Optimal spending functions for asymmetric group sequential designs

We present optimized group sequential designs where testing of a single parameter theta is of interest. We require specification of a loss function and of a prior distribution for theta. For the examples presented, we pre-specify Type I and II error rates and minimize the expected sample size over the prior distribution for theta. Minimizing the square of sample size rather than the sample size is found to produce designs with slightly less aggressive interim stopping rules and smaller maximum sample sizes with essentially identical expected sample size. We compare optimal designs using Hwang-Shih-DeCani and Kim-DeMets spending functions to fully optimized designs not restricted by a spending function family. In the examples selected, we also examine when there might be substantial benefit gained by adding an interim analysis. Finally, we provide specific optimal asymmetric spending function designs that should be generally useful and simply applied when a design with minimal expected sample size is desired.     

A Confidence Interval for the Wallace Coefficient of Concordance and Its Application to Microbial Typing Methods

Very diverse research fields frequently deal with the analysis of multiple clustering results, which should imply an objective detection of overlaps and divergences between the formed groupings. The congruence between these multiple results can be quantified by clustering comparison measures such as the Wallace coefficient (W). Since the measured congruence is dependent on the particular sample taken from the population, there is variability in the estimated values relatively to those of the true population. In the present work we propose the use of a confidence interval (CI) to account for this variability when W is used. The CI analytical formula is derived assuming a Gaussian sampling distribution and recurring to the algebraic relationship between W and the Simpson''s index of diversity. This relationship also allows the estimation of the expected Wallace value under the assumption of independence of classifications. We evaluated the CI performance using simulated and published microbial typing data sets. The simulations showed that the CI has the desired 95% coverage when the W is greater than 0.5. This behaviour is robust to changes in cluster number, cluster size distributions and sample size. The analysis of the published data sets demonstrated the usefulness of the new CI by objectively validating some of the previous interpretations, while showing that other conclusions lacked statistical support.     

Fine-mapping of a region of variation in recombination rate on BTA23 to the D23S22-D23S23 interval using sperm typing and meiotic breakpoint analysis.

Meiotic recombination rate (theta) within chromosome segments of similar physical size is known to vary widely throughout the genome. This variation has a genetic component, occurring between the sexes and among individuals of the same sex. We reported previously the existence of variation in theta between males in the DYA-PRL interval on bovine chromosome 23 (BTA23). This region contains the bovine major histocompatibility complex and has been shown to contain recombination hotspots in humans and mice. The aim of this study was to map more finely the interval(s) on BTA23 where variation in theta occurs using sperm typing and meiotic breakpoint analysis. By adding a marker (DRB3) between DYA and PRL, the DYA-PRL interval was subdivided into two adjacent intervals, thus permitting evaluation and comparison of theta among five bulls. Significant variation in theta was found for both intervals; theta(DYA-DRB3) ranged from 13.2 to 28.1%, and theta(DRB3-PRL) ranged from 2.4 to 13.0%. The variation in theta was individual- and region-specific. A meiotic breakpoint strategy employing PCR amplification products from recombinant sperm was then used to refine the chromosomal location associated with variation in theta within the DYA-DRB3 interval. The subinterval D23S22-D23S23 exhibited the greatest degree of variation among bulls having high and low theta within the DYA-DRB3 interval. To confirm this result, theta(D23S22-D23S23) was directly evaluated in three additional randomly chosen bulls using sperm typing. The region showing variation in theta was narrowed to the D23S22-D23S23 subinterval, ranging from 4.6 to 9.2%. Identification of the molecular basis for variation in theta may be useful for map-dependent applications, such as marker-assisted selection and positional cloning of genes affecting physiologically important traits.     

Sample size determination for confidence intervals on the population mean and on the difference between two population means

Sample size determination is usually based on the premise that a hypothesis test is to be used. A confidence interval can sometimes serve better than a hypothesis test. In this paper a method is presented for sample size determination based on the premise that a confidence interval for a simple mean, or for the difference between two means, with normally distributed data is to be used. For this purpose, a concept of power relevant to confidence intervals is given. Some useful tables giving required sample size using this method are also presented.     

