Critical values for unit root and cointegration test statistics - the use of response surface equations

This note considers the value of surface response equations which can be used to calculate critical values for a range of unit root and cointegration tests popular in applied economic research.

The convenient calculation of some test statistics in models of discrete choice

The paper considers the use of artificial regression in calculating different types of score test when the log

Incongruence between test statistics and P values in medical papers

Given an observed test statistic and its degrees of freedom, one may compute the observed P value with most statistical packages. It is unknown to what extent test statistics and P values are congruent in published medical papers. Methods:We checked the congruence of statistical results reported in all the papers of volumes 409–412 of Nature (2001) and a random sample of 63 results from volumes 322–323 of BMJ (2001). We also tested whether the frequencies of the last digit of a sample of 610 test statistics deviated from a uniform distribution (i.e., equally probable digits).Results: 11.6% (21 of 181) and 11.1% (7 of 63) of the statistical results published in Nature and BMJ respectively during 2001 were incongruent, probably mostly due to rounding, transcription, or type-setting errors. At least one such error appeared in 38% and 25% of the papers of Nature and BMJ, respectively. In 12% of the cases, the significance level might change one or more orders of magnitude. The frequencies of the last digit of statistics deviated from the uniform distribution and suggested digit preference in rounding and reporting.Conclusions: this incongruence of test statistics and P values is another example that statistical practice is generally poor, even in the most renowned scientific journals, and that quality of papers should be more controlled and valued

Incongruence between test statistics and P values in medical papers

Given an observed test statistic and its degrees of freedom, one may compute the observed P value with most statistical packages. It is unknown to what extent test statistics and P values are congruent in published medical papers. Methods: We checked the congruence of statistical results reported in all the papers of volumes 409–412 of Nature (2001) and a random sample of 63 results from volumes 322–323 of BMJ (2001). We also tested whether the frequencies of the last digit of a sample of 610 test statistics deviated from a uniform distribution (i.e., equally probable digits).Results: 11.6% (21 of 181) and 11.1% (7 of 63) of the statistical results published in Nature and BMJ respectively during 2001 were incongruent, probably mostly due to rounding, transcription, or type-setting errors. At least one such error appeared in 38% and 25% of the papers of Nature and BMJ, respectively. In 12% of the cases, the significance level might change one or more orders of magnitude. The frequencies of the last digit of statistics deviated from the uniform distribution and suggested digit preference in rounding and reporting.Conclusions: this incongruence of test statistics and P values is another example that statistical practice is generally poor, even in the most renowned scientific journals, and that quality of papers should be more controlled and valued

Bayes Factors Based on Test Statistics

Traditionally, the use of Bayes factors has required the specification of proper prior distributions on model parameters implicit to both null and alternative hypotheses. In this paper, I describe an approach to defining Bayes factors based on modeling test statistics. Because the distributions of test statistics do not depend on unknown model parameters, this approach eliminates the subjectivity normally associated with the definition of Bayes factors. For standard test statistics, including the _2, F, t and z statistics, the values of Bayes factors that result from this approach can be simply expressed in closed form.

MODIFIED TEST STATISTICS BY INTER-VOXEL VARIANCE SHRINKAGE WITH AN APPLICATION TO fMRI

Functional Magnetic Resonance Imaging (fMRI) is a non-invasive technique which is commonly used to quantify changes in blood oxygenation and flow coupled to neuronal activation. One of the primary goals of fMRI studies is to identify localized brain regions where neuronal activation levels vary between groups. Single voxel t-tests have been commonly used to determine whether activation related to the protocol differs across groups. Due to the generally limited number of subjects within each study, accurate estimation of variance at each voxel is difficult. Thus, combining information across voxels in the statistical analysis of fMRI data is desirable in order to improve efficiency. Here we construct a hierarchical model and apply an Empirical Bayes framework on the analysis of group fMRI data, employing techniques used in high throughput genomic studies. The key idea is to shrink residual variances by combining information across voxels, and subsequently to construct an improved test statistic in lieu of the classical t-statistic. This hierarchical model results in a shrinkage of voxel-wise residual sample variances towards a common value. The shrunken estimator for voxelspecific variance components on the group analyses outperforms the classical residual error estimator in terms of mean squared error. Moreover, the shrunken test-statistic decreases false positive rate when testing differences in brain contrast maps across a wide range of simulation studies. This methodology was also applied to experimental data regarding a cognitive activation task.

Meta-analysis of gene-level associations for rare variants based on single-variant statistics.

