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.     

On the exact null distribution of the generalised likelihood ratio test for analysing unreplicated factorial designs

Al-Shiha and Yang (1999) proposed a multistage procedure for analysing unreplicated factorial experiments, which is based on the statistic that is derived from the generalised likelihood ratio test statistic under the assumption of normality. It was shown by their simulation study that the method is quite competitive with Lenth's (1989) method. In their paper, because of the difficulty of determining the null distribution analytically, the quantiles of the null distribution were empirically simulated. In this paper, we give the exact null distribution of their test statistic, which makes it possible to calculate the critical values of the test.     

To permute or not to permute

Permutation test is a popular technique for testing a hypothesis of no effect, when the distribution of the test statistic is unknown. To test the equality of two means, a permutation test might use a test statistic which is the difference of the two sample means in the univariate case. In the multivariate case, it might use a test statistic which is the maximum of the univariate test statistics. A permutation test then estimates the null distribution of the test statistic by permuting the observations between the two samples. We will show that, for such tests, if the two distributions are not identical (as for example when they have unequal variances, correlations or skewness), then a permutation test for equality of means based on difference of sample means can have an inflated Type I error rate even when the means are equal. Our results illustrate permutation testing should be confined to testing for non-identical distributions. CONTACT: calian@raunvis.hi.is.     

Levene tests of homogeneity of variance for general block and treatment designs

This article develops a weighted least squares version of Levene's test of homogeneity of variance for a general design, available both for univariate and multivariate situations. When the design is balanced, the univariate and two common multivariate test statistics turn out to be proportional to the corresponding ordinary least squares test statistics obtained from an analysis of variance of the absolute values of the standardized mean-based residuals from the original analysis of the data. The constant of proportionality is simply a design-dependent multiplier (which does not necessarily tend to unity). Explicit results are presented for randomized block and Latin square designs and are illustrated for factorial treatment designs and split-plot experiments. The distribution of the univariate test statistic is close to a standard F-distribution, although it can be slightly underdispersed. For a complex design, the test assesses homogeneity of variance across blocks, treatments, or treatment factors and offers an objective interpretation of residual plots.     

Microarray Data Analysis

The analysis of differential gene expression in microarray experiments requires the development of adequate statistical tools. This article describes a simple statistical method for detecting differential expression between two conditions with a low number of replicates. When comparing two group means using a traditional t-test, gene-specific variance estimates are unstable and can lead to wrong conclusions. We construct a likelihood ratio test while modelling these variances hierarchically across all genes, and express it as a t-test statistic. By borrowing information across genes we can take advantage of their large numbers, and still yield a gene-specific test statistic. We show that this hierarchical t-test is more powerful than its traditional version and generates less false positives in a simulation study, especially with small sample sizes. This approach can be extended to cases where there are more than two groups.     

Designing fermentation media: A comparison of neural networks to factorial design

Summary Neural networks were compared to factorial experiments as techniques for designing fermentation media. The production of intracellular oil byRhodotorula gracilis (Rhodosporidium toruloides) was used as a model system. Investigating three factors, the molasses, ammonium nitrate and yeast extract concentrations, each at three concentrations, 27 experiments were required for a complete factorial expriment. In contrast, neural networks could be trained on 10 experiments and predict the test experiments with reasonable accuracy. This represents a 63% saving in the number of experiments that need to be conducted. Thus neural networks are a useful tool in developing fermentation medium.     

A cautionary note on exact unconditional inference for a difference between two independent binomial proportions

Fisher's exact test for comparing response proportions in a randomized experiment can be overly conservative when the group sizes are small or when the response proportions are close to zero or one. This is primarily because the null distribution of the test statistic becomes too discrete, a partial consequence of the inference being conditional on the total number of responders. Accordingly, exact unconditional procedures have gained in popularity, on the premise that power will increase because the null distribution of the test statistic will presumably be less discrete. However, we caution researchers that a poor choice of test statistic for exact unconditional inference can actually result in a substantially less powerful analysis than Fisher's conditional test. To illustrate, we study a real example and provide exact test size and power results for several competing tests, for both balanced and unbalanced designs. Our results reveal that Fisher's test generally outperforms exact unconditional tests based on using as the test statistic either the observed difference in proportions, or the observed difference divided by its estimated standard error under the alternative hypothesis, the latter for unbalanced designs only. On the other hand, the exact unconditional test based on the observed difference divided by its estimated standard error under the null hypothesis (score statistic) outperforms Fisher's test, and is recommended. Boschloo's test, in which the p-value from Fisher's test is used as the test statistic in an exact unconditional test, is uniformly more powerful than Fisher's test, and is also recommended.     