Estimating effective population size or mutation rate with microsatellites

Microsatellites are short tandem repeats that are widely dispersed among eukaryotic genomes. Many of them are highly polymorphic; they have been used widely in genetic studies. Statistical properties of all measures of genetic variation at microsatellites critically depend upon the composite parameter theta = 4Nmicro, where N is the effective population size and micro is mutation rate per locus per generation. Since mutation leads to expansion or contraction of a repeat number in a stepwise fashion, the stepwise mutation model has been widely used to study the dynamics of these loci. We developed an estimator of theta, theta; (F), on the basis of sample homozygosity under the single-step stepwise mutation model. The estimator is unbiased and is much more efficient than the variance-based estimator under the single-step stepwise mutation model. It also has smaller bias and mean square error (MSE) than the variance-based estimator when the mutation follows the multistep generalized stepwise mutation model. Compared with the maximum-likelihood estimator theta; (L) by, theta; (F) has less bias and smaller MSE in general. theta; (L) has a slight advantage when theta is small, but in such a situation the bias in theta; (L) may be more of a concern.     

Genetic analysis of the population structure of socially organized oystercatchers (Haematopus ostralegus) using microsatellites

On the island of Schiermonnikoog (The Netherlands), the breeding population of oystercatchers can be divided into two groups: 'residents' and 'leapfrogs', based on their distinct social characteristics and limited probabilities of status change between breeding seasons. In order to investigate whether this social organization has caused local genetic differentiation, leapfrogs and residents were compared at eight polymorphic microsatellite loci. No significant genetic subdivision between residents and leapfrogs was observed (theta = 0.0000; 95% confidence interval (CI), -0.0027-0.0033), indicating that the oystercatcher population on the island of Schiermonnikoog has to be considered as one panmictic unit. Investigation of three additional locations in the northern part of The Netherlands did not reveal significant genetic population subdivision either (theta = -0.0005; 95% CI, -0.0045-0.0037), despite the fact that adult osytercatchers show extreme fidelity to their breeding localities. These results indicate panmixis and considerable levels of gene flow within the northern part of The Netherlands. Thus, the results from genetical analyses do not seem to be in agreement with observational data on the dispersal behaviour of breeding individuals. It is argued that the lack of population structure, locally on Schiermonnikoog as well as across larger geographical distances, is to be attributed to high levels of gene flow through dispersal of juvenile birds.     

A new method for choosing sample size for confidence interval-based inferences

Scientists often need to test hypotheses and construct corresponding confidence intervals. In designing a study to test a particular null hypothesis, traditional methods lead to a sample size large enough to provide sufficient statistical power. In contrast, traditional methods based on constructing a confidence interval lead to a sample size likely to control the width of the interval. With either approach, a sample size so large as to waste resources or introduce ethical concerns is undesirable. This work was motivated by the concern that existing sample size methods often make it difficult for scientists to achieve their actual goals. We focus on situations which involve a fixed, unknown scalar parameter representing the true state of nature. The width of the confidence interval is defined as the difference between the (random) upper and lower bounds. An event width is said to occur if the observed confidence interval width is less than a fixed constant chosen a priori. An event validity is said to occur if the parameter of interest is contained between the observed upper and lower confidence interval bounds. An event rejection is said to occur if the confidence interval excludes the null value of the parameter. In our opinion, scientists often implicitly seek to have all three occur: width, validity, and rejection. New results illustrate that neglecting rejection or width (and less so validity) often provides a sample size with a low probability of the simultaneous occurrence of all three events. We recommend considering all three events simultaneously when choosing a criterion for determining a sample size. We provide new theoretical results for any scalar (mean) parameter in a general linear model with Gaussian errors and fixed predictors. Convenient computational forms are included, as well as numerical examples to illustrate our methods.     

Mapping of the von Hippel-Lindau disease locus to a small region of chromosome 3p by genetic linkage analysis.

Genetic linkage studies were performed in 22 families with von Hippel-Lindau (VHL) disease by using polymorphic DNA markers from distal chromosome 3p. Linkage was detected between VHL disease and the markers D3S18 (Zmax = 6.6 at theta = 0.0, confidence interval (CI) 0.00-0.06), RAF1 (Zmax = 5.9 at theta = 0.06, CI 0.01-0.16), and THRB (Zmax 3.4 at theta = 0.11). Multipoint linkage analysis localized the VHL disease gene within a small region (approximately 8 cM) of 3p25-p26 between RAF1 and (D3S191, D3S225) and close to the D3S18 locus. There was no evidence of locus heterogeneity, and families with and without pheochromocytoma showed linkage to D3S18. The identification of DNA markers flanking the VHL disease gene allows reliable presymptomatic and prenatal diagnosis to be offered to informative families.     

