Using a Normal Approximation to Test for the Binomial Distribution

The commonly used method to test for the binomial distribution is the x²-test. In this paper, we introduce an alternative method to test for the binomial distribution. Suppose N is the number of sample groups with n individuals each, x_ij is the jth sample in ith group, a Bernoulli variable with parameter and VVI=s²/[m(1 - m)/n]. Then it is well know that the asymptotic distribution of the statistic (N - 1) VVI is x²(N - 1) under the hypothesis p₁ = p₂ = … = p_N. Here we find that VVI has an asymptotic normal distribution N(1, 2(1 - 1/n)/(N - 1)). Unlike the x²-statistic, the variance of the normal test statistic is a function of n. This method is convenient in detecting spatial patterns and dispersion in the study of diseased organisms (e.g., plants) in field samples.     

An Exact Unconditional test for the Hardy-Weinberg Equilibrium

An exact test based on the unconditional distribution of a test statistic for the Hardy-Weinberg equilibrium is introduced. This test is usually more powerful and requires less extensive tabulations compared with the (ordinary) exact conditional test. Tables of critical values are given for N = 5(1)100 and α =·10, ·05, ·01. Sample sizes required to attain a given power with the unconditional and conditional exact tests are provided.     

The Exact Distribution of the Anderson-Kannemann Statistic

The Anderson-Kannemann test is a rank test for treatment effects in a randomized block design with K treatments and N blocks. In this paper, an algorithm for computing the exact distribution of the Anderson-Kannemann test statistic under the null hypothesis is deduced. Then, the exact distribution is compared with the asymptotic χ²-distribution, and it is shown that the exact distribution is approximated fairly well by the asymptotic distribution. Tables of the exact distribution are given for K = 3, N = 3(1)15; K = 4, N = 3(1)11; K = 5, N = 3(1)7; and K = 6, N = 3(1)5.     

On Exact Unconditional Test for Linear Trend in Dose‐Response Studies

The one‐degree‐of‐freedom Cochran‐Armitage (C‐A) test statistic for linear trend has been widely applied in various dose‐response studies (e.g., anti‐ulcer medications and short‐term antibiotics, animal carcinogenicity bioassays and occupational toxicant studies). This approximate statistic relies, however, on asymptotic theory that is reliable only when the sample sizes are reasonably large and well balanced across dose levels. For small, sparse, or skewed data, the asymptotic theory is suspect and exact conditional method (based on the C‐A statistic) seems to provide a dependable alternative. Unfortunately, the exact conditional method is only practical for the linear logistic model from which the sufficient statistics for the regression coefficients can be obtained explicitly. In this article, a simple and efficient recursive polynomial multiplication algorithm for exact unconditional test (based on the C‐A statistic) for detecting a linear trend in proportions is derived. The method is applicable for all choices of the model with monotone trend including logistic, probit, arcsine, extreme value and one hit. We also show that this algorithm can be easily extended to exact unconditional power calculation for studies with up to a moderately large sample size. A real example is given to illustrate the applicability of the proposed method.     

A novel peak detection approach with chemical noise removal using short‐time FFT for prOTOF MS data

Peak detection is a pivotal first step in biomarker discovery from MS data and can significantly influence the results of downstream data analysis steps. We developed a novel automatic peak detection method for prOTOF MS data, which does not require a priori knowledge of protein masses. Random noise is removed by an undecimated wavelet transform and chemical noise is attenuated by an adaptive short‐time discrete Fourier transform. Isotopic peaks corresponding to a single protein are combined by extracting an envelope over them. Depending on the S/N, the desired peaks in each individual spectrum are detected and those with the highest intensity among their peak clusters are recorded. The common peaks among all the spectra are identified by choosing an appropriate cut‐off threshold in the complete linkage hierarchical clustering. To remove the 1 Da shifting of the peaks, the peak corresponding to the same protein is determined as the detected peak with the largest number among its neighborhood. We validated this method using a data set of serial peptide and protein calibration standards. Compared with MoverZ program, our new method detects more peaks and significantly enhances S/N of the peak after the chemical noise removal. We then successfully applied this method to a data set from prOTOF MS spectra of albumin and albumin‐bound proteins from serum samples of 59 patients with carotid artery disease compared to vascular disease‐free patients to detect peaks with S/N≥2. Our method is easily implemented and is highly effective to define peaks that will be used for disease classification or to highlight potential biomarkers.     