A KPSS test for stationarity for spatial point processes

Summary .   We propose a formal method to test stationarity for spatial point processes. The proposed test statistic is based on the integrated squared deviations of observed counts of events from their means estimated under stationarity. We show that the resulting test statistic converges in distribution to a functional of a two-dimensional Brownian motion. To conduct the test, we compare the calculated statistic with the upper tail critical values of this functional. Our method requires only a weak dependence condition on the process but does not assume any parametric model for it. As a result, it can be applied to a wide class of spatial point process models. We study the efficacy of the test through both simulations and applications to two real data examples that were previously suspected to be nonstationary based on graphical evidence. Our test formally confirmed the suspected nonstationarity for both data.     

A Bootstrap Procedure for Adaptive Selection of the Test Statistic in Flexible Two‐Stage Designs

Adaptive two‐stage designs allow a data‐driven change of design characteristics during the ongoing trial. One of the available options is an adaptive choice of the test statistic for the second stage of the trial based on the results of the interim analysis. Since there is often only a vague knowledge of the distribution shape of the primary endpoint in the planning phase of a study, a change of the test statistic may then be considered if the data indicate that the assumptions underlying the initial choice of the test are not correct. Collings and Hamilton proposed a bootstrap method for the estimation of the power of the two‐sample Wilcoxon test for shift alternatives. We use this approach for the selection of the test statistic. By means of a simulation study, we show that the gain in terms of power may be considerable when the initial assumption about the underlying distribution was wrong, whereas the loss is relatively small when in the first instance the optimal test statistic was chosen. The results also hold true for comparison with a one‐stage design. Application of the method is illustrated by a clinical trial example.     

Statistical methods of translating microarray data into clinically relevant diagnostic information in colorectal cancer

MOTIVATION: It is a common practice in cancer microarray experiments that a normal tissue is collected from the same individual from whom the tumor tissue was taken. The indirect design is usually adopted for the experiment that uses a common reference RNA hybridized both to normal and tumor tissues. However, it is often the case that the test material is not large enough for the experimenter to extract enough RNA to conduct the microarray experiment. Hence, collecting n cases does not necessarily end up with a matched pair sample of size n. Instead we usually have a matched pair sample of size n1, and two independent samples of sizes n2 and n3, respectively, for 'reference versus normal tissue only' and 'reference versus tumor tissue only' hybridizations (n=n1 + n2 + n3). Standard statistical methods need to be modified and new statistical procedures are developed for analyzing this mixed dataset. RESULTS: We propose a new test statistic, t3, as a means of combining all the information in the mixed dataset for detecting differentially expressed (DE) genes between normal and tumor tissues. We employed the extended receiver operating characteristic approach to the mixed dataset. We devised a measure of disagreement between a RT-PCR experiment and a microarray experiment. Hotelling's T2 statistic is employed to detect a set of DE genes and its prediction rate is compared with the prediction rate of a univariate procedure. We observe that Hotelling's T2 statistic detects DE genes more efficiently than a univariate procedure and that further research is warranted on the formal test procedure using Hotelling's T2 statistic. CONTACT: bskim@yonsei.ac.kr.     

Construction of null statistics in permutation-based multiple testing for multi-factorial microarray experiments

MOTIVATION: The parametric F-test has been widely used in the analysis of factorial microarray experiments to assess treatment effects. However, the normality assumption is often untenable for microarray experiments with small replications. Therefore, permutation-based methods are called for help to assess the statistical significance. The distribution of the F-statistics across all the genes on the array can be regarded as a mixture distribution with a proportion of statistics generated from the null distribution of no differential gene expression whereas the other proportion of statistics generated from the alternative distribution of genes differentially expressed. This results in the fact that the permutation distribution of the F-statistics may not approximate well to the true null distribution of the F-statistics. Therefore, the construction of a proper null statistic to better approximate the null distribution of F-statistic is of great importance to the permutation-based multiple testing in microarray data analysis. RESULTS: In this paper, we extend the ideas of constructing null statistics based on pairwise differences to neglect the treatment effects from the two-sample comparison problem to the multifactorial balanced or unbalanced microarray experiments. A null statistic based on a subpartition method is proposed and its distribution is employed to approximate the null distribution of the F-statistic. The proposed null statistic is able to accommodate unbalance in the design and is also corrected for the undue correlation between its numerator and denominator. In the simulation studies and real biological data analysis, the number of true positives and the false discovery rate (FDR) of the proposed null statistic are compared with those of the permutated version of the F-statistic. It has been shown that our proposed method has a better control of the FDRs and a higher power than the standard permutation method to detect differentially expressed genes because of the better approximated tail probabilities.     