The performance of a two‐stage analysis of ABAB/BABA crossover trials

Freeman has considered the following two‐stage procedure for finding a confidence interval for the treatment difference theta, using data from an AB/BA crossover trial. In the first stage, a preliminary test of the null hypothesis that the differential carryover is zero is carried out. If this hypothesis is accepted then the confidence interval for theta is constructed assuming that the differential carryover is zero. If, on the other hand, this hypothesis is rejected then this confidence interval is constructed using only data from the first period. Freeman has shown that this confidence interval has minimum coverage probability far below nominal. He therefore concludes that this confidence interval should not be used. In the present paper, we analyze the performance of a similar two‐stage procedure for an ABAB/BABA crossover trial. This trial differs in very significant ways from an AB/BA crossover trial, including the fact that for an ABAB/BABA crossover trial there is an unbiased estimator of the differential carryover that is unaffected by between‐subject variation. Despite these great differences, we arrive at the same conclusion as Freeman. Namely, that the confidence interval resulting from the two‐stage procedure should not be used.     

Using lod scores to detect sex differences in male-female recombination fractions

Human recombination fraction (RF) can differ between males and females, but investigators do not always know which disease genes are located in genomic areas of large RF sex differences. Knowledge of RF sex differences contributes to our understanding of basic biology and can increase the power of a linkage study, improve gene localization, and provide clues to possible imprinting. One way to detect these differences is to use lod scores. In this study we focused on detecting RF sex differences and answered the following questions, in both phase-known and phase-unknown matings: (1) How large a sample size is needed to detect a RF sex difference? (2) What are "optimal" proportions of paternally vs. maternally informative matings? (3) Does ascertaining nonoptimal proportions of paternally or maternally informative matings lead to ascertainment bias? Our results were as follows: (1) We calculated expected lod scores (ELODs) under two different conditions: "unconstrained," allowing sex-specific RF parameters (theta(female), theta(male)); and "constrained," requiring theta(female) = theta(male). We then examined the DeltaELOD (identical with difference between maximized constrained and unconstrained ELODs) and calculated minimum sample sizes required to achieve statistically significant DeltaELODs. For large RF sex differences, samples as small as 10 to 20 fully informative matings can achieve statistical significance. We give general sample size guidelines for detecting RF differences in informative phase-known and phase-unknown matings. (2) We defined p as the proportion of paternally informative matings in the dataset; and the optimal proportion p(circ) as that value of p that maximizes DeltaELOD. We determined that, surprisingly, p(circ) does not necessarily equal (1/2), although it does fall between approximately 0.4 and 0.6 in most situations. (3) We showed that if p in a sample deviates from its optimal value, no bias is introduced (asymptotically) to the maximum likelihood estimates of theta(female) and theta(male), even though ELOD is reduced (see point 2). This fact is important because often investigators cannot control the proportions of paternally and maternally informative families. In conclusion, it is possible to reliably detect sex differences in recombination fraction.     

The quantitative LOD score: test statistic and sample size for exclusion and linkage of quantitative traits in human sibships.

We present a test statistic, the quantitative LOD (QLOD) score, for the testing of both linkage and exclusion of quantitative-trait loci in randomly selected human sibships. As with the traditional LOD score, the boundary values of 3, for linkage, and -2, for exclusion, can be used for the QLOD score. We investigated the sample sizes required for inferring exclusion and linkage, for various combinations of linked genetic variance, total heritability, recombination distance, and sibship size, using fixed-size sampling. The sample sizes required for both linkage and exclusion were not qualitatively different and depended on the percentage of variance being linked or excluded and on the total genetic variance. Information regarding linkage and exclusion in sibships larger than size 2 increased as approximately all possible pairs n(n-1)/2 up to sibships of size 6. Increasing the recombination (theta) distance between the marker and the trait loci reduced empirically the power for both linkage and exclusion, as a function of approximately (1-2theta)4.     

Sample size determination for establishing equivalence/noninferiority via ratio of two proportions in matched-pair design

In this article, we propose approximate sample size formulas for establishing equivalence or noninferiority of two treatments in match-pairs design. Using the ratio of two proportions as the equivalence measure, we derive sample size formulas based on a score statistic for two types of analyses: hypothesis testing and confidence interval estimation. Depending on the purpose of a study, these formulas can be used to provide a sample size estimate that guarantees a prespecified power of a hypothesis test at a certain significance level or controls the width of a confidence interval with a certain confidence level. Our empirical results confirm that these score methods are reliable in terms of true size, coverage probability, and skewness. A liver scan detection study is used to illustrate the proposed methods.     

