Estimation of a parameter and its exact confidence interval following sequential sample size reestimation trials

For confirmatory trials of regulatory decision making, it is important that adaptive designs under consideration provide inference with the correct nominal level, as well as unbiased estimates, and confidence intervals for the treatment comparisons in the actual trials. However, naive point estimate and its confidence interval are often biased in adaptive sequential designs. We develop a new procedure for estimation following a test from a sample size reestimation design. The method for obtaining an exact confidence interval and point estimate is based on a general distribution property of a pivot function of the Self-designing group sequential clinical trial by Shen and Fisher (1999, Biometrics55, 190-197). A modified estimate is proposed to explicitly account for futility stopping boundary with reduced bias when block sizes are small. The proposed estimates are shown to be consistent. The computation of the estimates is straightforward. We also provide a modified weight function to improve the power of the test. Extensive simulation studies show that the exact confidence intervals have accurate nominal probability of coverage, and the proposed point estimates are nearly unbiased with practical sample sizes.     

Simple dynamic model of the human blood pressure regulatory system as based on the orthostatic load sequence]

The two statistical parameter estimation methods, the recursive least squares and the recursive generalized least squares, are dealt with briefly. An additional noncorrelated disturbance is necessary for unbiased parameter estimation in the closed-loop system. The disturbance is realized by an orthostatic load sequence shaped according to the experimental programme. Men and women were subjected to head-up tilt between 10 degrees and 55 degrees. The disturbance, mean blood pressure and heart rate were measured. These discrete data were used for parameter estimation of transfer functions.     

Parameter Estimation in Systematic Sampling

First it is shown that an estimate of the variance of the sample-mean in systematic sampling from a non-autocorrelated population with linear trend, which is published in textbooks, isn't a suitable estimate: It is biased and not dependent on the essential parameter, the slope of the linear trend. In section 2 an unbiased estimate of the variance is given. As estimate of the sample-mean we take the same as usually used in literature. In section 3 a centric estimate of the sample-mean is introduced, which takes into consideration the slope of the trendline. It is shown that this estimate is unbiased; an unbiased estimate of its variance is given.     

A software tool for standardised non-destructive biomass estimation in short rotation forestry

Following an evaluation of the various methods available for non-destructive biomass estimation in short rotation forestry, a standardised procedure was defined and incorporated into a computer programme (BioEst). Special efforts were made to ensure that the system can be used by people who are unfamiliar with computers and mathematics. BioEst provides an interface between a calliper and a spreadsheet programme which was written in Microsoft Excel macro language. Therefore, it is simple to modify the programme and create personal protocols. BioEst can be run on a portable PC with Microsoft Excel for Windows. The computer continuously recalculates an estimate of the amount of biomass per hectare, as well as some summary statistics, when fed data on shoot diameter obtained by making row-section-wise measurements with a standard digital calliper. BioEst is available without cost from the author.     

Estimating synonymous and nonsynonymous substitution rates

Partitioning the total substitution rate into synnonymous and nonsynonymous components is a key aspect of many analyses in molecular evolution. Numerous methods exist for estimating these rates. However, until recently none of the estimation procedures were based on a sound statistical footing. In this paper, the evolutionary model of Muse and Gaut (1994) is used as the basis for two sets of parameters quantifying silent and replacement substitution rates. The parameters are shown to be equal when the four nucleotides are equally frequent and unequal otherwise. Maximum-likelihood estimation of these parameters is described, and the performance of these estimates is compared to that of existing estimation procedures. It is shown that the estimates of Nei and Gojobori (1986) are not unbiased for either set of parameters, although they provide very good estimates for one set as long as sequence divergence is not too high. However, some disturbing properties are found for the Nei and Gojobori estimates. In particular, it is shown that the expected value of the Nei and Gojobori estimate of silent substitution rate is a function of both the silent and replacement substitution rates. The maximum-likelihood estimates have no such problems.      

