them in the base population. The a priori unknown distribution of the test statistic under the null hypothesis, no linkage, was obtained via Monte Carlo simulation. Significant TDT statistics were found for markers AFABP and SW818 in the F-line, indicating the presence of a closely linked QTL affecting growth performance. In the L-line, none of the markers studied showed significance. This study emphasizes the potential of the TDT as a quick and simple approach to screen for QTL in situations where marker genotypes are available on selected individuals. The results suggest that previously identified QTL in crosses of genetically diverse breeds may also segregate in commercial selection lines.     

Nonparametric bayesian estimation of positive false discovery rates

We propose a Dirichlet process mixture model (DPMM) for the P-value distribution in a multiple testing problem. The DPMM allows us to obtain posterior estimates of quantities such as the proportion of true null hypothesis and the probability of rejection of a single hypothesis. We describe a Markov chain Monte Carlo algorithm for computing the posterior and the posterior estimates. We propose an estimator of the positive false discovery rate based on these posterior estimates and investigate the performance of the proposed estimator via simulation. We also apply our methodology to analyze a leukemia data set.     

Assessing Equivalence Tests with Respect to their Expected p‐Value

Monte‐Carlo simulation methods are commonly used for assessing the performance of statistical tests under finite sample scenarios. They help us ascertain the nominal level for tests with approximate level, e.g. asymptotic tests. Additionally, a simulation can assess the quality of a test on the alternative. The latter can be used to compare new tests and established tests under certain assumptions in order to determinate a preferable test given characteristics of the data. The key problem for such investigations is the choice of a goodness criterion. We expand the expected p‐value as considered by Sackrowitz and Samuel‐Cahn (1999) to the context of univariate equivalence tests. This presents an effective tool to evaluate new purposes for equivalence testing because of its independence of the distribution of the test statistic under null‐hypothesis. It helps to avoid the often tedious search for the distribution under null‐hypothesis for test statistics which have no considerable advantage over yet available methods. To demonstrate the usefulness in biometry a comparison of established equivalence tests with a nonparametric approach is conducted in a simulation study for three distributional assumptions.     

Testing hypotheses about interclass correlations from familial data

Testing problems concerning interclass correlations from familial data are considered in the case where the number of siblings varies among families. Under the assumption of multivariate normality, two test procedures are proposed for testing the hypothesis that an interclass correlation is equal to a specified value. To compare the properties of the tests, including a likelihood ratio test, Monte Carlo experiments are performed. Several test statistics are derived for testing whether two variables about a parent and child are uncorrelated. The proposed tests are compared with previous test procedures, using Monte Carlo simulation. A general procedure for finding confidence intervals for interclass correlations is also derived.     

Avoiding fallacious significance tests in stepwise regression: a Monte Carlo method applied to a meteorological theory for the Canadian lynx cycle

Stepwise regression is often used in ecology to identify critical factors. From a large number of possible predictors, the procedure selects the subset generating the highest coefficient of determination,R ². This work presents a method for testing the significance of this coefficient. Monte Carlo simulations are used to calculate the statistical distribution ofR ² under the null hypothesis that the response variable is independent of the predictors. The method is illustrated by an application to a previously published analysis of the Canadian lynx population cycle where more than 75% of the variance could be explained by four meteorological factors.     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

On the assessment of statistical significance in disease-gene discovery.

One of the major challenges facing genome-scan studies to discover disease genes is the assessment of the genomewide significance. The assessment becomes particularly challenging if the scan involves a large number of markers collected from a relatively small number of meioses. Typically, this assessment has two objectives: to assess genomewide significance under the null hypothesis of no linkage and to evaluate true-positive and false-positive prediction error rates under alternative hypotheses. The distinction between these goals allows one to formulate the problem in the well-established paradigm of statistical hypothesis testing. Within this paradigm, we evaluate the traditional criterion of LOD score 3.0 and a recent suggestion of LOD score 3.6, using the Monte Carlo simulation method. The Monte Carlo experiments show that the type I error varies with the chromosome length, with the number of markers, and also with sample sizes. For a typical setup with 50 informative meioses on 50 markers uniformly distributed on a chromosome of average length (i.e., 150 cM), the use of LOD score 3.0 entails an estimated chromosomewide type I error rate of.00574, leading to a genomewide significance level >.05. In contrast, the corresponding type I error for LOD score 3.6 is.00191, giving a genomewide significance level of slightly <.05. However, with a larger sample size and a shorter chromosome, a LOD score between 3.0 and 3.6 may be preferred, on the basis of proximity to the targeted type I error. In terms of reliability, these two LOD-score criteria appear not to have appreciable differences. These simulation experiments also identified factors that influence power and reliability, shedding light on the design of genome-scan studies.     