脑电信号的高阶奇异谱分析

奇异谱分析是脑电信号分析的一种新方法,脑电信号的奇异谱可以反映脑电的特征,它有助于研究大脑的动力学行为。奇异谱分析方法是基于二阶统计的方法,反映的是信号时间上和空间上的一种线性相关关系。而脑电信号属于非线性信号,其内在的非线性关系很难通过奇异谱得到真实的反映,从而会丢失某些有用的信息。提出一种新的基于高阶统计的脑电奇异谱分析方法,并将其运用于正常脑电和癫痫患者的脑电分析中。大量的实测信号样本仿真实验结果表明,正常脑电和癫痫脑电的奇异谱有明显的不同。此外,基于高阶统计的奇异谱和基于二阶统计的奇异谱相比更能反映出信号的细节。     

生物多样性和均匀度显著性的随机化检验及计算软件

多样性指数和均匀度以其简单易用而被广泛应用于群落生物学和生物多样性等研究中,然而由于缺乏合适的统计检验方法等原因,其分析的可信性往往较低,因而限制了其应用。鉴于生物多样性研究中广泛应用主观和直接的比较不,有必要建立和使用较为严格的多样性统计检验。本研究建立和应用了如下随机化检验方法：单群落多样性指数和均匀度的显著性检验,单群落多样性指数和均匀度的置认区间,群落间多群样和均匀度的差异显著性检验。随机化方法已被成功地应用于群落生态学研究,其原理是：随机排序某一向量中的元素,或随机交换两向量中的对应元素。计算该随机化数据的多样性和均匀度,重复该过程多次,统计和计算显著性检验的p值,由向量中的对应元素。计算该随机化数据的多样性和均匀度,重复该过程多次,统计和显著性检验的p值。由此可确定多样性和差异的统计显著性。同时,研制了相应的Internet计算软件BiodiverisytTest。该软件由7个Java类和1个HTML文件组成,可运行于多种操作系统和网络浏览器上,可读取多种类型的ODBC数据库文件如Access,Excel,FoxPro,Dbase等。该软件中包括Shannon-Wiener多样性指数,Simpson多样性指数,McIntosh多样性指数,Berger-Parker多样性指数,Hrlbert多样性指数以及Brillouin多样性指数。基于Shannon-Wiener多样性指数和Berger-Parker多样性指数,用BiodiversityTest软件对水稻田节肢动物群落多样性（15个地点,17个功能群,125个节肢动物种）进行了比较和分析。结果显示,两组结果可较好地反映水稻节肢动物群落多样性的差异显著性,这些检验方法可有效地反映多样性指数和均匀度的变化。与水稻田节肢动物群落间多样性的直接比较法相比,该随机化检验方法获得更客观的结果。本算法与软件有助于改进生物多样性研究中使用的某些不甚严格的分析方法,为随机化检验方法在生物多样性研究中的进一步应用提供了一种可用的工具。     

On the use of permutation in and the performance of a class of nonparametric methods to detect differential gene expression

MOTIVATION: Recently a class of nonparametric statistical methods, including the empirical Bayes (EB) method, the significance analysis of microarray (SAM) method and the mixture model method (MMM), have been proposed to detect differential gene expression for replicated microarray experiments conducted under two conditions. All the methods depend on constructing a test statistic Z and a so-called null statistic z. The null statistic z is used to provide some reference distribution for Z such that statistical inference can be accomplished. A common way of constructing z is to apply Z to randomly permuted data. Here we point our that the distribution of z may not approximate the null distribution of Z well, leading to possibly too conservative inference. This observation may apply to other permutation-based nonparametric methods. We propose a new method of constructing a null statistic that aims to estimate the null distribution of a test statistic directly. RESULTS: Using simulated data and real data, we assess and compare the performance of the existing method and our new method when applied in EB, SAM and MMM. Some interesting findings on operating characteristics of EB, SAM and MMM are also reported. Finally, by combining the idea of SAM and MMM, we outline a simple nonparametric method based on the direct use of a test statistic and a null statistic.     

