Statistical methods to decide whether mutagenicity test data from Drosophila assays indicate a positive, negative, or inconclusive result

Two alternative hypotheses are used to distinguish among the possibilities of a positive, inconclusive, or negative result in Drosophila mutagenicity tests. In the null hypothesis one assumes that there is no difference in the mutation frequency between control and treated series. The alternative hypothesis postulates a priori that the treatment results in an increased mutation frequency that is m times the spontaneous frequency. To test against the hypotheses, the conditional binomial test according to Kastenbaum and Bowman or the chi 2 test for proportions may be applied. These 2 methods are in principle equivalent. An alternative method which is based on determining confidence limits of observed mutation frequencies also leads to the same conclusions. The practical calculations are formulated and an application is shown with a test example demonstrating the genotoxicity of the pyrrolizidine alkaloid 7-acetylintermedine in the somatic wing mosaic test. In the Appendix, the calculus for the 3 testing methods is explained with a numerical example.     

Theoretical and empirical power of regression and maximum-likelihood methods to map quantitative trait loci in general pedigrees

Both theoretical calculations and simulation studies have been used to compare and contrast the statistical power of methods for mapping quantitative trait loci (QTLs) in simple and complex pedigrees. A widely used approach in such studies is to derive or simulate the expected mean test statistic under the alternative hypothesis of a segregating QTL and to equate a larger mean test statistic with larger power. In the present study, we show that, even when the test statistic under the null hypothesis of no linkage follows a known asymptotic distribution (the standard being chi(2)), it cannot be assumed that the distribution under the alternative hypothesis is noncentral chi(2). Hence, mean test statistics cannot be used to indicate power differences, and a comparison between methods that are based on simulated average test statistics may lead to the wrong conclusion. We illustrate this important finding, through simulations and analytical derivations, for a recently proposed new regression method for the analysis of general pedigrees to map quantitative trait loci. We show that this regression method is not necessarily more powerful nor computationally more efficient than a maximum-likelihood variance-component approach. We advocate the use of empirical power to compare trait-mapping methods.     

A Bayesian analysis of the two-period crossover design for clinical trials

Statisticians have been critical of the use of the two-period crossover designs for clinical trials because the estimate of the treatment difference is biased when the carryover effects of the two treatments are not equal. In the standard approach, if the null hypothesis of equal carryover effects is not rejected, data from both periods are used to estimate and test for treatment differences; if the null hypothesis is rejected, data from the first period alone are used. A Bayesian analysis based on the Bayes factor against unequal carryover effects is given. Although this Bayesian approach avoids the "all-or-nothing" decision inherent in the standard approach, it recognizes that with small trials it is difficult to provide unequivocal evidence that the carryover effects of the two treatments are equal, and thus that the interpretation of the difference between treatment effects is highly dependent on a subjective assessment of the reality or not of equal carryover effects.     

The transmission/disequilibrium test: history, subdivision, and admixture.

Disease association with a genetic marker is often taken as a preliminary indication of linkage with disease susceptibility. However, population subdivision and admixture may lead to disease association even in the absence of linkage. In a previous paper, we described a test for linkage (and linkage disequilibrium) between a genetic marker and disease susceptibility; linkage is detected by this test only if association is also present. This transmission/disequilibrium test (TDT) is carried out with data on transmission of marker alleles from parents heterozygous for the marker to affected offspring. The TDT is a valid test for linkage and association, even when the association is caused by population subdivision and admixture. In the previous paper, we did not explicitly consider the effect of recent history on population structure. Here we extend the previous results by examining in detail the effects of subdivision and admixture, viewed as processes in population history. We describe two models for these processes. For both models, we analyze the properties of (a) the TDT as a test for linkage (and association) between marker and disease and (b) the conventional contingency statistic used with family data to test for population association. We show that the contingency test statistic does not have a chi 2 distribution if subdivision or admixture is present. In contrast, the TDT remains a valid chi 2 statistic for the linkage hypothesis, regardless of population history.     

