A New Approach to Equivalence Assessment in Standard Comparative Bioavailability Trials by Means of the Mann-Whitney Statistic

By a suitable transformation of the pairs of observations obtained in the successive periods of the trial, bioequivalence assessment in a standard comparative bioavailability study reduces to testing for equivalence of two continuous distributions from which unrelated samples are available. Let the two distribution functions be given by F(x) = P[X ≤ x], G(y) = P[Y ≤ y] with (X, Y) denoting an independent pair of real-valued random variables. An intuitively appealing way of putting the notion of equivalence of F and G into nonparametric terms can be based on the distance of the functional P[X > Y] from the value it takes if F and G coincide. This leads to the problem of testing the null hypothesis H_o P[X > Y] ≤ 1/2 - ε₁ or P[X > Y] ≥ 1/2 + ε₂ versus H₁ : 1/2 ? ε₁ < P[X > Y] < 1/2 + ∈₂, with sufficiently small ε₁, ε₂ ∈ (0, 1/2). The testing procedure we derive for (₀, H₁) and propose to term Mann-Whitney test for equivalence, consists of carrying out in terms of the U-statistics estimator of P[X > Y] the uniformly most powerful level a test for an interval hypothesis about the mean of a Gaussian distribution with fixed variance. The test is shown to be asymptotically distribution-free with respect to the significance level. In addition, results of an extensive simulation study are presented which suggest that the new test controls the level even with sample sizes as small as 10. For normally distributed data, the loss in power as against the optimal parametric procedure is found to be almost as small as in comparisons between the Mann-Whitney and the t-statistic in the conventional one or two-sided setting, provided the power of the parametric test does not fall short of 80%.     

A Mixed Model for Binary Data: The Modified X2-test

In this paper, a statistical model for clinical trials is presented for the special situation that a varying and unstructered number of binary responses is obtained from each subject. The assumptions of the model are the following: 1.) For each subject there is a (constant) individual Bernoulli parameter determining the distribution of the binary responses of this subject. 2.) The Bernoulli parameters associated with the subjects are realizations of independent random variables with distributions P_g in treatment group g(g = 1, 2, …, G). 3.) Given the value of the Bernoulli parameter, the observations are stochastically independent within each subject. Under these assumptions, a test statistic is derived to test the hypothesis H₀:E(P₁) = E(P₂) = … = E(P_G). It is proven and demonstrated by simulations, that the test statistic asymptotically (i.e. for a large number of subjects) follows the X²-distribution.     

The Point Evaluation Method for Approximating Function Means,Variances and Covariances

A perennial problem in statistics is the determination of biases, variances and covariances for functions of random variables X₁, X₂, …, X_n which themselves have a known distribution. A common approach is through equations based upon Taylor series approximations but a “point evaluation” method may sometimes be a useful alternative. This involves approximating the multivariate distribution of the X variables by the 2ⁿ points given by X₁=μ₁±₁, X₂ = μ₂ ±₂, …, X_n = = μ_n μ_n, where μ_i is the mean and σ_i the standard deviation of Xi, with appropriate point weights. An advantage over the Taylor series approach is that function derivatives do not have to be explicitely calculated. The point evaluation method is particularly useful in cases where the X variables are uncorrelated. Then the evaluation of the 2ⁿ points can be replaced by the evaluation of 2n points. The point evaluation method is illustrated with powers of a normally distributed variable, and with estimation of gene frequencies from ABO blood group frequencies.     

The Estimation of a Multivariate Fitness Function from Several Samples Taken from a Population

The situation is considered where the multivariate distribution of certain variables X₁, X₂, …, X_p is changing with time in a population because natural selection related to the X's is taking place. It is assumed that random samples taken from the population at times t₁, t₂, …, t_s are available and it is desirable to estimate the fitness function w_t(x₁, x₂,…,x_p) which shows how the number of individuals with X_i = x_i, i = 1, 2, …, p at time t is related to the number of individuals with the same X values at time zero. Tests for population changes are discussed and indices of the selection on the population dispersion and the population mean are proposed. The situation with a multivariate normal distribution is considered as a special case. A maximum likelihood method that can be applied with any form of population distribution is proposed for estimating w_t. The methods discussed in the paper are illustrated with data on four dimensions of male Egyptian skulls covering a time span from about 4500 B.C. to about 300 A.D. In this case there seems to have been very little selection on the population dispersion but considerable selection on means.     