The Sampling Distribution of Linkage Disequilibrium under an Infinite Allele Model without Selection

The sampling distributions of several statistics that measure the association of alleles on gametes (linkage disequilibrium) are estimated under a two-locus neutral infinite allele model using an efficient Monte Carlo method. An often used approximation for the mean squared linkage disequilibrium is shown to be inaccurate unless the proper statistical conditioning is used. The joint distribution of linkage disequilibrium and the allele frequencies in the sample is studied. This estimated joint distribution is sufficient for obtaining an approximate maximum likelihood estimate of C = 4Nc, where N is the population size and c is the recombination rate. It has been suggested that observations of high linkage disequilibrium might be a good basis for rejecting a neutral model in favor of a model in which natural selection maintains genetic variation. It is found that a single sample of chromosomes, examined at two loci cannot provide sufficient information for such a test if C less than 10, because with C this small, very high levels of linkage disequilibrium are not unexpected under the neutral model. In samples of size 50, it is found that, even when C is as large as 50, the distribution of linkage disequilibrium conditional on the allele frequencies is substantially different from the distribution when there is no linkage between the loci. When conditioned on the number of alleles at each locus in the sample, all of the sample statistics examined are nearly independent of theta = 4N mu, where mu is the neutral mutation rate.     

Sample size under inverse negative binomial group testing for accuracy in parameter estimation

Background

The group testing method has been proposed for the detection and estimation of genetically modified plants (adventitious presence of unwanted transgenic plants, AP). For binary response variables (presence or absence), group testing is efficient when the prevalence is low, so that estimation, detection, and sample size methods have been developed under the binomial model. However, when the event is rare (low prevalence <0.1), and testing occurs sequentially, inverse (negative) binomial pooled sampling may be preferred.

Methodology/Principal Findings

This research proposes three sample size procedures (two computational and one analytic) for estimating prevalence using group testing under inverse (negative) binomial sampling. These methods provide the required number of positive pools (), given a pool size (k), for estimating the proportion of AP plants using the Dorfman model and inverse (negative) binomial sampling. We give real and simulated examples to show how to apply these methods and the proposed sample-size formula. The Monte Carlo method was used to study the coverage and level of assurance achieved by the proposed sample sizes. An R program to create other scenarios is given in Appendix S2.

Conclusions

The three methods ensure precision in the estimated proportion of AP because they guarantee that the width (W) of the confidence interval (CI) will be equal to, or narrower than, the desired width (), with a probability of . With the Monte Carlo study we found that the computational Wald procedure (method 2) produces the more precise sample size (with coverage and assurance levels very close to nominal values) and that the samples size based on the Clopper-Pearson CI (method 1) is conservative (overestimates the sample size); the analytic Wald sample size method we developed (method 3) sometimes underestimated the optimum number of pools.     

Interval estimation of the LD50 based on an up-and-down experiment

It is well known that an up-and-down method can be more efficient than fixed-sample methods in estimating the LD50 of a quantal response curve. A problem that has not been addressed by many is that of obtaining a confidence interval for the LD50 from the up-and-down method. Dixon and Mood (1948, Journal of the American Statistical Association 43, 109-126) proposed a confidence interval using a maximum likelihood approach, but not much is known about its properties. In this paper, a new confidence interval for the LD50 based on turning points is obtained, which uses the concept of phi-mixing. Simulation results indicate that the coverage probabilities of both methods tend to be less than the nominal level unless the sample size is large. Even so, when the tolerance distribution is normal, the proposed confidence interval is found to be superior to Dixon's interval in terms of the coverage, the width, and stability. The advantages of the method do not appear to hold in the presence of nonnormal tolerance distribution.     

Tuning of Strouhal number for high propulsive efficiency accurately predicts how wingbeat frequency and stroke amplitude relate and scale with size and flight speed in birds

The wing kinematics of birds vary systematically with body size, but we still, after several decades of research, lack a clear mechanistic understanding of the aerodynamic selection pressures that shape them. Swimming and flying animals have recently been shown to cruise at Strouhal numbers (St) corresponding to a regime of vortex growth and shedding in which the propulsive efficiency of flapping foils peaks (St approximately fA/U, where f is wingbeat frequency, U is cruising speed and A approximately bsin(theta/2) is stroke amplitude, in which b is wingspan and theta is stroke angle). We show that St is a simple and accurate predictor of wingbeat frequency in birds. The Strouhal numbers of cruising birds have converged on the lower end of the range 0.2 < St < 0.4 associated with high propulsive efficiency. Stroke angle scales as theta approximately 67b-0.24, so wingbeat frequency can be predicted as f approximately St.U/bsin(33.5b-0.24), with St0.21 and St0.25 for direct and intermittent fliers, respectively. This simple aerodynamic model predicts wingbeat frequency better than any other relationship proposed to date, explaining 90% of the observed variance in a sample of 60 bird species. Avian wing kinematics therefore appear to have been tuned by natural selection for high aerodynamic efficiency: physical and physiological constraints upon wing kinematics must be reconsidered in this light.     