Maximally selected chi2 statistics for k x 2 tables

It is common in epidemiologic analyses to summarize continuous outcomes as falling above or below a threshold. With such a dichotomized outcome, the usual chi2 statistics for association or trend can be used to test for equality of proportions across strata of the study population. However, if the threshold is chosen to maximize the test statistic, the nominal chi2 reference distributions are incorrect. In this paper, the asymptotic distributions of maximally selected chi2 statistics for association and for trend for the k x 2 table are derived. The methodology is illustrated with data from an AIDS clinical trial. The results of simulation experiments that assess the accuracy of the asymptotic distributions in moderate sample sizes are also reported.     

On computation of p-values in parametric linkage analysis

Parametric linkage analysis is usually used to find chromosomal regions linked to a disease (phenotype) that is described with a specific genetic model. This is done by investigating the relations between the disease and genetic markers, that is, well-characterized loci of known position with a clear Mendelian mode of inheritance. Assume we have found an interesting region on a chromosome that we suspect is linked to the disease. Then we want to test the hypothesis of no linkage versus the alternative one of linkage. As a measure we use the maximal lod score Z(max). It is well known that the maximal lod score has asymptotically a (2 ln 10)(-1) x (1/2 chi2(0) + 1/2 chi2(1)) distribution under the null hypothesis of no linkage when only one point (one marker) on the chromosome is studied. In this paper, we show, both by simulations and theoretical arguments, that the null hypothesis distribution of Zmax has no simple form when more than one marker is used (multipoint analysis). In fact, the distribution of Zmax depends on the number of families, their structure, the assumed genetic model, marker denseness, and marker informativity. This means that a constant critical limit of Zmax leads to tests associated with different significance levels. Because of the above-mentioned problems, from the statistical point of view the maximal lod score should be supplemented by a p-value when results are reported.     

Approximate likelihood-ratio test for branches: A fast, accurate, and powerful alternative

We revisit statistical tests for branches of evolutionary trees reconstructed upon molecular data. A new, fast, approximate likelihood-ratio test (aLRT) for branches is presented here as a competitive alternative to nonparametric bootstrap and Bayesian estimation of branch support. The aLRT is based on the idea of the conventional LRT, with the null hypothesis corresponding to the assumption that the inferred branch has length 0. We show that the LRT statistic is asymptotically distributed as a maximum of three random variables drawn from the chi(0)2 + chi(1)2 distribution. The new aLRT of interior branch uses this distribution for significance testing, but the test statistic is approximated in a slightly conservative but practical way as 2(l1- l2), i.e., double the difference between the maximum log-likelihood values corresponding to the best tree and the second best topological arrangement around the branch of interest. Such a test is fast because the log-likelihood value l2 is computed by optimizing only over the branch of interest and the four adjacent branches, whereas other parameters are fixed at their optimal values corresponding to the best ML tree. The performance of the new test was studied on simulated 4-, 12-, and 100-taxon data sets with sequences of different lengths. The aLRT is shown to be accurate, powerful, and robust to certain violations of model assumptions. The aLRT is implemented within the algorithm used by the recent fast maximum likelihood tree estimation program PHYML (Guindon and Gascuel, 2003).     

Polymorphism of genes of the renin-angiotensin system ACE, AT1R, and AT2R in patients with pulmonary tuberculosis