The Reliability of Tests of Hypotheses when Overdispersed Logistic-Linear Models are Fitted by Maximum Quasi-Likelihood

When overdispersed logistic-linear models are fitted by maximum quasi-likelihood hypotheses can be tested by comparing either the Wald statistic, or the quasi-likelihood score statistic, or the quasi likelihood-ratio statistic, with the approximating null X² distribution. This paper reports a simulation study of the reliability of these tests. Some factors affecting their relative reliabilities are identified. An extended quasi-likelihood ratio test is also considered.     

A bivariate median test

A bivariate median test is proposed which makes use of the geometric median or mediancentre. Simulations have shown that the resulting test statistic is X²-distributed. The test may be extended to more than two groups, but not to more than two variates.     

An improved method for estimating R-phycoerythrin and R-phycocyanin contents from crude aqueous extracts of <Emphasis Type="Italic">Porphyra</Emphasis> (Bangiales,Rhodophyta)

One frequently-cited method for determining phycoerythrin (PE) and phycocyanin (PC) contents from crude aqueous extracts of red seaweeds utilizes peaks and troughs of absorbance spectra. The trough absorbance values are used to establish a linear or logarithmic baseline attributable to background scatter of particulate cellular debris not removed by centrifugation. Pigment contents are calculated by subtracting baseline values from PE and PC absorbance peaks. The baseline correction is intended to make the method independent of centrifugation time and/or speed. However, when crude extracts of Porphyra were analyzed using this protocol, R-PE and R-PC estimates were significantly affected by centrifugation time, suggesting that the method was not reliable for the genus. The present study has shown that with sufficient centrifugation, background scatter in Porphyra extracts can be removed, the remaining spectrum representing the overlapping absorbance peaks of water-soluble pigments in the extract. Using fourth derivative analysis of Porphyra extract absorbance spectra, peaks corresponding to chlorophyll, R-PE, R-PC, and allophycocyanin (APC) were identified. Dilute solutions of purified R-PE, R-PC and chlorophyll were scanned separately to identify spectral overlaps and develop new equations for phycobilin quantification. The new equations were used to estimate R-PE and R-PC contents of Porphyra extracts and purified R-PE, R-PC and chlorophyll solutions were mixed according to concentrations corresponding to the sample estimates. Absorbances and fourth derivative spectra of the sample extract and purified pigment mixtures were compared and found to coincide. The newly derived equations are more accurate for determining R-PE and R-PC of Porphyra than previously published methods.     

Multivariate Homogeneity Testing Using an Extended Concept of Nearest Neighbors

Given independent multivariate random samples {X_ij: j = 1, …, n_i} from F_i, for i = 1,2, a test is desired for H₀: F₁ = F₂ against general alternatives. Consider the k · (n₁ + n₂) possible ways of choosing one observation from the combined samples and then one of its k nearest neighbors, and let S_k be the proportion of these choices in which the point and neighbor are in the same sample. Schilling (1986) proposed S_k as a test statistic, but did not indicate how to determine k. We suggest as test statistic W = N Σ kS_k, which we show is equivalent to a sum of N Wilcoxon rank sums, and also to a sum of two two-sample U-statistics of degrees (1, 2) and (2, 1). Simulation with multivariate normal data suggests that our test is generally more powerful than Schilling's test using k = 1, 2, or 3. We illustrate its use with Fisher's iris data.     

Extended Rank Analysis of Covariance as a More Efficient Matched Analysis Considering Trend Information

Classical matched analysis, regarded as analysis of covariance (ANOCOVA) in a broad sense, makes no attempt in modeling and may therefore be inefficient. In this paper, we discuss the relative efficiencies of the ERMP (extended rank and matched‐pair) test (Chen and Quade , 2000) to standard matched methods, and extend it to the case of multivariate covariables X . Taking advantage of trend information between the response Y and the covariables X by ranking after matching, ERMP test achieves better efficiency than a proposed class of weighted matched statistics. When Y is dichotomous, the optimal weighted matched statistic is equivalent to the Mantel‐Haenszel statistic. Example and simulation results also suggest the conclusion.     

