Inference based on type II censored samples

A method for drawing inferences about the mean of a normal population from small censored samples is discussed. The modified standardized statistic presented takes into account the correlation between the estimates of the mean and standard deviation and thus gives asymmetrical confidence intervals in contrast to the common standardized variate. A bias correction is suggested which improves the performance of the proposed statistic for small samples.     

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.     

Comparative quantitative studies of mental and physical development of children up to age 3

Biomathematical-methodical aspects are exposed of quantitative recording of mean growth courses which will be taken as basis for comparisons between different sample curves. In the paper the body length growth process of children in the age period from 0 to 3 years is considered. There are samples of children growing up under different social conditions and data of corresponding samples collected about 20 years ago. After remarks on the correlation between physical and psychical development of infants several possibilities for representing mean body length growth curves are discussed. The advantages of empirical regression are explained as the best suited method for a modelfree data evaluation. The nonparametric location trend test of Cox and Stuart is used for statistical proving of global difference or parallelism in the course of 2 mean growth curves. Level differences in the global sense between 2 compared curves may be tested by constructing a Scheffé confidence region for a properly defined constrast. Local level differences can be proved by the t-test for those pairs of corresponding mean values for which the conditions of applying this test are fulfilled. The calculation of a curve-related normal belt as a succession of 95%-tolerance regions (without confidence probability) is demonstrated by an example.     

Testing Equality of Means and Simultaneous Confidence Intervals in Repeated Measures with Missing Data

In this paper, repeated measures with intraclass correlation model is considered when the observations are missing at random. An exact test for the equality of the mean components and simultaneous confidence intervals (Scheffé and Bonferroni inequality types) are given for linear contrasts of the mean components when the missing observations are of a monotone type. When the missing observations are not of the monotone type, the maximum likelihood estimates are obtained numerically by iterative methods given in Srivastava and Carter (1986). These estimators are then used to obtain asymptotic tests and confidence intervals for the equality of mean components and linear contrasts, respectively. An example is given to illustrate the method.     

Phylogenetic dating with confidence intervals using mean path lengths

The mean path length (MPL) method, a simple method for dating nodes in a phylogenetic tree, is presented. For small trees the age estimates and corresponding confidence intervals, calibrated with fossil data, can be calculated by hand, and for larger trees a computer program gives the results instantaneously (a Pascal program is available upon request). Necessary input data are a rooted phylogenetic tree with edge lengths (internode lengths) approximately corresponding to the number of substitutions between the nodes. Given this, the MPL method produces relative age estimates with confidence intervals for all nodes of the tree. With the age of one or several nodes of the tree being known from reference fossils, the relative age estimates induce absolute age estimates and confidence intervals of the nodes of the tree. The MPL method relies on the assumptions that substitutions occur randomly and independently in different sites in the DNA sequence and that the substitution rates are approximately constant in time, i.e., assuming a molecular clock. A method is presented for identification of the nodes in the tree at which significant deviations from the clock assumption occur, such that dating may be done using different rates in different parts of the tree. The MPL method is illustrated with the Liliales, a group of monocot flowering plants.     

Higher order inference for the consensus mean in inter-laboratory studies

In inter-laboratory studies, a fundamental problem of interest is inference concerning the consensus mean, when the measurements are made by several laboratories which may exhibit different within-laboratory variances, apart from the between laboratory variability. A heteroscedastic one-way random model is very often used to model this scenario. Under such a model, a modified signed log-likelihood ratio procedure is developed for the interval estimation of the common mean. Furthermore, simulation results are presented to show the accuracy of the proposed confidence interval, especially for small samples. The results are illustrated using an example on the determination of selenium in non-fat milk powder by combining the results of four methods. Here, the sample size is small, and the confidence limits for the common mean obtained by different methods produce very different results. The confidence interval based on the modified signed log-likelihood ratio procedure appears to be quite satisfactory.     

Interval Estimation for Ratios in Bioequivalence Trials

The problem for assessment of equivalence in variability of bioavailability between two drug products is considered. An exact confidence region for the ratio between intrasubject variabilities is derived when the intersubject variance is known. When the intersubject variance is unknown, a large sample approximation is considered. The proposed method for assessing equivalence in variability of bioavailability appears to be asymptotically uncorrelated with the sample mean ratio for average bioavailabilty. As a result, the proposed method in conjunction with the sample mean ratio method can be utilized for assessing population bioequivalence. An example concerning a bioequivalence trial with 24 healthy volunteers is presented to illustrate the proposed method.     

Confidence intervals following group sequential tests in clinical trials

Tsiatis, Rosner, and Mehta (1984, Biometrics 40, 797-803) proposed a procedure for constructing confidence intervals following group sequential tests of a normal mean. This method is first extended for group sequential tests for which the sample sizes between interim analyses are not identical or the times are not equally spaced. Then properties of this confidence interval estimation procedure are studied by simulation. The extension accommodates the flexible procedure by Lan and DeMets (1983, Biometrika 70, 659-663) for constructing discrete group sequential boundaries to form a structure for monitoring and estimation following a class of group sequential tests. Finally, it is demonstrated how to combine the procedures by Lan and DeMets and by Tsiatis, Rosner, and Mehta using a FORTRAN program.     