Optimized group sequential study designs for tests of genetic linkage and association in complex diseases

The study of genetic linkage or association in complex traits requires large sample sizes, as the expected effect sizes are small and extremely low significance levels need to be adopted. One possible way to reduce the numbers of phenotypings and genotypings is the use of a sequential study design. Here, average sample sizes are decreased by conducting interim analyses with the possibility to stop the investigation early if the result is significant. We applied optimized group sequential study designs to the analysis of genetic linkage (one-sided mean test) and association (two-sided transmission/disequilibrium test). For designs with two and three stages at overall significance levels of.05 and.0001 and a power of.8, we calculated necessary sample sizes, time points, and critical boundaries for interim and final analyses. Monte Carlo simulation analyses were performed to confirm the validity of the asymptotic approximation. Furthermore, we calculated average sample sizes required under the null and alternative hypotheses in the different study designs. It was shown that the application of a group sequential design led to a maximal increase in sample size of 8% under the null hypothesis, compared with the fixed-sample design. This was contrasted by savings of up to 20% in average sample sizes under the alternative hypothesis, depending on the applied design. These savings affect the amounts of genotyping and phenotyping required for a study and therefore lead to a significant decrease in cost and time.     

Directional Selection and Variation in Finite Populations

Predictions are made of the equilibrium genetic variances and responses in a metric trait under the joint effects of directional selection, mutation and linkage in a finite population. The "infinitesimal model" is analyzed as the limiting case of many mutants of very small effect, otherwise Monte Carlo simulation is used. If the effects of mutant genes on the trait are symmetrically distributed and they are unlinked, the variance of mutant effects is not an important parameter. If the distribution is skewed, unless effects or the population size is small, the proportion of mutants that have increasing effect is the critical parameter. With linkage the distribution of genotypic values in the population becomes skewed downward and the equilibrium genetic variance and response are smaller as disequilibrium becomes important. Linkage effects are greater when the mutational variance is contributed by many genes of small effect than few of large effect, and are greater when the majority of mutants increase rather than decrease the trait because genes that are of large effect or are deleterious do not segregate for long. The most likely conditions for "Muller's ratchet" are investigated.     

A comparison of likelihood ratio tests and Rao's score test for three separable covariance matrix structures

The problem of testing the separability of a covariance matrix against an unstructured variance‐covariance matrix is studied in the context of multivariate repeated measures data using Rao's score test (RST). The RST statistic is developed with the first component of the separable structure as a first‐order autoregressive (AR(1)) correlation matrix or an unstructured (UN) covariance matrix under the assumption of multivariate normality. It is shown that the distribution of the RST statistic under the null hypothesis of any separability does not depend on the true values of the mean or the unstructured components of the separable structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Monte Carlo simulations are then used to study the comparative behavior of the null distribution of the RST statistic, as well as that of the LRT statistic, in terms of sample size considerations, and for the estimation of the empirical percentiles. Our findings are compared with existing results where the first component of the separable structure is a compound symmetry (CS) correlation matrix. It is also shown by simulations that the empirical null distribution of the RST statistic converges faster than the empirical null distribution of the LRT statistic to the limiting χ² distribution. The tests are implemented on a real dataset from medical studies.     

