The evaluation of bioequivalence using nonparametric procedures

Applications of nonparametric methods to the evaluation of bioequiv-alence for two treatments are presented for independent samples and for a crossover design. Included are procedures for testing for equivalence in location, in dispersion, and in general. Also presented are procedures for the calculation of confidence limits. A general strategy for the evaluation of bioequivalence is developed which involves both hypothesis testing and the calculation of confidencelimits for parameters which characterize departures from equivalene.     

Partitioning with Respect to a Control: Unequal Sample Sizes Case

This article describes a method for partitioning with respect to a control for the situation in which the treatment sample sizes are unequal and also for the situation where the treatment sample sizes are equal except for a few missing values. Calculation of the critical values required for finding confidence limits is discussed and tables are presented for the “almost equal” sample size case. An application of this method to length of stay data for congestive heart failure patients is also provided.     

Statistical methods of risk assessment for continuous variables

Adverse health effects for continuous responses are not as easily defined as adverse health effects for binary responses. Kodell and West (1993) developed methods for defining adverse effects for continuous responses and the associated risk. Procedures were developed for finding point estimates and upper confidence limits for additional risk under the assumption of a normal distribution and quadratic mean response curve with equal variances at each dose level. In this paper, methods are developed for point estimates and upper confidence limits for additional risk at experimental doses when the equal variance assumption is relaxed. An interpolation procedure is discussed for obtaining information at doses other than the experimental doses. A small simulation study is presented to test the performance of the methods discussed.     

P-values for two-sided comparisons with a control

Critical values for Dunnett's multiple comparison procedure for simultaneously comparing the means of several experimental treatments with the mean of a control treatment are available in many references. However, these values are primarily applicable to one-way analysis of variance with equal sample sizes for the experimental treatments. Even for this design, these critical values do not suffice for situations where the corresponding p-values are desired. Here the technique is presented for calculation of the p-value for multiple comparisons with a control when a general linear model is used for the analysis. A FORTRAN program using IMSL subroutines is described for the calculations.     

Censored Exponential Data: Large Deviations for MLEs and Posterior Distributions

In this article, we consider several statistical models for censored exponential data. We prove a large deviation result for the maximum likelihood estimators (MLEs) of each model, and a unique result for the posterior distributions which works well for all the cases. Finally, comparing the large deviation rate functions for MLEs and posterior distributions, we show that a typical feature fails for one model; moreover, we illustrate the relation between this fact and a well-known result for curved exponential models.     

Tests for symmetry of two-piece normal distribution

In this paper, classical optimum tests for symmetry of two-piece normal distribution is derived. Uniformly most powerful one-sided test for the skewness parameter is obtained when the location and scale parameters are known and is compared with sequential probability ratio test. An ad-hoc test for symmetry and likelihood ratio test for symmetry for large samples, can be found in literature for this distribution. But in this paper, we derive exact likelihood ratio test for symmetry, when location parameter is known. The exact power of the test is evaluated for different sample sizes.     

Estimation in linear mixed models for longitudinal data under linear restricted conditions

In the paper, we consider a linear mixed model (LMM) for longitudinal data under linear restriction and find the estimators for the parameters of interest. The strong consistency and asymptotic normality of the estimators are obtained under some regularity conditions. Besides, we derive the strong consistent estimator of the fourth moment for the error which is useful for statistical inference for random effects and error variance. Simulations and an example are reported for illustration.     

A selection criterion for choosing between functional forms of income distribution

The value of Elderton's k - criterion can be bounded for a given pdf. This paper presents these bounds for the most widely used distribution functions for income and computes the value of K for household incomes as reported in the Consumer Population Survey. The variance for theK are also estimated using the bootstrap, jackknife and delta methods. We find that the K can be used to narrow the field of potential pdfs for income and that all three methods for estimating the variance coincide in flagging low precision in the estimated K.     

Comparing several experimental groups with a control in the multivariate case

This paper presents a multivariate extension of Dunnett's test for comparing simultaneously k treatment group means with a single control group mean. A test based on Hotelling T²statistics is presented and approximate critical values are evaluated for the case of equal numbers of observations in each group, for the .05 and .01 levels of significance, for 1 to 5 variates, for 1 to 10 treatment groups, and for varying degrees of freedom. The accuracy of the procedure for generating approximate critical values is assessed via simulation studies conducted for selected cases and an example is presented using real data.     

On a data-dependent choice of the tuning parameter appearing in certain goodness-of-fit tests

We propose a data-dependent method for choosing the tuning parameter appearing in many recently developed goodness-of-fit test statistics. The new method, based on the bootstrap, is applicable to a class of distributions for which the null distribution of the test statistic is independent of unknown parameters. No data-dependent choice for this parameter exists in the literature; typically, a fixed value for the parameter is chosen which can perform well for some alternatives, but poorly for others. The performance of the new method is investigated by means of a Monte Carlo study, employing three tests for exponentiality. It is found that the Monte Carlo power of these tests, using the data-dependent choice, compares favourably to the maximum achievable power for the tests calculated over a grid of values of the tuning parameter.     

Performance of traditional f tests in repeated measures designs under covariance heterogeneity

This study investigates the performance of two traditional F tests, one for main effects and the other for interaction in repeated measures designs under several conditions of covariance heterogeneity. Overall, the test for interaction is more vulnerable than the one for main effects. Distortion in the level of significance is less serious for the case of equal group size.     

Uniform validity of saddlepoint expansion on compact sets

This paper shows that Daniels's (1954) saddlepoint expansion for the density of a sample mean is, for all practical purposes, always uniformly valid on compact subsets in the interior of the domain of the mean. This uniform validity is the key for establishing the relation between the saddlepoint expansion for the density function and Lugannani and Rice's expansion for the tail probability, and for establishing the validity of a high-order asymptotic expansion for the density of a standardized mean.     