The goal of this work was to verify the hypothesis about the possible role of some genes of the renin-angiotensin system in the innate immunity to tuberculosis. The insertion/deletion polymorphism (I/D) of the gene of the angiotensin-converting enzyme (ACE) is known to have an effect on the concentration of the angiotensin II in human body and also an indirect effect on various branches of metabolism. On the one hand, people with homozygote deletion of the ACE gene (DD genotype) are vulnerable to adiposity, arterial hypertension, hypercholesterolemia, and a number of other pathological conditions. On the other hand, it was shown that hypocholesterolemia is the general phenomenon for the patients with pulmonary tuberculosis (Perez-Guzman C. et al., Chest (2005)). In this work, we studied the I/D polymorphism of the gene ACE (genotypes DD, ID, and II), single nucleotide polymorphism (SNP) of the gene AT1R (1166 A/C), and SNP in 3123 positions of the gene AT2R (3123 A/C) in 200 patients with tuberculosis, 202 patients with essential hypertension, and 208 apparently healthy subjects. A group of patients with essential hypertension was used as a contrast group. According to the hypothesis stated above, the excess in the number of patients with the DD genotype (ACE) should be statistically significant in the group of patients with hypertension as compared to the group of patients with tuberculosis (chi2 = 9.64; chi2 = 0.0019; OR = 2.0; CI 1.2-3.3). There was a trend toward an increase in the rate of the DD genotype in the group of patients with tuberculosis relative to healthy subjects. Similar trend was observed in healthy subjects relative to the group of patients with hypertension. However, this difference was found to be statistically insignificant. The genotypes and allelotypes were compared in the group of patients with tuberculosis versus both the two control groups (healthy subjects and patients with hypertension). The significant difference from control was observed only in male rather than female patients with tuberculosis. It was shown that the greatest contribution to the distinction between groups was due to the genes ACE and AT2R. The combination of the genotypes of genes ACE and AT2R (ID+3123C) was met significantly more frequently in male patients with tuberculosis as compared to control group of healthy subjects (chi2 = 9.70; chi2 = 0.002; OR = 2.3; CI 1.2-4.3). The results obtained in this work are discussed in terms of the hypothesis stated above.     

A new statistical test for linkage heterogeneity.

A new, statistical test for linkage heterogeneity is described. It is a likelihood-ratio test based on a beta distribution for the prior distribution of the recombination fraction among families (or individuals). The null distribution for this statistic (called the B-test) is derived under a broad range of circumstances. Two other heterogeneity test statistics--the admixture test or A-test first described by Smith and Morton's test (here referred to as the K-test)--are also examined. The probability distribution for the K-test statistic is very sensitive to family size, whereas the other two statistics are not. All three statistics are somewhat sensitive to the magnitude of the recombination fraction theta. Critical values for each of the test statistics are given. A conservative approximation for both the A-test and B-test is given by a chi 2 distribution when P/2 instead of P is used for the observed significance level. In terms of power, the B-test performs best among the three tests over a broad range of alternate heterogeneity hypotheses--except for the specific case of admixture with loose linkage, in which the A-test performs best. Overall, the difference in power among the three tests is not large. An application to some recently published data on the fragile-X syndrome and X-chromosome markers is given.     

An Investigation on the Allelic Chi‐Square Test Used in Genetic Association Studies

Case‐control studies are primary study designs used in genetic association studies. Sasieni (Biometrics 1997, 53, 1253–1261) pointed out that the allelic chi‐square test used in genetic association studies is invalid when Hardy‐Weinberg equilibrium (HWE) is violated in a combined population. It is important to know how much type I error rate is deviated from the nominal level under violated HWE. We examine bounds of type I error rate of the allelic chi‐square test. We also investigate power of the goodness‐of‐fit test for HWE which can be used as a guideline for selecting an appropriate test between the allelic chi‐square test and the modified allelic chi‐square test, the latter of which was proposed for cases of violated HWE. In small samples, power is not large enough to detect the Wright's inbreeding model of small values of inbreeding coefficient. Therefore, when the null hypothesis of HWE is barely accepted, the modified test should be considered as an alternative method. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear tests for identifying differentially expressed genes or genetic networks

MOTIVATION: One of the recently developed statistics for identifying differentially expressed genetic networks is Hotelling T2 statistic, which is a quadratic form of difference in linear functions of means of gene expressions between two types of tissue samples, and so their power is limited. RESULTS: To improve the power of test statistics, a general statistical framework for construction of non-linear tests is presented, and two specific non-linear test statistics that use non-linear transformations of means are developed. Asymptotical distributions of the non-linear test statistics under the null and alternative hypothesis are derived. It has been proved that under some conditions the power of the non-linear test statistics is higher than that of the T2 statistic. Besides theory, to evaluate in practice the performance of the non-linear test statistics, they are applied to two real datasets. The preliminary results demonstrate that the P-values of the non-linear statistics for testing differential expressions of the genetic networks are much smaller than those of the T2 statistic. And furthermore simulations show the Type I errors of the non-linear statistics agree with the threshold used and the statistics fit the chi2 distribution. SUPPLEMENTARY INFORMATION: Supplementary data are available on Bioinformatics online.     