Methods of assessing population increase in aphids and the effect of growth stage of the host plant on population growth rates

Realistic values of population growth rates are needed when used in forecasting programmes, e.g., in a programme of integrated control. Therefore, comparisons were made in a chrysanthemum – aphid system between different methods of assessing population growth rates. The reproductive performances of the aphid species Aphis gossypii and Myzus persicae were measured on two chrysanthemum cultivars using three plant growth stages (young vegetative, budding and flowering). In the first set of experiments, development time and reproduction were used to estimate the population growth rate r_m. The mean relative growth rates (MRGR) were also assessed. It was shown for the first time that the relationship between r_m and MRGR was influenced by aphid species. In a second experiment, the aphid population increase on a whole plant was measured and r_m was estimated by calculating the slope of the (ln transformed) population increase. It is shown that population growth rate is affected by the growth stage of the plant, and that cultivar and aphid species interact with plant growth stage in influencing population growth rate. Thus, no single growth stage of chrysanthemum for maximal aphid population growth can be assigned, but the budding and flowering stage are the most suitable in three out of four aphid × cultivar combinations. Comparison between the results from both experiments demonstrates clearly that more realistic values for r_m are obtained when measured on whole plants.     

Estimating mutation rate and generation time from longitudinal samples of DNA sequences

We present in this paper a simple method for estimating the mutation rate per site per year which also yields an estimate of the length of a generation when mutation rate per site per generation is known. The estimator, which takes advantage of DNA polymorphisms in longitudinal samples, is unbiased under a number of population models, including population structure and variable population size over time. We apply the new method to a longitudinal sample of DNA sequences of the env gene of human immunodeficiency virus type 1 (HIV-1) from a single patient and obtain 1.62 x 10(-2) as the mutation rate per site per year for HIV-1. Using an independent data set to estimate the mutation rate per generation, we obtain 1.8 days as the length of a generation of HIV-1, which agrees well with recent estimates based on viral load data. Our estimate of generation time differs considerably from a recent estimate by Rodrigo et al. when the same mutation rate per site per generation is used. Some factors that may contribute to the difference among different estimators are discussed.     

Unbiased estimation of selected treatment means in two-stage trials

Straightforward estimation of a treatment's effect in an adaptive clinical trial can be severely hindered when it has been chosen from a larger group of potential candidates. This is because selection mechanisms that condition on the rank order of treatment statistics introduce bias. Nevertheless, designs of this sort are seen as a practical and efficient way to fast track the most promising compounds in drug development. In this paper we extend the method of Cohen and Sackrowitz (1989) who proposed a two-stage unbiased estimate for the best performing treatment at interim. This enables their estimate to work for unequal stage one and two sample sizes, and also when the quantity of interest is the best, second best, or j -th best treatment out of k. The implications of this new flexibility are explored via simulation.     

Skin colorimetry and admixture measurement: Some further considerations

Several techniques for using skin colorimetric data to measure admixture contributions to a hybrid population have been reviewed by Lees and Relethford ('78), who find this general approach to be useful. This paper extends their evaluation, providing a method for obtaining an error term and considering at greater length certain potential sources of estimate bias. Questions are raised about their discussion of the appropriate employment of distance statistics in this context, and an alternative procedure for obtaining a single estimate of admixture proportions jointly over all reflectance measurements is offered. While it is shown that the hybrid reflectance curves do not support entirely an assumption of unlinked loci contributing equally and additively to skin color, preliminary findings indicate that admixture estimates made from these data are apparently unbiased and reasonably efficient.     

An analytical model for the estimation of chromosome substitution effects in the offspring of individuals heterozygous at a segregating marker locus

Summary Use of marker genes for quantitative traits has been suggested as a supplement to selection for livestock species. Linkage relationships can be estimated by using data from offspring of a heterozygous parent, if offspring can be positively assigned segregation of one or the other of the marker alleles. In field data, some data on offspring can be characterized and used to estimate the difference in chromosome substitution effects, but other matings result in uncertain transfer of the marker alleles. In this study, an alternative estimation procedure is proposed that would allow incorporation of data on all offspring of a heterozygous parent, even those where chromosome segregation is ambiguous. If the frequency of the marker alleles is known in the population of mates of a heterozygous individual, the mean and variance of the heterozygous offspring can be used in a generalized leastsquares model to estimate the chromosome substitution effect. When gene frequencies are not known, maximum likelihood estimates can be obtained from the data for use in a conditional estimate. Monte Carlo simulations of data following the assumed genetic model were analyzed as proposed, and parameter estimates were characterized. Estimates of chromosome substitution effects were reasonable approximations of input values. Distributions of t-statistics testing the null hypothesis of no difference between marked chromosome segments were unbiased, with only slightly larger variance than expected. Addition of data from heterozygous offspring improved the efficiency of detection of chromosome substitution effects by more than four times when marker gene frequencies were low.     

Deviations from Michaelis-Menten kinetics. Computation of the probabilities of obtaining complex curves from simple kinetic schemes.