Testing for Treatment Effects on Subsets of Endpoints

Multiple endpoints are tested to assess an overall treatment effect and also to identify which endpoints or subsets of endpoints contributed to treatment differences. The conventional p‐value adjustment methods, such as single‐step, step‐up, or step‐down procedures, sequentially identify each significant individual endpoint. Closed test procedures can also detect individual endpoints that have effects via a step‐by‐step closed strategy. This paper proposes a global‐based statistic for testing an a priori number, say, r of the k endpoints, as opposed to the conventional approach of testing one (r = 1) endpoint. The proposed test statistic is an extension of the single‐step p‐value‐based statistic based on the distribution of the smallest p‐value. The test maintains strong control of the FamilyWise Error (FWE) rate under the null hypothesis of no difference in any (sub)set of r endpoints among all possible combinations of the k endpoints. After rejecting the null hypothesis, the individual endpoints in the sets that are rejected can be tested further, using a univariate test statistic in a second step, if desired. However, the second step test only weakly controls the FWE. The proposed method is illustrated by application to a psychosis data set.     

Effects of climate,natural enemies and biocides on three citrus mites in coastal New South Wales

Infestations of the citrus rust mitePhyllocoptruta oleivora (Ashmead), the brown citrus rust miteTegolophus australis Keifer, and the citrus red mitePanonychus citri (McGregor), did not cause economic damage to orange (Citrus sinensis (L.) Osbeck) in two coastal New South Wales orchards during a three-year study. The seasonal abundance of the mites and their phytoseiid and coccinellid predators was defined using periodic regressions of transformed counts on time. Population peaks coincided with seasonal rises and falls in temperature and evaporation in late-summer/autumn and in spring/early-summer. Population troughs in late-spring/summer coincided with peaks in temperature and evaporation and reproductive dormancy in some of the coccinellids. Population troughs in winter generally coincided with troughs in evaporation and temperature, particularly in July when the average temperature was about 10°C. Partial correlation coefficients were used to determine the relationships between the predators and their prey. These were mostly nonsignificant atP=0.05. Fortnightly and daily fluctuations in weather did not have any obvious effects on numbers of the mites. The effects of fungicides and pesticides applied during the study were not clear.     

gener: a server‐based analysis of pollen pool structure

The server‐based program gener performs the two‐generation analysis of pollen flow for data consisting of mother/offspring arrays using genetic markers. The gener program decomposes the genetic variance sampled by maternal individuals within and among pollen pool components of genetic variance and is accessible from http://dyerlab.bio.vcu.edu . These estimates are used to construct the test statistic, Φ_ft , whose significance is tested via permutation. The Φ_ft statistic can subsequently be used to quantify genetic effective pollen donor population size (N_ep), effective mating area and dispersal distance. Furthermore, the gener program can calculate Φ_ft values for all pairs of substrata within the data set.     

Using Discrete Distributions to Compute Powers for Goodness-of-Fit Test Statistics in a One-Way Multinomial Setting

Both the Bionomial and Poisson distributions are employed in this study to compute approximate powers for goodness-of-fit test statistics. The procedure adopted involves simulating 1000 samples from each of these distributions. These samples are then employed to compute both the randomized nominal critical values and estimated powers. The type I error rates returned from the use of the randomized critical levels Cα fall within the acceptable regions. We illustrate the use of the procedure with the Pearson's X² test statistic and show that this can readily be extended to any of the other well known goodness-of-fit test statistics.     

Novel products in hyaluronan digested by bovine testicular hyaluronidase

When the products of hyaluronan (HA) digested by bovine testicular hyaluronidase (BTH) were analyzed by high-performance liquid chromatography (HPLC), minor peaks were detected just before the main even-numbered oligosaccharide peaks. The amount of each minor peak was dependent on the reaction conditions for transglycosylation, rather than hydrolysis, by the BTH. Mainly based on HPLC and MS analysis, each minor peak was found to correspond to its oligosaccharide with one N-acetyl group removed from the reducing terminal N-acetylglucosamine. Enzymatic studies showed that the N-deacetylation activity was closely related to reaction temperature, pH, and the concentration of NaCl contained in the buffer, and glycosaminoglycan types and chain lengths of substrates. These findings strongly suggest that the N-deacetylation reaction in minor peaks was due to a novel enzyme contaminant in the BTH, N-deacetylase, that carries out N-deacetylation at the reducing terminal N-acetylglucosamine of oligosaccharides and is dependent on HA hydrolysis by BTH.     