Correlation-based inference for linkage disequilibrium with multiple alleles

The correlation between alleles at a pair of genetic loci is a measure of linkage disequilibrium. The square of the sample correlation multiplied by sample size provides the usual test statistic for the hypothesis of no disequilibrium for loci with two alleles and this relation has proved useful for study design and marker selection. Nevertheless, this relation holds only in a diallelic case, and an extension to multiple alleles has not been made. Here we introduce a similar statistic, R(2), which leads to a correlation-based test for loci with multiple alleles: for a pair of loci with k and m alleles, and a sample of n individuals, the approximate distribution of n(k - 1)(m - 1)/(km)R(2) under independence between loci is chi((k-1)(m-1))(2). One advantage of this statistic is that it can be interpreted as the total correlation between a pair of loci. When the phase of two-locus genotypes is known, the approach is equivalent to a test for the overall correlation between rows and columns in a contingency table. In the phase-known case, R(2) is the sum of the squared sample correlations for all km 2 x 2 subtables formed by collapsing to one allele vs. the rest at each locus. We examine the approximate distribution under the null of independence for R(2) and report its close agreement with the exact distribution obtained by permutation. The test for independence using R(2) is a strong competitor to approaches such as Pearson's chi square, Fisher's exact test, and a test based on Cressie and Read's power divergence statistic. We combine this approach with our previous composite-disequilibrium measures to address the case when the genotypic phase is unknown. Calculation of the new multiallele test statistic and its P-value is very simple and utilizes the approximate distribution of R(2). We provide a computer program that evaluates approximate as well as "exact" permutational P-values.     

Locomotor-respiratory coupling during wheelchair propulsion.

Visceral movement due to impact loading is believed to play a role in the locomotor-respiratory coupling (LRC) that has been detected in a number of mammalian species. In the bird and bat species in which LRC has been described, the effect of the wing muscles on the timing of respiration appears to be a dominant influence. To test the hypothesis that LRC occurs in humans propelling wheelchairs (where there is no impact loading and the arms are used for locomotion), we studied 10 wheelchair athletes on a motorized treadmill at three speeds. Each subject's data were analyzed by spectral analysis (based on the fast Fourier transform), which detected apparent LRC (rates within 1% of a single-digit integer ratio) in 12 (40%) of the 30 test settings. However, a control analysis, in which each subject's arm-thrust rates were compared with another subject's breathing rates, revealed apparent (but false) coupling in 8 (27%), not significantly less often (using the chi 2 test). These findings appear to refute the hypothesis that LRC occurs during wheelchair propulsion. These data are consistent with the theory that the visceral piston is important to LRC and suggest that rhythmic arm movements are insufficient to induce the phenomenon in this setting.     

Sample size determination for confidence intervals on the population mean and on the difference between two population means

Sample size determination is usually based on the premise that a hypothesis test is to be used. A confidence interval can sometimes serve better than a hypothesis test. In this paper a method is presented for sample size determination based on the premise that a confidence interval for a simple mean, or for the difference between two means, with normally distributed data is to be used. For this purpose, a concept of power relevant to confidence intervals is given. Some useful tables giving required sample size using this method are also presented.     

Statistical quality control of variety trial data

I propose detection criteria for identifying an abnormal or erroneous data vector provided by a single variety trial in a longer series of variety trials. The test criteria are based on the linear effects estimated separately for each studied trial using global variance components estimated from the whole series of variety trials. The criteria comprise three mutually independent test statistics. The first one is a quadratic form in the estimated fixed effects, the second one is a quadratic form in the estimated realized linear random effects not including the residual effects, and the third one is a quadratic form in the estimated realized residual effects. Under the null hypothesis defining a valid data vector, the three quadratics have independent chi2 distributions. Under natural alternative hypotheses, they have noncentral chi2 distributions. Decomposing the total variation of the data vector studied into quadratic forms due to different types of the realized linear effects intuitively justifies the resulting test criteria. The decomposition may also be used to show that the resulting tests are likelihood ratio tests. I further present computational procedures that allow us to dispense with the need for prior estimation of the linear effects.     

A note on the conditional approach to interval estimation in the calibration problem.