A Distribution-Free Two-Sample Equivalence Test Allowing for Tied Observations

A new testing procedure is derived which enables to assess the equivalence of two arbitrary noncontinuous distribution functions from which unrelated samples are taken as the data to be analyzed. The equivalence region is defined to consist of all pairs (F,G) of distribution functions such that for independent X ∼ F, Y ∼ G the conditional probability of {X > Y} given {X ≠ Y} lies in some short interval around 1/2. The test rejects the null hypothesis of nonequivalence if and only if the standardized distance between the U-statistics estimator of P[X > Y ∣ X ≠ Y] and the center of the equivalence interval (1/2 — ε₁, 1/2 + ε₂) does not exceed a critical upper bound which has to be computed as the α-quantile of a χ²-distribution with one degree of freedom and a random noncentrality parameter proportional to the squared length of that interval. The test is shown to maintain the asymptotic significance level under very weak regularity conditions. Results of an extensive simulation study suggest that its level properties are very satisfactory in small samples as well. The power turns out to be inversely related to the rate P[X = Y] of ties between observations from different samples.     

On Encoding Values for Data Recorded as X ≥ C

In many medical and other contexts, an ordered variable is measured and the limits of measuring ability will be reached. In such cases, the data may be reported as X ≥ C. For example, in a cell count, a laboratory technician may report the data as “count ≥ 100.” If the data are a dependent variable, censored data methods are the standard techniques. However, if X is an independent variable, some methods need to be devised to handle the problem. A simple method is to encode the values X ≥ C to some value X₀. A candidate for this is the conditional mean (or median) of X given X ≥ C. This paper gives some results, and examines some robustness questions for the proposed method.     

A Novel Approach to Optimize and Formulate Fast Disintegrating Tablets for Nausea and Vomiting

The aim of this study was to optimize and formulate fast disintegrating tablets (FDTs) for nausea and vomiting using aminoacetic acid, carmellose and sodium alginate with enough mechanical strength. Ondansetron HCl (water soluble) or domperidone (water insoluble) drug were added to FDTs and their disintegration behaviour was evaluated. Plackett Burman Screening Design was used to screen the independent active process variables [concentration of aminoacetic acid (X ₁), concentration of carmellose (X ₂) and tablet crushing strength (X ₃)] which were found to actively influence the dependent variables [disintegration time in the mouth (DT), wetting time (WT), and water absorption ratio (WAR)] for both the drugs. Also, the coefficients of active variables (DT, WT and WAR) of FDTs containing domperidone was found to be significantly different (P < 0.05) from the coefficients of active factors (X ₁, X ₂ and X ₃) containing ondansetron HCl FDTs. Further, FDTs containing domperidone was prepared according to central composite design for estimating the effect of active factors (X ₁, X ₂, X ₃) in extended spherical domain. The regression analysis of quadratic fit revealed that DT, WT and WAR were 98% correlated with active factors (X ₁, X ₂ or X ₃). The optimized domperidone FDTs were further compared with superdisintegrants (croscarmellose sodium or crospovidone). The data revealed that optimized domperidone FDTs were better than domperidone FDTs containing croscarmellose or crospovidone. Hence, this novel excipients combination can be used for delivery of water insoluble drugs in place of superdisintegrants.     

Multivariate Probability in Terms of Marginal Probability and Correlation Coefficient

This paper is motivated by a practical problem relating to student performance in a number of subjects of equal standing. Its mathematical formulation is to find an approximation to a multivariate probability of the form Pr {X₁⩾a, X₂⩾a, …, X_N⩾a} for arbitrary a and N, in terms of p = Pr {X₁⩾a} and q = Corr (X_i, X_j), i≠j, where X_i, i = 1, …, N are exchangeable random variables with mean 0 and variance unity.     