The Method of Unweighted Means in Unbalanced Analysis of Variance

For factorial experiments with unbalanced data the method of unweighted means is an alternate method of analysis which is computationally simpler than likelihood ratio based F-tests. The quality of the null approximation to the F-distribution of the statistic for the method of unweighted means and the power of this test relative to the likelihood ratio test are discussed. Recommendations are made of when in the course of data analysis the superior operating characteristics of the likelihood ratio test may not outweigh the computational simplicity of the method of unweighted means test.     

Sensitivity analyses informed by tests for bias in observational studies

In an observational study, the treatment received and the outcome exhibited may be associated in the absence of an effect caused by the treatment, even after controlling for observed covariates. Two tactics are common: (i) a test for unmeasured bias may be obtained using a secondary outcome for which the effect is known and (ii) a sensitivity analysis may explore the magnitude of unmeasured bias that would need to be present to explain the observed association as something other than an effect caused by the treatment. Can such a test for unmeasured bias inform the sensitivity analysis? If the test for bias does not discover evidence of unmeasured bias, then ask: Are conclusions therefore insensitive to larger unmeasured biases? Conversely, if the test for bias does find evidence of bias, then ask: What does that imply about sensitivity to biases? This problem is formulated in a new way as a convex quadratically constrained quadratic program and solved on a large scale using interior point methods by a modern solver. That is, a convex quadratic function of N variables is minimized subject to constraints on linear and convex quadratic functions of these variables. The quadratic function that is minimized is a statistic for the primary outcome that is a function of the unknown treatment assignment probabilities. The quadratic function that constrains this minimization is a statistic for subsidiary outcome that is also a function of these same unknown treatment assignment probabilities. In effect, the first statistic is minimized over a confidence set for the unknown treatment assignment probabilities supplied by the unaffected outcome. This process avoids the mistake of interpreting the failure to reject a hypothesis as support for the truth of that hypothesis. The method is illustrated by a study of the effects of light daily alcohol consumption on high-density lipoprotein (HDL) cholesterol levels. In this study, the method quickly optimizes a nonlinear function of $N=800$ variables subject to linear and quadratic constraints. In the example, strong evidence of unmeasured bias is found using the subsidiary outcome, but, perhaps surprisingly, this finding makes the primary comparison insensitive to larger biases.     

Estimation of the Number of Classes in a Population

The problem of estimating M, the number of classes in a population, is formulated as an occupancy problem in which N items are drawn from M urns. Under the assumption of a uniform distribution for the number of classes in the population, the sufficient statistic for M, which is the number of distinct classes observed, does not depend upon the number of repetitions in the sample. Point and interval estimates of M are developed using maximum likelihood and the method of moments. Both techniques give rise to the same basic equation which requires a simple iterative solution. These same techniques are used in the more general situation in which the classes can be further subdivided according to type.     

A New Statistical Goodness-of-Fit Test Based on Graphical Representation

A statistical goodness-of-fit test, based on representing the sample observations by linked vectors, is developed. The direction and the length of the linked vectors are defined as functions of the expected values of the order statistics and sample order statistics, respectively. The underlying method can be used to test distributional assumptions for any location-scale family. A test statistic Q_n is introduced and some of its properties are studied. It is shown that the proposed test can be generalized to test if two or more independent samples come from the same distribution. The test procedure provides a graphical method of identifying the true distribution when the null hypothesis is rejected.     

Comparing Power Behaviours for Some Goodness-of-Fit Test Statistics in Sparse Multinomials

This paper is concerned with the power behaviour of four goodness-of-fit test statistics in sparse multinomials with k cells. Most previous work has been concerned only with both Pearson's X² and the likelihood ratio test statistics. We consider in this study, two additional test statistics, namely, the Cressie-Read test statistic – I(2/3) and the modified Freeman-Tukey test (FT) statistic. Because k ≥ 10 in this study, a Monte Carlo procedure based on 1000 simulated samples is used to estimate the powers for the four test statistics. Alternatives on various line segments are employed. Results suggest that none of the test statistics completely dominate the other and that the choice of which test to use depends on the nature of the alternative hypothesis. These results are consistent with those obtained by West and Kempthorne (1972), although, the Pearson's χ² test statistic may be preferred because of its closer approximation to the χ² distribution in terms of the attained α levels.