The performances of the chi-square test and complexity measures for signal recognition in biological sequences

With large amounts of experimental data, modern molecular biology needs appropriate methods to deal with biological sequences. In this work, we apply a statistical method (Pearson's chi-square test) to recognize the signals appear in the whole genome of the Escherichia coli. To show the effectiveness of the method, we compare the Pearson's chi-square test with linguistic complexity on the complete genome of E. coli. The results suggest that Pearson's chi-square test is an efficient method for distinguishing genes (coding regions) form pseudogenes (noncoding regions). On the other hand, the performance of the linguistic complexity is much lower than the chi-square test method. We also use the Pearson's chi-square test method to determine which parts of the Open Reading Frame (ORF) have significant effect on discriminating genes form pseudogenes. Moreover, different complexity measures and Pearson's chi-square test applied on the genes with high value of Pearson's chi-square statistic. We also compute the measures on homologous of these genes. The results illustrate that there is a region near the start codon with high value of chi-square statistic and low complexity that is conserve between homologous genes.     

A reduction factor in goodness-of-fit and independence tests for clustered and weighted observations

Situations often arise in a large-scale household survey where a complex probability sample of clusters rather than of individuals is drawn from a large population. Typically, the clusters of such complex samples include a number of correlated members. The responses of these members are then weighted to obtain estimates for the population. Such weighted data are commonly published by the National Center for Health Statistics and other U.S. federal agencies. Frequently, problems arise when such data are tested by usual chi-square test statistics for goodness of fit or independence. Researchers have discovered that the usual chi-square tests provide spuriously inflated results when applied to cluster samples and that new methods are required to correct such problems. This paper proposes a strategy for a goodness-of-fit or independence test based on correlated and weighted data arising in cluster samples, and provides a factor that validly reduces the inflation of the usual chi-square statistics. This method is applied to the chronic condition data collected from the St Paul-Minneapolis, Minnesota, primary sampling unit (PSU) during the 1975 National Health Interview Survey (NHIS). This analysis, together with simulation studies presented elsewhere, provides evidence that the usual chi-square statistics from such data can be corrected for the impacts of clustering and weighting by use of the proposed reduction factor.     

The choice of a statistic for testing hypotheses regarding seasonality

This paper discusses statistical methods for testing hypotheses of seasonality in births across one year, using monthly frequencies. It argues that the Freedman test, a variant of the Kolmogorov-Smirnov one-sample test, and the Edwards test and its modifications are most appropriate for this purpose. These tests take into account the order of the monthly frequencies and have a higher power than the ordinary chi-square goodness-of-fit test, regarding seasonality. The paper addresses some points mentioned in earlier papers by McCullough, and O'Brien and Holbert.     

On the "NPD ratio" as a test for crossover interference

The "NPD ratio," widely used by yeast geneticists, is of limited applicability and is prone to falsely indicate significant crossover interference in a chi-square test. A simple, better chi-square test for interference in two-factor crosses is described.     

One Degree of Freedom for Non-Additivity in a Contingency Table and Its Multivariate Extension

The paper develops a chi-square test with one degree of freedom for row-column interaction in a contingency table. Use is made of the JOHNSON-GRAYBILL [1972] model which is herein generalized to a twoway multiple response situation. Some of the merits of the new test over the better-known tests for the same hypothesis are discussed.     

Maximally selected chi-square statistics for ordinal variables

The association between a binary variable Y and a variable X having an at least ordinal measurement scale might be examined by selecting a cutpoint in the range of X and then performing an association test for the obtained 2 x 2 contingency table using the chi-square statistic. The distribution of the maximally selected chi-square statistic (i.e. the maximal chi-square statistic over all possible cutpoints) under the null-hypothesis of no association between X and Y is different from the known chi-square distribution. In the last decades, this topic has been extensively studied for continuous X variables, but not for non-continuous variables of at least ordinal measurement scale (which include e.g. classical ordinal or discretized continuous variables). In this paper, we suggest an exact method to determine the finite-sample distribution of maximally selected chi-square statistics in this context. This novel approach can be seen as a method to measure the association between a binary variable and variables having an at least ordinal scale of different types (ordinal, discretized continuous, etc). As an illustration, this method is applied to a new data set describing pregnancy and birth for 811 babies.     

The Relative Power of Two Asymptotic Multivariate Tests for Homogeneity in a Contingency Table

In a three way contingency table two multivariate tests for homogeneity have been proposed by the author (1983) a the “catanova” test, which is a trace “metric” test and b the “multinova” test which is determinant based. Both tests are asymptotically distributed as chi-square. In this paper, the power values of the tests are compared and conditions are given for preference of each test.     