Multiple sex chromosome system of X1X1X2X2/X1X2)Y type in lutjanid fish, Lutjanus quinquelineatus (Perciformes)

The karyotype and other chromosomal markers as revealed by C-banding and Ag-staining were studied in Lutjanus quinquelineatus and L. kasmira (Lutjanidae, Perciformes). While in latter species, the karyotype was invariably composed of 48 acrocentric chromosomes in both sexes, in L. quinquelineatus the female karyotype had exclusively 48 acrocentric chromosomes (2n = 48) but that of the male consisted of one large metacentric and 46 acrocentric chromosomes (2n = 47). The chromosomes in the first meiotic division in males showed 22 bivalents and one trivalent, which was formed by an end-to-end association and a chiasmatic association. Multiple sex chromosome system of X₁X₁X₂X₂/X₁X₂Y type resulting from single Robertsonian fusion between the original Y chromosome and an autosome was hypothesized to produce neo-Y sex chromosome. The multiple sex chromosome system of L. quinquelineatus appears to be at the early stage of the differentiation. The positive C-banded heterochromatin was situated exclusively in centromeric regions of all chromosomes in both species. Similarly, nucleolus organizer region sites were identified in the pericentromeric region of one middle-sized pair of chromosomes in both species. The cellular DNA contents were the same (3.3 pg) between the sexes and among this species and related species.     

SYNTHETIC AGROPYRON-ELYMUS HYBRIDS: III. ELYMUS CANADENSIS X AGROPYRON CANINUM,A. TRACHYCAULUM,AND A. STRIATUM

Hybrids were produced with relative ease from controlled crosses of Elymus canadensis L. with European Agropyron caninum (L.) Beauv., North American A. trachycaulum (Link) Malte ex H. F. Lewis, and Asian A. striatum Nees ex Steud. All hybrids appeared to be completely sterile and were, for the most part, morphologically intermediate between their parents. The E. canadensis × A. caninum hybrids were exceptionally vigorous and leafy and may have some potential as forage grasses if fertility can be achieved. All parent plants were tetraploid, 2n = 28, and they behaved cytologically as alloploids. Chromosome pairing in the hybrids indicated that both E. canadensis genomes were closely homologous with those of A. trachycaulum and somewhat less homologous with those of A. caninum. Interchanged and inverted chromosome segments apparently constitute the major differences between E. canadensis, A. trachycaulum, and A. caninum genomes; however, cryptic structural differences must also exist. Partial homologies were detected between one A. striatum and E. canadensis genome, but their other genomes were distinctly different. The genome relations between the parent species were expressed in terms of the following genome formulas: E. canadensis, S₁S₁X₁X₁; A. trachycaulum, S₂S₂X₂X₂; A. caninum, S₃S₃X₃X₃ : and A. striatum S₄S₄YY or X₄X₄YY, where “S” refers to a genome derived from A. spicatum and “X” and “Y” are genomes of unknown origin.     

Formulation of Dacarbazine-loaded Cubosomes—Part II: Influence of Process Parameters

The purpose of this study was to investigate the combined influence of three-level, three-factor variables on the formulation of dacarbazine (a water-soluble drug) loaded cubosomes. Box–Behnken design was used to obtain a second-order polynomial equation with interaction terms to predict response values. In this study, the selected and coded variables X ₁, X ₂, and X ₃ representing the amount of monoolein, polymer, and drug as the independent variables, respectively. Fifteen runs of experiments were conducted, and the particle size (Y ₁) and encapsulation efficiency (Y ₂) were evaluated as dependent variables. We performed multiple regression to establish a full-model second-order polynomial equation relating independent and dependent variables. A second-order polynomial regression model was constructed for Y ₁ and confirmed by performing checkpoint analysis. The optimization process and Pareto charts were obtained automatically, and they predicted the levels of independent coded variables X ₁, X ₂, and X ₃ (−1, 0.53485, and −1, respectively) and minimized Y ₁ while maximizing Y ₂. These corresponded to a cubosome formulation made from 100 mg of monoolein, 107 mg of polymer, and 2 mg with average diameter of 104.7 nm and an encapsulation efficiency of 6.9%. The Box–Behnken design proved to be a useful tool to optimize the particle size of these drug-loaded cubosomes. For encapsulation efficiency (Y ₂), further studies are needed to identify appropriate regression model.     