Reliability of Confidence Interval Estimators under Various Nested Design Parental Sample Sizes

The coverage probabilities of several confidence limit estimators of genetic parameters, obtained from North Carolina I designs, were assessed by means of Monte Carlo simulations. The reliability of the estimators was compared under three different parental sample sizes. The coverage of confidence intervals set on the Normal distribution, and using standard errors either computed by the “delta” method or derived using an approximation for the variance of a variance component estimated by means of a linear combination of mean squares, was affected by the number of males and females included in the experiment. The “delta” method was found to provide reliable standard errors of the genetic parameters only when at least 48 males were each mated to six different females randomly selected from the reference population. Formulae are provided for obtaining “delta” method standard errors, and appropriate statistical software procedures are discussed. The error rates of confidence limits based on the Normal distribution and using standard errors obtained by an approximation for the variance of a variance component varied widely. The coverage of F-distribution confidence intervals for heritability estimates was not significantly affected by parental sample size and consistently provided a mean coverage near the stated coverage. For small parental sample sizes, confidence intervals for heritability estimates should be based on the F-distribution.     

Nomograms for obtaining a necessary, minimum sample size. I. When distribution of data is normal.

Four different nomograms were devised to obtain a necessary, minimum sample size at a 95% confidence rate when data were distributed normally. They corresponded to four different cases, the population of which was either infinite or finite and the permitted error of which was either mu--m or mu--m/s, where mu was population mean, m sample mean and s sample standard deviation. They could also be used for obtaining the confidence limits of the population mean from the data after having carried out a work.     

Statistics of divergence times.

Given the number of nucleotide substitutions between two species (K) and the substitution rate nu, the expectation of the corresponding divergence time is usually calculated as K/(2 nu). This is strictly true only if nu is regarded as a constant because the ratio of two random variables, such as K/(2 nu), has distributional properties different from those of the distribution of K. Therefore, both the mean and any confidence interval for divergence times are unknown in this situation. We model the distribution of K and nu using the Gamma distribution and calculate the mean and 95% confidence interval for the corresponding divergence time. These calculations are compared with results obtained by bootstrapping sequence data from the model plant Arabidopsis thaliana and its relatives. We show that for nonoverlapping pairs of phylogenetic distances, our method approaches the bootstrap results very closely. In contrast, regarding the mutation rate as a constant leads to strong underestimation of the confidence interval. An implementation of our method of computing divergence times is accessible through a web interface at http://www.soft.ice.mpg.de/cite.     

Simultaneous confidence intervals from randomization tests with application in testing bioequivalence with multiple endpoints

We propose a method to construct simultaneous confidence intervals for a parameter vector from inverting a series of randomization tests (RT). The randomization tests are facilitated by an efficient multivariate Robbins–Monro procedure that takes the correlation information of all components into account. The estimation method does not require any distributional assumption of the population other than the existence of the second moments. The resulting simultaneous confidence intervals are not necessarily symmetric about the point estimate of the parameter vector but possess the property of equal tails in all dimensions. In particular, we present the constructing the mean vector of one population and the difference between two mean vectors of two populations. Extensive simulation is conducted to show numerical comparison with four methods. We illustrate the application of the proposed method to test bioequivalence with multiple endpoints on some real data.     

Inferences on the common mean of several log-normal populations: the generalized variable approach

This paper proposes a novel approach for the confidence interval estimation and hypothesis testing of the common mean of several log-normal populations using the concept of generalized variable. Simulation studies demonstrate that the proposed approach can provide confidence intervals with satisfying coverage probabilities and can perform hypothesis testing with satisfying type-I error control even at small sample sizes. Overall, it is superior to the large sample approach. The proposed method is illustrated using two examples.     

Hts1 Encodes Both the Cytoplasmic and Mitochondrial Histidyl-Trna Synthetase of Saccharomyces Cerevisiae: Mutations Alter the Specificity of Compartmentation

Wright's method of estimating the number of genes contributing to the difference in a quantitative character between two populations involves observing the means and variances of the two parental populations and their hybrid populations. Although simple, Wright's method provides seriously biased estimates, largely due to linkage and unequal effects of alleles. A method is suggested to evaluate the bias of Wright's estimate, which relies on estimation of the mean recombination frequency between a pair of loci and a composite parameter of variability of allelic effects and frequencies among loci. Assuming that the loci are uniformly distributed in the genome, the mean recombination frequency can be calculated for some organisms. Theoretical analysis and an analysis of the Drosophila data on distributions of effects of P element inserts on bristle numbers indicate that the value of the composite parameter is likely to be about three or larger for many quantitative characters. There are, however, some serious problems with the current method, such as the irregular behavior of the statistic and large sampling variances of estimates. Because of that, the method is generally not recommended for use unless several favorable conditions are met. These conditions are: the two parental populations are many phenotypic standard deviations apart, linkage is not tight, and the sample size is very large. An example is given on the fruit weight of tomato from a cross with parental populations differing in means by more than 14 phenotypic standard deviations. It is estimated that the number of loci which account for 95% of the genic variance in the F2 population is 16, with a 95% confidence interval of 7-28, and the effect of the leading locus is 13% of the parental difference, with 95% confidence interval 8.5-25.7%.     