Some Statistical Improvements for Estimating Population Size and Mutation Rate from Segregating Sites in DNA Sequences

In population genetics, under a neutral Wright-Fisher model, the scaling parameter straight theta=4Nmu represents twice the average number of new mutants per generation. The effective population size is N and mu is the mutation rate per sequence per generation. Watterson proposed a consistent estimator of this parameter based on the number of segregating sites in a sample of nucleotide sequences. We study the distribution of the Watterson estimator. Enlarging the size of the sample, we asymptotically set a Central Limit Theorem for the Watterson estimator. This exhibits asymptotic normality with a slow rate of convergence. We then prove the asymptotic efficiency of this estimator. In the second part, we illustrate the slow rate of convergence found in the Central Limit Theorem. To this end, by studying the confidence intervals, we show that the asymptotic Gaussian distribution is not a good approximation for the Watterson estimator.     

Linkage data on MN and the Hb beta locus.

A sample of 28 informative families was studied for linkage between Hb beta and MN. Values of the neuterized recombination fraction from these and other families from the literature excluded a recombination fraction of less than .30 between these loci. Our results support different recombination values for males and females (theta equals .34 abd .50, respectively). A simple approach to estimate the sample size required as well as a study of the relationship between sibship size and sample size under conditions of loose linkage are also presented.     

Study of agreement between LDL size as measured by nuclear magnetic resonance and gradient gel electrophoresis

LDL particle size can be measured by gradient gel electrophoresis (GGE) and NMR. The agreement between the two methods has not been extensively evaluated. Therefore, we measured LDL size by NMR and GGE in 324 individuals (152 with type 1 diabetes and 172 controls). The Spearman correlation between both methods was 0.39 [95% confidence interval (CI) = 0.29, 0.48]. The average difference was 5.38 nm (NMR being smaller), but it increased with increasing LDL size. Less than 50% of people classified as pattern B on GGE were classified as pattern B on NMR (kappa = 0.31; 95% CI = 0.17, 0.45). Agreement was lower for diabetic subjects compared with controls, for women compared with men, and for subjects with triglycerides less than 1.30 mmol/l compared with subjects with triglycerides greater than 1.30 mmol/l. External validation showed that cholesteryl ester transfer rate was related to LDL size on GGE in all subgroups and to LDL size on NMR only in men and nondiabetic subjects. Our findings show that agreement between NMR- and GGE-based LDL size is far from perfect and is not consistent across subgroups of patients. In particular, the two methods should not be assumed to be interchangeable in women and diabetic subjects. Whether NMR or GGE predicts cardiovascular disease risk better has not yet been evaluated.     

Effective stage in the cell cycle for control of the budding direction of cdc mutants of Saccharomyces cerevisiae using electric stimulus

Cell division cycle (cdc) mutants of Saccharomyces cerevisiae were used to determine the most effective stage for the directional control of cell budding using an electric stimulus. The selected mutants were cdc 35 and cdc 28, which could be reversibly arrested before spindle pole body satellite formation (SPBSF) and spindle pole body duplication (SPBD), respectively. The budding direction (theta) was defined so that the direction parallel to that of the electric field was 0 degree. Considering the symmetry of the experimental conditions, the range of theta was defined as 0-90 degrees. The electric stimulus applied in the present study was alternating pulses (pulse height, +/- 15 V; pulse width at half pulse height, 5 microseconds; frequency; 10 kHz). The peak height of the cross membrane potential was estimated as 472 mV, which was sufficient to induce considerable strain in the cell membrane. In the case of cdc 35, the 95% confidence interval (95% CI) of the budding direction was 7-25 degrees when subjected to electric stimulus, while the 95% CI of the budding direction without electric stimulus was 35-57 degrees. In the case of cdc 28, 95% CI values of the budding direction with and without electric stimulus were 1229 degrees and 23-56 degrees, respectively. These results demonstrate that the stage after SPBD is effective for the directional control of yeast cell budding using an electric stimulus. Simultaneously, an electric stimulus reduced the cell budding time of both the cdc mutants used. Therefore, the electric stimulus was also effective in promoting cell cycle progression under the present conditions.