A Statistical Test of a Neutral Model Using the Dynamics of Cytonuclear Disequilibria

In this paper we use cytonuclear disequilibria to test the neutrality of mtDNA markers. The data considered here involve sample frequencies of cytonuclear genotypes subject to both statistical sampling variation as well as genetic sampling variation. First, we obtain the dynamics of the sample cytonuclear disequilibria assuming random drift alone as the source of genetic sampling variation. Next, we develop a test statistic using cytonuclear disequilibria via the theory of generalized least squares to test the random drift model. The null distribution of the test statistic is shown to be approximately chi-squared using an asymptotic argument as well as computer simulation. Power of the test statistic is investigated under an alternative model with drift and selection. The method is illustrated using data from cage experiments utilizing different cytonuclear genotypes of Drosophila melanogaster. A program for implementing the neutrality test is available upon request.     

The analysis of a factorial series of insecticide tests

When a set of insecticidal toxicity tests yields parallel regression lines for the relationship between mortality probit and log dose, the potencies of the materials or conditions under test may be compared purely in terms of log L.D. 50's. The purpose of this paper is to suggest that, when tests have been made with all combinations of several different factors, standard methods for the statistical analysis of factorial experiments may be adapted to the examination of the relative potencies.
Data obtained by Potter & Gillham (1046), in a 2' factorial experiment on alternative storage conditions for insects before and after spraying and the adjuvant action of terpineol in a pyrethrins spray, are used in an example of the computations. Details are given of the test of parallelism of the regression lines, the factorial analysis of the log L.D. '50s, the estimation of the mean effects and interactions and their standard errors, the significance tests, and the preparation of summary tables.     

Testing equality of survival functions of quality-adjusted lifetime

We present a method for comparing the survival functions of quality-adjusted lifetime from two treatments. This test statistic becomes the ordinary log-rank test when quality-adjusted lifetime is the same as the survival time. Simulation experiments are conducted to examine the behavior of our proposed test statistic under both null and alternative hypotheses. In addition, we apply our method to a breast cancer trial for comparing the distribution of quality-adjusted lifetime between two treatment regimes.     

Application of Taguchi method in optimization of cervical ring cage

The Taguchi method is a statistical approach to overcome the limitation of the factorial and fractional factorial experiments by simplifying and standardizing the fractional factorial design. The objective of the current study is to illustrate the procedures and strengths of the Taguchi method in biomechanical analysis by using a case study of a cervical ring cage optimization. A three-dimensional finite element (FE) model of C(5)-C(6) with a generic cervical ring cage inserted was modelled. Taguchi method was applied in the optimization of the cervical ring cage in material property and dimensions for producing the lowest stress on the endplate to reduce the risk of cage subsidence, as in the following steps: (1) establishment of objective function; (2) determination of controllable factors and their levels; (3) identification of uncontrollable factors and test conditions; (4) design of Taguchi crossed array layout; (5) execution of experiments according to trial conditions; (6) analysis of results; (7) determination of optimal run; (8) confirmation of optimum run. The results showed that a cage with larger width, depth and wall thickness can produce the lower von Mises stress under various conditions. The contribution of implant materials is found trivial. The current case study illustrates that the strengths of the Taguchi method lie in (1) consistency in experimental design and analysis; (2) reduction of time and cost of experiments; (3) robustness of performance with removing the noise factors. The Taguchi method will have a great potential application in biomechanical field when factors of the issues are at discrete level.     

A Robust Statistical Method for Detecting Differentially Expressed Genes

DNA microarray technology allows researchers to monitor the expressions of thousands of genes under different conditions, and to measure the levels of thousands of different DNA molecules at a given point in the life of an organism, tissue or cell. A wide variety of different diseases that are characterised by unregulated gene expression, DNA replication, cell division and cell death, can be detected early using microarrays. One of the major objectives of microarray experiments is to identify differentially expressed genes under various conditions. The detection of differential gene expression under two different conditions is very important in biological studies, and allows us to identify experimental variables that affect different biological processes. Most of the tests available in the literature are based on the assumption of normal distribution. However, the assumption of normality may not be true in real-life data, particularly with respect to microarray data.A test is proposed for the identification of differentially expressed genes in replicated microarray experiments conducted under two different conditions. The proposed test does not assume the distribution of the parent population; thus, the proposed test is strictly nonparametric in nature. We calculate the p-value and the asymptotic power function of the proposed test statistic. The proposed test statistic is compared with some of its competitors under normal, gamma and exponential population setup using the Monte Carlo simulation technique. The application of the proposed test statistic is presented using microarray data. The proposed test is robust and highly efficient when populations are non-normal.