Confidence intervals for the mean of diagnostic test charge data containing zeros

In this paper, we consider the problem of interval estimation for the mean of diagnostic test charges. Diagnostic test charge data may contain zero values, and the nonzero values can often be modeled by a log-normal distribution. Under such a model, we propose three different interval estimation procedures: a percentile-t bootstrap interval based on sufficient statistics and two likelihood-based confidence intervals. For theoretical properties, we show that the two likelihood-based one-sided confidence intervals are only first-order accurate and that the bootstrap-based one-sided confidence interval is second-order accurate. For two-sided confidence intervals, all three proposed methods are second-order accurate. A simulation study in finite-sample sizes suggests all three proposed intervals outperform a widely used minimum variance unbiased estimator (MVUE)-based interval except for the case of one-sided lower end-point intervals when the skewness is very small. Among the proposed one-sided intervals, the bootstrap interval has the best coverage accuracy. For the two-sided intervals, when the sample size is small, the bootstrap method still yields the best coverage accuracy unless the skewness is very small, in which case the bias-corrected ML method has the best accuracy. When the sample size is large, all three proposed intervals have similar coverage accuracy. Finally, we analyze with the proposed methods one real example assessing diagnostic test charges among older adults with depression.     

Confidence intervals for heritability for two-factor mating design single environment linear models

Summary Precision measurement is an essential part of heritability estimate interpretation. Approximate standard errors are commonly used as measures of precision for heritability on a progeny mean basis (H). Their derivation, however, is not inferred from the distribution theory for H. F-distribution based exact confidence intervals have been derived for some onefactor mating design H estimators. Extension of the confidence interval results from one-factor to twofactor mating designs is reported in this paper. Functions of heritability on a full-sib or half-sib progeny mean basis from nested or factorial mating design parameters were distributed according to the F-distribution. Exact confidence intervals were derived for heritability on a full-sib progeny mean basis. Exact confidence intervals for heritability on a half-sib progeny basis were adapted from previous results. Maize (Zea mays L.) data were used to estimate confidence intervals. Complete equations were given for interpolation in F-tables.Oregon Agricultural Experiment Station Technical Paper No. 7659     

Confidence intervals for a normal mean following a group sequential test

This paper presents methodology based on the likelihood-ratio test, developed for construction of confidence intervals for a normal mean, following a group sequential test. Examples show that the confidence intervals produced by this method have accurate nominal probability of coverage and have generally shorter average length than that produced by Tsiatis, Rosner, and Mehta (1984, Biometrics 40, 797-803).     

Simultaneous confidence corridors for mean functions in functional data analysis of imaging data

Motivated by recent work involving the analysis of biomedical imaging data, we present a novel procedure for constructing simultaneous confidence corridors for the mean of imaging data. We propose to use flexible bivariate splines over triangulations to handle an irregular domain of the images that is common in brain imaging studies and in other biomedical imaging applications. The proposed spline estimators of the mean functions are shown to be consistent and asymptotically normal under some regularity conditions. We also provide a computationally efficient estimator of the covariance function and derive its uniform consistency. The procedure is also extended to the two-sample case in which we focus on comparing the mean functions from two populations of imaging data. Through Monte Carlo simulation studies, we examine the finite sample performance of the proposed method. Finally, the proposed method is applied to analyze brain positron emission tomography data in two different studies. One data set used in preparation of this article was obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database.     

A New Algorithm for Explicit Determination of Variance Component Estimations in Mixed Models and Mixed Classifications of Balanced ANOVA

This paper deals with the balanced case of the analysis of variance. The use of a classification function leads to an easy determination of all possible sources of variation of any mixed classification. For mixed models a new method is derived, which allows to represent explicit the ANOVA-estimations of the variance components respectively the estimation of the mean sum of squares of the fixed effects for all sources of variation. Thereby the corresponding F-quotients and the approximate confidence intervals of variance components are received in a simple way.     

The analysis of cancer chemoprevention experiments

This paper is concerned with the analysis of certain cancer chemoprevention experiments that involve Type I censoring. In experiments of this nature, two common response variables are the number of induced cancers and the rate at which they develop. In this study we assume that the number of induced tumors and their times to detection are described by the Poisson and gamma distributions, respectively. Using the method of maximum likelihood, we discuss a procedure for estimating the parameters characterizing these two distributions. We apply standard techniques in order to construct a confidence region and conduct a hypothesis test concerning the parameters of interest. We discuss a method for comparing the effects of two different treatments using the likelihood ratio principle. A technique for isolating group differences in terms of the mean number of promoted tumors and the mean time to detection is described. Using the techniques developed in this paper, we reanalyze an existing data set in the cancer chemoprevention literature and obtain contrasting results.