1. It is possible to calculate the intrinsic probability associated with any curve shape that is allowed for rational functions of given degree when the coefficients are independent or dependent random variables with known probability distributions. 2. Computations of such probabilities are described when the coefficients of the rational function are generated according to several probability distribution functions and in particular when rate constants are varied randomly for several simple model mechanisms. 3. It is concluded that each molecular mechanism is associated with a specific set of curve-shape probabilities, and this could be of value in discriminating between model mechanisms. 4. It is shown how a computer program can be used to estimate the probability of new complexities such as extra inflexions and turning points as the degree of rate equations increases. 5. The probability of 3 : 3 rate equations giving 2 : 2 curve shapes is discussed for unrestricted coefficients and also for the substrate-modifier mechanisms. 6. The probability associated with the numerical values of coefficients in rate equations is also calculated for this mechanism, and a possible method for determining the approximate magnitude of product-release steps is given. 7. The computer programs used in the computations have been deposited as Supplement SUP 50113 (21 pages) with the British Library Lending Division, Boston Spa, Wetherby, West Yorkshire LS23 7BQ, U.K., from whom copies can be obtained on the terms indicated in Biochem, J. (1978) 169, 5.     

普通克立格法在昆虫生态学中的应用

地统计学是以区域化变量为基础,以变差函数为主要工龄,分析空间相关变量结构的统计方法。在对波动较大的实验变差函数进行拟合时,虽无法获得最优拟合,但运用人机对话的拟合方法来灵活选取参数,可以得到较理想的变差函数模型的参数。本文运用加权多项式回归法以及人机对话的方法,得到了较理想的1级与2级球状模型拟合结果,同时利用直线函数对实验变差函数进行了拟合,最后利用普通Kriging法,对待估计点进行各理论模型的最优、线性、无偏内插估计,得出克立格内插权重。将此方法应用于广东省四会市大沙镇富溪乡试验田稻飞观测数据,由待估点周围若干观测点的数据,有效地估计出待估点的昆虫分布密度,并讨论比较了不同理论模型的拟合效果以及估计误差。结果表明,2级球状模型的拟合最好,一级球状模型次之,直线函数的拟合最差,但直线函数计算最为简便。     

Group sequential methods for an ordinal logistic random-effects model under misspecification

Interim analyses in clinical trials are planned for ethical as well as economic reasons. General results have been published in the literature that allow the use of standard group sequential methodology if one uses an efficient test statistic, e.g., when Wald-type statistics are used in random-effects models for ordinal longitudinal data. These models often assume that the random effects are normally distributed. However, this is not always the case. We will show that, when the random-effects distribution is misspecified in ordinal regression models, the joint distribution of the test statistics over the different interim analyses is still a multivariate normal distribution, but a sandwich-type correction to the covariance matrix is needed in order to obtain the correct covariance matrix. The independent increment structure is also investigated. A bias in estimation will occur due to the misspecification. However, we will also show that the treatment effect estimate will be unbiased under the null hypothesis, thus maintaining the type I error. Extensive simulations based on a toenail dermatophyte onychomycosis trial are used to illustrate our results.     

Heat transfer and the energetic cost of singing by canaries <Emphasis Type="Italic">Serinus canaria</Emphasis>

Sexually selected displays, such as male passerine bird song, are predicted to be costly. However, most measurements calculated the rate of oxygen consumption during singing using respirometry have shown that bird song has a low energetic cost. Since birds are reluctant to sing when enclosed inside a respirometry chamber, the energetic cost of singing could differ from that under more normal circumstances. We used heat transfer modelling, based on thermal images, to estimate the energetic cost of singing by canaries (Serinus canaria) that were not enclosed in respirometry chambers. Metabolic rate calculated from heat transfer modelling was 0.70±0.02 W (N=10 birds) during singing, which was 14±5% greater than during standing (0.62±0.02 W). The energetic cost of singing did not differ significantly from that measured previously using respirometry when we took into account that birds sang for a greater proportion of the time during the current experiments. These conclusions were not sensitive to potential errors in the heat transfer model. Heat transfer modelling would be especially useful to obtain measurements of the energetic cost of activities that animals do not perform readily inside respirometry chambers, such as singing in birds.     