Various X2-Tests of Homogeneity: Some Comments

Methodological issues in the analysis of incidence rates or prevalence proportions for count data, presented in a form of a sequence of 2×2 tables, corresponding to levels (strata) of a specified variable (risk factor) X, are discussed. Suppose λ_1i and λ_2i are the incidence rates of an event D in the ith stratum for populations 1 and 2, respectively. The homogeneity (null) hypothesis is formulated in the form: H_0:λ1i=λ_2i for all i (i = 1, 2, …, I). Three X²-tests for H₀ and their theoretical bases are discussed: X_Total² which is sensitive to alternatives HA :λ1i± λ2i for at least some i; X_Comb² which is sensitive to alternatives H A : λ1iλ2i² or < λ_2i but not both for all i; and X_Diff² which is sensitive to alternatives H_A:λ1i>λ2i³ for some i and λ_1i′ < λ_2i′ for some i′ (i≠i′). These statistics satisfy the relation X_Total² = X_Comb² + X_Diff². Also, X′²-statistic for pooled data is calculated, which in conjunction with X_Comb² can serve for detecting confounding. Although most of these techniques are known, they are rather scattered in the literature, and not always considered jointly, as it is emphasized in the present paper. It is hoped that these comments will be helpful to biostatisticians as well as to epidemiologists and medical researchers in the analysis of mortality and morbidity data. For illustration, two examples with large sets of epidemiological data are given.     

Characterizations of the Exponential Distribution by Local Maximum Random Variables

Let {X_n, n≧1} be a sequence of independent and identically distributed random variables. Suppose n(K)=min n for which X_n>X_k, with K fixed and n>K. We will characterize the exponential distribution by considering the distributional properties of X_k and X_k and X_n(k).     

Chromosomal rearrangements and the first indication of an ♀X1X1X2X2/♂X1X2Y sex chromosome system in Rineloricaria fishes (Teleostei: Siluriformes)

Rineloricaria is the most diverse genus within the freshwater fish subfamily Loricariinae, and it is widely distributed in the Neotropical region. Despite limited cytogenetic data, records from southern and south-eastern Brazil suggest a high rate of chromosomal rearrangements in this genus, mirrored in remarkable inter- and intraspecific karyotype variability. In the present work, we investigated the karyotype features of Rineloricaria teffeana, an endemic representative from northern Brazil, using both conventional and molecular cytogenetic techniques. We revealed different diploid chromosome numbers (2n) between sexes (33♂/34♀), which suggests the presence of an ♀X₁X₁X₂X₂/♂X₁X₂Y multiple sex chromosome system. The male-limited Y chromosome was the largest and the only biarmed element in the karyotype, implying Y-autosome fusion as the most probable mechanism behind its origination. C-banding revealed low amounts of constitutive heterochromatin, mostly confined to the (peri)centromeric regions of most chromosomes (including the X₂ and the Y) but also occupying the distal regions of a few chromosomal pairs. The chromosomal localization of the 18S ribosomal DNA (rDNA) clusters revealed a single site on chromosome pair 4, which was adjacent to the 5S rDNA cluster. Additional 5S rDNA loci were present on the autosome pair 8, X₁ chromosome, and in the presumed fusion point on the Y chromosome. The probe for telomeric repeat motif (TTAGGG)_n revealed signals of variable intensities at the ends of all chromosomes except for the Y chromosome, where no detectable signals were evidenced. Male-to-female comparative genomic hybridization revealed no sex-specific or sex-biased repetitive DNA accumulations, suggesting a presumably low level of neo-Y chromosome differentiation. We provide evidence that rDNA sites might have played a role in the formation of this putative multiple sex chromosome system and that chromosome fusions originate through different mechanisms among different Rineloricaria species.     

Characterization of the WEIBULL Distribution by Properties of Order Statistics

Let X_1:n, X_2:n, X_3:n…, X_n:n be the order statistics of n independent random variables with the common (absolutely continuous strictly increasing) distribution function F. The main results given in this article are:

• 1 For any fixed r and two distinct numbers s₁ and s₂ (1<r<s₁<s₂≦n) the distributions of V_i and W_i (defined in (1.11) and (1.12) are identical for i = 1,2 iff F(x) is WEIBULL (1.2).
• 2 The statistics D₁ and D₂ (as in (1.8) and (1.9)) are independent iff F(x) is WEIBULL (1.2).
• 3 The statistics U_i (1≦j≦n?1) and X_i:n (i ≦ j) are independent iff F(x) is WEIBULL (1.2).
• 4 Let X, X₁, X₂, …, X_k be random variables such that

These conditions are necessary and sufficient for F(x) to be WEIBULL .     