In the calibration problem, the need to construct a confidence interval to estimate the unknown chi 0 arises when the null hypothesis of zero slope is rejected. Otherwise, the resulting confidence interval will be infinite to reflect the fact that the slope of the regression line may be zero. Under the condition of rejecting the hypothesis of zero slope, we study the properties of the conditional coverage rate of the calibration confidence interval. The conditional coverage rate (P1) is a function of the slope, distance between chi 0 and the mean of the trailing sample means, the sum of squares of chi, and n. When the true slope is close to 0 and chi 0 is away from means, P1 can go down to 0. On the other hand, as the power of testing zero slope reaches 1, with or without chi 0 close to means, P1 will tend to the desired nominal coverage rate. In summary, one should choose a reasonably small alpha in testing zero slope to avoid constructing a confidence interval for chi 0 when the true slope is 0. In addition, it is desirable to have high power in testing zero slope so that the resulting confidence interval will maintain the desired coverage rate when using the conditional approach in the calibration problem.     

An optical method for the determination of the isotopic composition of 1H2O/2H2O mixtures: Eu3+ luminescence lifetimes

A spectroscopic method employing pulsed dye laser instrumentation is described for the determination of the 1H2O/2H2O composition of aqueous solutions by the measurement of reciprocal excited state lifetimes of EuEDTA-. The reciprocal lifetimes, gamma-1, of the 1H2O/2H2O mixtures increase linearly with the mole fraction of 1H2O. For EuEDTA- the relationship between gamma-1 and the mole fraction, chi H, of 1H2O in 1H2O/2H2O mixtures is expressed by the equation chi H = 0.37 gamma-1-0.152, with a sensitivity in chi H of +/- 0.02. The reciprocal lifetimes are independent of pH in the range 5.1 to 10.5, changes in ionic strength, and the type of buffer used in EuEDTA- containing solutions.     

Testing for Homogeneity in Meta‐Analysis I. The One‐Parameter Case: Standardized Mean Difference

Summary Meta‐analysis seeks to combine the results of several experiments in order to improve the accuracy of decisions. It is common to use a test for homogeneity to determine if the results of the several experiments are sufficiently similar to warrant their combination into an overall result. Cochran’s Q statistic is frequently used for this homogeneity test. It is often assumed that Q follows a chi‐square distribution under the null hypothesis of homogeneity, but it has long been known that this asymptotic distribution for Q is not accurate for moderate sample sizes. Here, we present an expansion for the mean of Q under the null hypothesis that is valid when the effect and the weight for each study depend on a single parameter, but for which neither normality nor independence of the effect and weight estimators is needed. This expansion represents an order O(1/n) correction to the usual chi‐square moment in the one‐parameter case. We apply the result to the homogeneity test for meta‐analyses in which the effects are measured by the standardized mean difference (Cohen’s d‐statistic). In this situation, we recommend approximating the null distribution of Q by a chi‐square distribution with fractional degrees of freedom that are estimated from the data using our expansion for the mean of Q. The resulting homogeneity test is substantially more accurate than the currently used test. We provide a program available at the Paper Information link at the Biometrics website http://www.biometrics.tibs.org for making the necessary calculations.     

Tests for linkage and association in nuclear families.

The transmission/disequilibrium test (TDT) originally was introduced to test for linkage between a genetic marker and a disease-susceptibility locus, in the presence of association. Recently, the TDT has been used to test for association in the presence of linkage. The motivation for this is that linkage analysis typically identifies large candidate regions, and further refinement is necessary before a search for the disease gene is begun, on the molecular level. Evidence of association and linkage may indicate which markers in the region are closest to a disease locus. As a test of linkage, transmissions from heterozygous parents to all of their affected children can be included in the TDT; however, the TDT is a valid chi2 test of association only if transmissions to unrelated affected children are used in the analysis. If the sample contains independent nuclear families with multiple affected children, then one procedure that has been used to test for association is to select randomly a single affected child from each sibship and to apply the TDT to those data. As an alternative, we propose two statistics that use data from all of the affected children. The statistics give valid chi2 tests of the null hypothesis of no association or no linkage and generally are more powerful than the TDT with a single, randomly chosen, affected child from each family.