Improving the calculation of statistical significance in genome-wide scans

Calculations of the significance of results from linkage analysis can be performed by simulation or by theoretical approximation, with or without the assumption of perfect marker information. Here we concentrate on theoretical approximation. Our starting point is the asymptotic approximation formula presented by Lander and Kruglyak (1995, Nature Genetics, 11, 241--247), incorporating the effect of finite marker spacing as suggested by Feingold et al. (1993, American Journal of Human Genetics, 53, 234--251). We consider two distinct ways in which this formula can be improved. Firstly, we present a formula for calculating the crossover rate rho for a pedigree of general structure. For a pedigree set, these values may then be weighted into an overall crossover rate which can be used as input to the original approximation formula. Secondly, the unadjusted p -value formula is based on the assumption of a Normally distributed nonparametric linkage (NPL) score. This leads to conservative or anti-conservative p -values of varying magnitude depending on the pedigree set structure. We adjust for non-Normality by calculating the marginal distribution of the NPL score under the null hypothesis of no linkage with an arbitrarily small error. The NPL score is then transformed to have a marginal standard Normal distribution and the transformed maximal NPL score, together with a slightly corrected value of the overall crossover rate, is inserted into the original formula in order to calculate the p -value. We use pedigrees of seven different structures to compare the performance of our suggested approximation formula to the original approximation formula, with and without skewness correction, and to results found by simulation. We also apply the suggested formula to two real pedigree set structure examples. Our method generally seems to provide improved behavior, especially for pedigree sets which show clear departure from Normality, in relation to the competing approximations.     

The Null Distributions of Test Statistics in Genomewide Association Studies

Genomewide association (GWA) studies assay hundreds of thousands of single nucleotide polymorphisms (SNPs) simultaneously across the entire genome and associate them with diseases, other biological or clinical traits. The association analysis usually tests each SNP as an independent entity and ignores the biological information such as linkage disequilibrium. Although the Bonferroni correction and other approaches have been proposed to address the issue of multiple comparisons as a result of testing many SNPs, there is a lack of understanding of the distribution of an association test statistic when an entire genome is considered together. In other words, there are extensive efforts in hypothesis testing, and almost no attempt in estimating the density under the null hypothesis. By estimating the true null distribution, we can apply the result directly to hypothesis testing; better assess the existing approaches of multiple comparisons; and evaluate the impact of linkage disequilibrium on the GWA studies. To this end, we estimate the empirical null distribution of an association test statistic in GWA studies using simulated population data. We further propose a convenient and accurate method based on adaptive spline to estimate the empirical value in GWA studies and validate our findings using a real data set. Our method enables us to fully characterize the null distribution of an association test that not only can be used to test the null hypothesis of no association, but also provides important information about the impact of density of the genetic markers on the significance of the tests. Our method does not require users to perform computationally intensive permutations, and hence provides a timely solution to an important and difficult problem in GWA studies.     

A Monte Carlo method for combined segregation and linkage analysis.

We introduce a Monte Carlo approach to combined segregation and linkage analysis of a quantitative trait observed in an extended pedigree. In conjunction with the Monte Carlo method of likelihood-ratio evaluation proposed by Thompson and Guo, the method provides for estimation and hypothesis testing. The greatest attraction of this approach is its ability to handle complex genetic models and large pedigrees. Two examples illustrate the practicality of the method. One is of simulated data on a large pedigree; the other is a reanalysis of published data previously analyzed by other methods.     

A simple permutation-type method for testing circular uniformity with correlated angular measurements

A simple method is provided for testing uniformity on the circle that allows dependence among repeated angular measurements on the same subject. Our null hypothesis is that the distribution of repeated angles is unaffected by rotation. This null can be evaluated with any test of uniformity by using a null reference distribution obtained by simulation, where each subject's vector of angles is rotated by a random amount. A new weighted version of the univariate Rayleigh test of circular uniformity is proposed.     

A group sequential type design for three‐arm non‐inferiority trials with binary endpoints

The three‐arm design with a test treatment, an active control and a placebo group is the gold standard design for non‐inferiority trials if it is ethically justifiable to expose patients to placebo. In this paper, we first use the closed testing principle to establish the hierarchical testing procedure for the multiple comparisons involved in the three‐arm design. For the effect preservation test we derive the explicit formula for the optimal allocation ratios. We propose a group sequential type design, which naturally accommodates the hierarchical testing procedure. Under this proposed design, Monte Carlo simulations are conducted to evaluate the performance of the sequential effect preservation test when the variance of the test statistic is estimated based on the restricted maximum likelihood estimators of the response rates under the null hypothesis. When there are uncertainties for the placebo response rate, the proposed design demonstrates better operating characteristics than the fixed sample design.     