Computing power and sample size for case-control association studies with copy number polymorphism: application of mixture-based likelihood ratio test

Recent studies suggest that copy number polymorphisms (CNPs) may play an important role in disease susceptibility and onset. Currently, the detection of CNPs mainly depends on microarray technology. For case-control studies, conventionally, subjects are assigned to a specific CNP category based on the continuous quantitative measure produced by microarray experiments, and cases and controls are then compared using a chi-square test of independence. The purpose of this work is to specify the likelihood ratio test statistic (LRTS) for case-control sampling design based on the underlying continuous quantitative measurement, and to assess its power and relative efficiency (as compared to the chi-square test of independence on CNP counts). The sample size and power formulas of both methods are given. For the latter, the CNPs are classified using the Bayesian classification rule. The LRTS is more powerful than this chi-square test for the alternatives considered, especially alternatives in which the at-risk CNP categories have low frequencies. An example of the application of the LRTS is given for a comparison of CNP distributions in individuals of Caucasian or Taiwanese ethnicity, where the LRTS appears to be more powerful than the chi-square test, possibly due to misclassification of the most common CNP category into a less common category.     

A Modified Chi-Square Test for Testing Equality of Two Multinomial Populations Against an Ordering Restricted Alternative

A modified chi-square test for testing the equality of two multinomial populations against an ordering restricted alternative in one sample and two sample cases is constructed. The relation between a concept of dependence called dependence by chi-square and stochastic ordering is established. A tabulation of the asymptotic distribution of the test statistic under the null hypothesis is given. Simulations are used to compare the power of this test with the power of the likelihood ratio test of stochastic ordering of the two multinomial populations.     

Product Quality Research Institute evaluation of cascade impactor profiles of pharmaceutical aerosols: Part 2—Evaluation of a method for determining equivalence

The purpose of this article is to present the thought process, methods, and interim results of a PQRI Working Group, which was charged with evaluating the chi-square ratio test as a potential method for determining in vitro equivalence of aerodynamic particle size distribution (APSD) profiles obtained from cascade impactor measurements. Because this test was designed with the intention of being used as a tool in regulatory review of drug applications, the capability of the test to detect differences in APSD profiles correctly and consistently was evaluated in a systematic way across a designed space of possible profiles. To establish a “base line,” properties of the test in the simplest case of pairs of identical profiles were studied. Next, the test's performance was studied with pairs of profiles, where some difference was simulated in a systematic way on a single deposition site using realistic product profiles. The results obtained in these studies, which are presented in detail here, suggest that the chi-square ratio test in itself is not sufficient to determine equivalence of particle size distributions. This article, therefore, introduces the proposal to combine the chi-square ratio test with a test for impactor-sized mass based on Population Bioequivalence and describes methods for evaluating discrimination capabilities of the combined test. The approaches and results described in this article elucidate some of the capabilities and limitations of the original chi-square ratio test and provide rationale for development of additional tests capable of comparing APSD profiles of pharmaceutical aerosols. Published: January 19, 2007     

One Parameter Model for Three-Factor Interactions in Contingency Tables

This paper shows that the sum of products models for three and higher order interactions in contingency tables can be reparameterized in the spirit of TUKEY (1949) to yield chi-square tests with one degree of freedom. The merits of this new test over the other known tests for the same hypotheses are discussed.     

Asymptotic Efficiency of the Chi-Square Test for Heterogeneity of Proportions in Some Sparse-Data Cases

We consider the problem of testing for heterogeneity of K proportions when K is not small and the binomial sample sizes may not be large. We assume that the binomial proportions are normally distributed with variance σ². The asymptotic relative efficiency (ARE) of the usual chi-square test is found relative to the likelihood-based tests for σ²=0. The chi-square test is found to have ARE = 1 when the binomial sample sizes are all equal and high relative efficiency for other cases. The efficiency is low only in cases where there is insufficient data to use the chi-square test.     

A disequilibrium coefficient approach to Hardy-Weinberg testing

A comparison was made of various tests for Hardy-Weinberg equilibrium, with emphasis on methods for multiple alleles. For an overall test of deviations from equilibrium, the classical chi-square goodness-of-fit test generally performs well, with continuity corrections needed for extreme gene frequencies or extreme departures from equilibrium. For small samples, probability tests are preferable and for multiple alleles these probability tests may be performed on a sample of all possible sets of genotypic frequencies having a fixed set of sample gene frequencies. Numerical work showed that the continuity-corrected chi-square was the most conservative test procedure, and the uncorrected chi-square the least conservative. With multiple alleles, a better appreciation of the nature of departures from equilibrium is given by the use of disequilibrium coefficients, defined for each heterozygote as the difference between observed and expected frequencies. Likelihood-ratio tests can be used to test each of these coefficients individually but a satisfactory procedure is to divide the squared estimate of each coefficient by its estimated variance and regard the ratio as a single-degree-of-freedom chi-square. Numerical studies confirmed the validity of this approach, which has the great advantage of not requiring solutions of nonlinear equations.     