Efficient and easy-to-use sample size formulas in ratio-based non-inferiority tests

In many biomedical applications, tests for the classical hypotheses based on the difference of treatment means in a one-way layout can be replaced by tests for ratios (or tests for relative changes). This approach is well noted for its simplicity in defining the margins, as for example in tests for non-inferiority. Here, we derive approximate and efficient sample size formulas in a multiple testing situation and then thoroughly investigate the relative performance of hypothesis testing based on the ratios of treatment means when compared with differences of means. The results will be illustrated with an example on simultaneous tests for non-inferiority.     

Horwitz's rule, transforming both sides and the design of experiments for mechanistic models

Summary. The paper develops methods for the design of experiments for mechanistic models when the response must be transformed to achieve symmetry and constant variance. The power transformation that is used is partially justified by a rule in analytical chemistry. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples are given of optimum designs, not only for single-response models, but also for experiments in which multivariate responses are measured and for experiments in which the model is defined by a set of differential equations which cannot be solved analytically. The extension to designs for checking models is discussed.     

Some results concerning off-training-set and IID error for the Gibbs and the Bayes optimal generalizers

In this paper we analyse the average behaviour of the Bayes-optimal and Gibbs learning algorithms. We do this both for off-training-set error and conventional IID (independent identically distributed) error (for which test sets overlap with training sets). For the IID case we provide a major extension to one of the better known results. We also show that expected IID test set error is a non-increasing function of training set size for either algorithm. On the other hand, as we show, the expected off-training-set error for both learning algorithms can increase with training set size, for non-uniform sampling distributions. We characterize the relationship the sampling distribution must have with the prior for such an increase. We show in particular that for uniform sampling distributions and either algorithm, the expected off-training-set error is a non-increasing function of training set size. For uniform sampling distributions, we also characterize the priors for which the expected error of the Bayes-optimal algorithm stays constant. In addition we show that for the Bayes-optimal algorithm, expected off-training-set error can increase with training set size when the target function is fixed, but if and only if the expected error averaged over all targets decreases with training set size. Our results hold for arbitrary noise and arbitrary loss functions.     

Correlation type gogdness-of-fit statistics with censored sampling

Some comments are made concerning the possible forms of a correlation coefficient type goodness-of-fit statistic, and their relationship with other goodness-of-fit statistics, Critical values for a correlation goodness-of-fit statistic and for the Cramer-von Mises statistic are provided for testing a completely-specified null hypothesis for both complete and censored sampling, Critical values for a correlation test statistic are provided for complete and censored sampling for testing the hypothesis of normality, two parameter exponentiality, Weibull (or, extreme value) and an exponential-power distribution, respectively. Critical values are also provided for a test of one-parameter exponentiality based on the Cramer-von Mises statistic     

Classification rules for identifying individuals at high risk of developing myocardial infarction based on ApoB,ApoA1 and the ratio were determined using a Bayesian approach

We have developed a new approach to determine the threshold of a biomarker that maximizes the classification accuracy of a disease. We consider a Bayesian estimation procedure for this purpose and illustrate the method using a real data set. In particular, we determine the threshold for Apolipoprotein B (ApoB), Apolipoprotein A1 (ApoA1) and the ratio for the classification of myocardial infarction (MI). We first conduct a literature review and construct prior distributions. We then develop classification rules based on the posterior distribution of the location and scale parameters for these biomarkers. We identify the threshold for ApoB and ApoA1, and the ratio as 0.908 (gram/liter), 1.138 (gram/liter) and 0.808, respectively. We also observe that the threshold for disease classification varies substantially across different age and ethnic groups. Next, we identify the most informative predictor for MI among the three biomarkers. Based on this analysis, ApoA1 appeared to be a stronger predictor than ApoB for MI classification. Given that we have used this data set for illustration only, the results will require further investigation for use in clinical applications. However, the approach developed in this article can be used to determine the threshold of any continuous biomarker for a binary disease classification.     

How does the DerSimonian and Laird procedure for random effects meta-analysis compare with its more efficient but harder to compute counterparts?

The procedure suggested by DerSimonian and Laird is the simplest and most commonly used method for fitting the random effects model for meta-analysis. Here it is shown that, unless all studies are of similar size, this is inefficient when estimating the between-study variance, but is remarkably efficient when estimating the treatment effect. If formal inference is restricted to statements about the treatment effect, and the sample size is large, there is little point in implementing more sophisticated methodology. However, it is further demonstrated, for a simple special case, that use of the profile likelihood results in actual coverage probabilities for 95% confidence intervals that are closer to nominal levels for smaller sample sizes. Alternative methods for making inferences for the treatment effect may therefore be preferable if the sample size is small, but the DerSimonian and Laird procedure retains its usefulness for larger samples.     

Union-Intersection testing for outliers in multivariate normal data

The union-intersection approach to multivariate test construction is used to develop an alternative to Wilks' likelihood ratio test statistic for testing for two or more outliers in multivariate normal data. It is shown that critical values of both statistics are poorly approximated by Bonferroni bounds. Simulated critical values are presented for both statistics for significance levels 1% and 5%, for sample sizes 10(5)30, 40, 50, 75 and 100 for 2, 3, 4 and 5 dimensions. A power comparison of the two tests in the slippage of the mean model for generating outliers indicates that the union-intersection test is the more powerful when the slippages are close to collinear. Although Wilks' test remains the preference for general use, the union-intersection test could be valuable when such special structure in the data is suspected.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.