A multivariate two-sample mean test for small sample size and missing data

We develop a new statistic for testing the equality of two multivariate mean vectors. A scaled chi-squared distribution is proposed as an approximating null distribution. Because the test statistic is based on componentwise statistics, it has the advantage over Hotelling's T2 test of being applicable to the case where the dimension of an observation exceeds the number of observations. An appealing feature of the new test is its ability to handle missing data by relying on only componentwise sample moments. Monte Carlo studies indicate good power compared to Hotelling's T2 and a recently proposed test by Srivastava (2004, Technical Report, University of Toronto). The test is applied to drug discovery data.     

Testing a Partial Ordering of Population Means with Application to Inference about Growth Habits of Cowpea Genotypes

Using general results available in the literature, we derive the likelihood ratio test for a particular partial ordering of means that naturally arises in a biological context. We then show that the conceptual and computational complexity of the derivation can be substantially reduced by equivalently deriving the test using the intersection-union principle for decomposing a complex null hypothesis into elemental forms. A Monte Carlo algorithm for obtaining the p-value of the test is proposed. The test procedure is illustrated with a data set of the competitive ability of several cowpea genotypes, where previous experiments have indicated the proposed partial order of the means. A simulation study is used to examine the power of the test.     

A Partition Function Approximation Using Elementary Symmetric Functions

In statistical mechanics, the canonical partition function can be used to compute equilibrium properties of a physical system. Calculating however, is in general computationally intractable, since the computation scales exponentially with the number of particles in the system. A commonly used method for approximating equilibrium properties, is the Monte Carlo (MC) method. For some problems the MC method converges slowly, requiring a very large number of MC steps. For such problems the computational cost of the Monte Carlo method can be prohibitive. Presented here is a deterministic algorithm – the direct interaction algorithm (DIA) – for approximating the canonical partition function in operations. The DIA approximates the partition function as a combinatorial sum of products known as elementary symmetric functions (ESFs), which can be computed in operations. The DIA was used to compute equilibrium properties for the isotropic 2D Ising model, and the accuracy of the DIA was compared to that of the basic Metropolis Monte Carlo method. Our results show that the DIA may be a practical alternative for some problems where the Monte Carlo method converge slowly, and computational speed is a critical constraint, such as for very large systems or web-based applications.     

Spatial point pattern analysis of available and exploited resources

A patchy spatial distribution of resources underpins many models of population regulation and species coexistence, so ecologists require methods to analyse spatially‐explicit data of resource distribution and use. We describe a method for analysing maps of resources and testing hypotheses about species' distributions and selectivity. The method uses point pattern analysis based on the L‐function, the linearised form of Ripley's K‐function. Monte Carlo permutations are used for statistical tests. We estimate the difference between observed and expected values of L(t), an approach with several advantages: 1) The results are easy to interpret ecologically. 2) It obviates the need for edge correction, which has largely precluded the use of L‐functions where plot boundaries are “real”. Including edge corrections may lead to erroneous conclusions if the underlying assumptions are invalid. 3) The null expectation can take many forms, we illustrate two models: complete spatial randomness (to describe the spatial pattern of resources in the landscape) and the underlying pattern of resource patches in the landscape (akin to a neutral landscape approach). The second null is particularly useful to test whether spatial patterns of exploited resource points simply reflect the spatial patterns of all resource points. We tested this method using over 100 simulated point patterns encompassing a range of patterns that might occur in ecological systems, and some very extreme patterns. The approach is generally robust, but Type II decision errors might arise where spatial patterns are weak and when trying to detect a clumped pattern of exploited points against a clumped pattern of all points. An empirical example of an intertidal lichen growing on barnacle shells illustrates how this technique might be used to test hypotheses about dispersal mechanisms. This approach can increase the value of survey data, by permitting quantification of natural resource patch distribution in the landscape as well as patterns of resource use by species.