Optimization and oxidative stability of the microencapsulated conjugated linoleic acid

We used response surface methodology to optimize the preparation conditions of conjugated linoleic acid (CLA) microcapsules for maximum entrapment efficiency. Three independent variables were used: the ratio of CLA core material to agar and waxy corn starch wall material (X₁), the temperature of dispersion fluid (X₂), and the concentration of emulsifier (X₃). The optimized values of X₁, X₂, and X₃ were found to be 3.82:6.18, 19.97 °C, and 0.34%, respectively. The CLA oxidation stability was significantly protected by microencapsulation. These results suggest that CLA-loaded microcapsules can be used as a means to enhance not only the entrapment efficiency but also the oxidative stability of CLA.     

A Note on the Characterization of the Normal Distribution

Suppose X₁, X₂,…, X_n are independent and identically distributed random variables with absolutely continuous distribution function F. It is known that if F is standard normal distribution then (i)∑X²_i is a chi-square with n degrees of freedom and (ii)nX¯² is a chi-square with 1 degrees of freedom where X¯=1/n ∑X_i. Here the above two properties are utilized to characterize the normal distribution.     

Estimation of Fixed and Random Effects in the Functional Nonlinear Errors-in-Variable Model

Consider an experiment where a nonlinear continuous functional relationship exists between y and X. Assume that this relationship has been measured at n replicated points of X from each of t treatments or populations. Assume further that the X are fixed unknown vectors and that the location parameter v is either a fixed unknown vector or a vector of random variables. In the first case various linear hypotheses are to be tested about v, such as tests for main effects and interaction; in the second case, the mean and variance of the random variable v are to be estimated. A two-step procedure based on asymptotic theory is presented to test hypotheses or develop estimates for the fixed effects or random effects functional errors-in-variable model. An example of a one-way random effects model is given.     

Characterization of the Mixtures of Gompertz Distributions by Conditional Expectation of Order Statistics

The mixtures of Gompertz random variables (Gompertz , 1825) X₁ and X₂ are identified in terms of relations between the conditional expectation of [exp (αX_2:2) — exp (αX_1:2)]^k given X_1:2 and the hazard rate function of the distribution, k is a positive integer and α < 0. Here X_1:2 and X_2:2 denote the corresponding order statistics. In addition, we also mention some related theorems to characterize the mixtures of Gompertz distributions. Finally, when the sample size is n, the above results are also valid and we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems).     

SYNTHETIC HYBRIDS OF NEW WORLD AND OLD WORLD AGROPYRONS II. AGROPYRON RIPARIUM X AGROPYRON REPENS

Five hybrids were obtained from 12 seeds formed in 26 emasculated florets of A. riparium pollinated by A. repens. The hybrid plants were morphologically intermediate between the parents for vegetative and spike characteristics, although they resembled A. repens more closely than A. riparium. The 28-chromosome A. riparium parent behaved cytologically as an allotetraploid and formed an average of 13.98 II and 0.04 I in 94 cells at metaphase I. An average of 20.27 II and 0.36 IV were observed at metaphase I in 55 cells of A. repens, which was designated as a segmental autoallohexaploid. The hybrids contained 35 chromosomes and averaged 6.75 I, 12.49 II, 1.05 III, 0.01 IV, and 0.01 V in 162 cells interpreted at metaphase I. Bivalent chromosome pairing in the hybrids was attributed to autosyndetic pairing of 2 A. repens genomes and allosyndetic pairing between 1 A. riparium genome and 1 A. repens genome. Multivalent chromosome associations were attributed to structural hybridity. A. repens and A. riparium apparently share a genome in common, and this genome is the one responsible for rhizomes in both species. A. riparium was given a genome formula of R₂R₂SS; whereas the A. repens genome formula was written as R₁R₁X₁X₁X₂X₂, and the hybrid genome formula was designated as R₁R₂X₁X₂S. The “S” genome of A. riparium was derived from A. spicatum, and the “R” genome is the genome shared by A. repens and A. riparium. The origin and distribution of the so-called “X” genomes of A. repens remain unknown. The hybrids produced from 3 to 10% stainable pollen; however, no seed was set on the hybrids during 2 years in the field.