On testing departure from the binomial and multinomial assumptions

This paper is concerned with testing the multinomial (binomial) assumption against the Dirichlet-multinomial (beta-binomial) alternatives. In particular, we discuss the distribution of the asymptotic likelihood ratio (LR) test and obtain the C(alpha) goodness-of-fit test statistic. The inadequacy of the regular chi-square approximation to the LR test is supported by some Monte Carlo experiments. The C(alpha) test is recommended based on empirical significance level and power and also computational simplicity. Two examples are given.     

基于统计差表与加权投票的高精度剪接位点预测

基于机器学习的高精度剪接位点识别是真核生物基因组注释的关键.本文采用卡方测验确定序列窗口长度,构建卡方统计差表提取位置特征,并结合碱基二联体频次表征序列;针对剪接位点正负样本高度不均衡这一情形,构建10个正负样本均衡的支持向量机分类器,进行加权投票决策,有效解决了不平衡模式分类问题. HS~3D数据集上的独立测试结果显示,供体、受体位点预测准确率分别达到93.39%、90.46%,明显高于参比方法.基于卡方统计差表的位置特征能有效表征DNA序列,在分子序列信号位点识别中具有应用前景.     

Product quality research institute evaluation of cascade impactor profiles of pharmaceutical aerosols, part 3: Final report on a statistical procedure for determining equivalence

The purpose of this article is to report final results of the evaluation of a chi-square ratio test proposed by the US Food and Drug Administration (FDA) for demonstrating equivalence of aerodynamic particle size distribution (APSD) profiles of nasal and orally inhaled drug products. A working group of the Product Quality Research Institute previously published results demonstrating some limitations of the proposed test. In an effort to overcome the test's limited discrimination, the group proposed a supplemental test, a population bioequivalence (PBE) test for impactor-sized mass (ISM). In this final report the group compares the chi-square ratio test to the ISM-PBE test and to the combination of both tests. The basis for comparison is a set of 55 realistic scenarios of cascade impactor data, which were evaluated for equivalence by the statistical tests and independently by the group members. In many instances, the combined application of these 2 tests appeared to increase the discriminating ability of the statistical procedure compared with the chi-square ratio test alone. In certain situations the chi-square ratio test alone was sufficient to determine equivalence of APSD profiles, while in other situations neither of the tests alone nor their combination was adequate. This report describes all of these scenarios and results. In the end, the group did not recommend a statistical test for APSD profile equivalence. The group did not investigate other in vitro tests, in vivo issues, or other statistical tests for APSD profile comparisons. The studied tests are not intended for routine quality control of APSD.     

Testing Hypotheses about Occupancy Rates for Clustered Populations

The use of the Pearson chi-square statistic for testing hypotheses on biological populations is not appropriate when the individuals are distributed by clusters. In the case where the clusters are distributed independently of each other, we propose an asymptotically chi-square distributed test statistic taking into account the cluster size distribution. An example provided by European Corn Borer eggs data is used to illustrate the test procedure.     

Power and sample size for testing homogeneity of relative risks in prospective studies

Power and sample-size formulas for testing the homogeneity of relative risks using the score method are presented. The homogeneity score test (Gart, 1985, Biometrika 72, 673-677) is formally equivalent to the Pearson chi-square test, although they look different. Results of this paper may be useful in assessing the validity of the model of a common relative risk before combining several 2 x 2 tables or in designing a prospective study for detecting heterogeneity of relative risks.     

A Report on a Monte Carlo Simulation of the Sampling Distribution of the Incidence Test and its Two Related Tests

A large-scale (5000 throws) Monte Carlo simulation experiment was carried out to study the nature of the sampling distributions of the incidence test and its two related tests, the FRIEDMAN test and the dual test, and to evaluate the goodness of the proposed gamma approximation relative to the conventional chi-square approximation. For all three tests, on the basis of the experimental results obtained for k = 3, 4, 5, and N = k(1) (120/k), the gamma approximation should be preferred over the chi-square. Exact companion tabulations show that use of the chi-square approximation entails an appreciable conservative bias. Extensive listings are provided of the moments of the experimental distributions of the three test statistics, and of the observed distributions of the corresponding gamma probabilities. The simulation experiment was run on a UNIVAC 1108 large-scale computer system.     

Estimating the size of closed populations using inverse multiple-recapture sampling

A log-linear model for estimating the size of a closed population is defined for inverse multiple-recapture sampling with dependent samples. Efficient estimators of the log-linear model parameters and the population size are obtained by the method of minimum chi-square. A chi-square test of the general linear hypothesis regarding the log-linear model parameters is defined.