An improved double sampling procedure based on the variance

Sample size calculations for a continuous outcome require specification of the anticipated variance; inaccurate specification can result in an underpowered or overpowered study. For this reason, adaptive methods whereby sample size is recalculated using the variance of a subsample have become increasingly popular. The first proposal of this type (Stein, 1945, Annals of Mathematical Statistics 16, 243-258) used all of the data to estimate the mean difference but only the first stage data to estimate the variance. Stein's procedure is not commonly used because many people perceive it as ignoring relevant data. This is especially problematic when the first stage sample size is small, as would be the case if the anticipated total sample size were small. A more naive approach uses in the denominator of the final test statistic the variance estimate based on all of the data. Applying the Helmert transformation, we show why this naive approach underestimates the true variance and how to construct an unbiased estimate that uses all of the data. We prove that the type I error rate of our procedure cannot exceed alpha.     

Spatially explicit maximum likelihood methods for capture-recapture studies

Live-trapping capture-recapture studies of animal populations with fixed trap locations inevitably have a spatial component: animals close to traps are more likely to be caught than those far away. This is not addressed in conventional closed-population estimates of abundance and without the spatial component, rigorous estimates of density cannot be obtained. We propose new, flexible capture-recapture models that use the capture locations to estimate animal locations and spatially referenced capture probability. The models are likelihood-based and hence allow use of Akaike's information criterion or other likelihood-based methods of model selection. Density is an explicit parameter, and the evaluation of its dependence on spatial or temporal covariates is therefore straightforward. Additional (nonspatial) variation in capture probability may be modeled as in conventional capture-recapture. The method is tested by simulation, using a model in which capture probability depends only on location relative to traps. Point estimators are found to be unbiased and standard error estimators almost unbiased. The method is used to estimate the density of Red-eyed Vireos (Vireo olivaceus) from mist-netting data from the Patuxent Research Refuge, Maryland, U.S.A. Estimates agree well with those from an existing spatially explicit method based on inverse prediction. A variety of additional spatially explicit models are fitted; these include models with temporal stratification, behavioral response, and heterogeneous animal home ranges.     

On responder analyses when a continuous variable is dichotomized and measurement error is present

In clinical studies results are often reported as proportions of responders, i.e. the proportion of subjects who fulfill a certain response criterion is reported, although the underlying variable of interest is continuous. In this paper, we consider the situation where a subject is defined as a responder if the (error-free) continuous measurements post-treatment are below a certain fraction of (error-free) continuous measurements obtained pre-treatment. Focus is on the one-sample case, but an extension to the two-sample case is also presented. The bias of different estimates for the proportion of responders is derived and compared. In addition, an asymptotically unbiased ML-type estimate for the proportion of responders is presented. The results are illustrated using data obtained in a clinical study investigating pre-menstrual dysphoric disorder (PMDD).     

Estimating the Prevalence of Nonpaternity in Germany

The prevalence of nonpaternity in human societies is difficult to establish. To obtain a current and fairly unbiased estimate of the nonpaternity rate in Germany, we analysed a dataset consisting of 971 children and their parents in whom human leukocyte antigen (HLA) typing had been carried out in the context of bone marrow transplantation. In this sample, nine exclusions (0.93%) could be identified on the basis of more than 300 HLA-haplotypes defined by four HLA genes. Given this number of exclusions, a maximum likelihood estimate of the nonpaternity rate in the population of 0.94% was obtained with asymptotic 95% confidence limits of 0.33% and 1.55%, respectively. This result is in accordance with recent surveys as well as findings from Switzerland for a comparable sample, and it suggests that earlier estimates of the nonpaternity rate which were often in excess of 10% may have been largely exaggerated.     

On fluctuation analysis: a new,simple and efficient method for computing the expected number of mutants

Fluctuation analysis, which is often used to demonstrate random mutagenesis in cell lines (and to estimate mutation rates), is based on the properties of a probability distribution known as the Luria-Delbrück distribution (and its generalizations). The two main new results reported in this paper are (i) a simple, completely general, and computationally efficient procedure for calculating probability distributions arising from fluctuation analysis and (ii) the formula for this procedure when cells in a colony have only grown for a finite number of generations after initial seeding. It is also shown that the procedure reduces to one that was developed earlier when an infinite number of generations is assumed. The derivation of the generating function of the distribution is also clarified. The results obtained should also be useful to experimentalists when only a relatively short time elapses between seeding and harvestint cultures for fluctuation analysis.     

Unbiased Estimation of Odds Ratio by Using Negative Binomial Plans

The use of the negative binomial distribution in both the numerator and denominator in prospective studies leads to an unbiased estimate of the odds ratio and an exact expression for its variance. Sample sizes that minimize the variance of odds ratio estimates are specified. The variance of the odds ratio estimate is shown to be close to the Cramér-Rao lower bound.