Meta-analysis of genome-wide association studies (GWASs) has led to the discoveries of many common variants associated with complex human diseases. There is a growing recognition that identifying "causal" rare variants also requires large-scale meta-analysis. The fact that association tests with rare variants are performed at the gene level rather than at the variant level poses unprecedented challenges in the meta-analysis. First, different studies may adopt different gene-level tests, so the results are not compatible. Second, gene-level tests require multivariate statistics (i.e., components of the test statistic and their covariance matrix), which are difficult to obtain. To overcome these challenges, we propose to perform gene-level tests for rare variants by combining the results of single-variant analysis (i.e., p values of association tests and effect estimates) from participating studies. This simple strategy is possible because of an insight that multivariate statistics can be recovered from single-variant statistics, together with the correlation matrix of the single-variant test statistics, which can be estimated from one of the participating studies or from a publicly available database. We show both theoretically and numerically that the proposed meta-analysis approach provides accurate control of the type I error and is as powerful as joint analysis of individual participant data. This approach accommodates any disease phenotype and any study design and produces all commonly used gene-level tests. An application to the GWAS summary results of the Genetic Investigation of ANthropometric Traits (GIANT) consortium reveals rare and low-frequency variants associated with human height. The relevant software is freely available.

A scaled difference chi-square test statistic for moment structure analysis

A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model say ${\cal M}_0$ implies on a less restricted one ${\cal M}_1$. If $T_0$ and $T_1$ denote the goodness-of-fit test statistics associated to ${\cal M}_0$ and ${\cal M}_1$, respectively, then typically the difference $T_d = T_0 - T_1$ is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the models ${\cal M}_0$ and ${\cal M}_1$. As in the case of the goodness-of-fit test, it is of interest to scale the statistic $T_d$ in order to improve its chi-square approximation in realistic, i.e., nonasymptotic and nonnormal, applications. In a recent paper, Satorra (1999) shows that the difference between two Satorra-Bentler scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are notavailable in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of models ${\cal M}_0$ and ${\cal M}_1$. A Monte Carlo study is provided to illustrate the performance of the competing statistics.

Fast and Rigorous Computation of Gene and Pathway Scores from SNP-Based Summary Statistics.

Integrating single nucleotide polymorphism (SNP) p-values from genome-wide association studies (GWAS) across genes and pathways is a strategy to improve statistical power and gain biological insight. Here, we present Pascal (Pathway scoring algorithm), a powerful tool for computing gene and pathway scores from SNP-phenotype association summary statistics. For gene score computation, we implemented analytic and efficient numerical solutions to calculate test statistics. We examined in particular the sum and the maximum of chi-squared statistics, which measure the strongest and the average association signals per gene, respectively. For pathway scoring, we use a modified Fisher method, which offers not only significant power improvement over more traditional enrichment strategies, but also eliminates the problem of arbitrary threshold selection inherent in any binary membership based pathway enrichment approach. We demonstrate the marked increase in power by analyzing summary statistics from dozens of large meta-studies for various traits. Our extensive testing indicates that our method not only excels in rigorous type I error control, but also results in more biologically meaningful discoveries.

A score test for binary data with patient non-compliance

A score test is developed for binary clinical trial data, which incorporates patient non-compliance while respecting randomization. It is assumed in this paper that compliance is all-or-nothing, in the sense that a patient either accepts all of the treatment assigned as specified in the protocol, or none of it. Direct analytic comparisons of the adjusted test statistic for both the score test and the likelihood ratio test are made with the corresponding test statistics that adhere to the intention-to-treat principle. It is shown that no gain in power is possible over the intention-to-treat analysis, by adjusting for patient non-compliance. Sample size formulae are derived and simulation studies are used to demonstrate that the sample size approximation holds. Copyright © 2003 John Wiley & Sons, Ltd.

Testing for Seasonal Unit Roots when Residuals Contain Serial Correlations under HEGY Test Framework

This paper introduces a corrected test statistic for testing seasonal unit roots when residuals contain serial correlations, based on the HEGY test proposed by Hylleberg,Engle, Granger and Yoo (1990). The serial correlations in the residuals of test regressionare accommodated by making corrections to the commonly used HEGY t statistics. Theasymptotic distributions of the corrected t statistics are free from nuisance parameters.The size and power properties of the corrected statistics for quarterly and montly data are investigated. Based on our simulations, the corrected statistics for monthly data havemore power compared with the commonly used HEGY test statistics, but they also have size distortions when there are strong negative seasonal correlations in the residuals.

Rank test of location optimal for hyperbolic secant distribution

There are at least two reasons for a symmetric, unimodal, diffuse tailed hyperbolic secant distribution to be interesting in real-life applications. It displays one of the common types of non normality in natural data and is closely related to the logistic and Cauchy distributions that often arise in practice. To test the difference in location between two hyperbolic secant distributions, we develop a simple linear rank test with trigonometric scores. We investigate the small-sample and asymptotic properties of the test statistic and provide tables of the exact null distribution for small sample sizes. We compare the test to the Wilcoxon two-sample test and show that, although the asymptotic powers of the tests are comparable, the present test has certain practical advantages over the Wilcoxon test.

Comparative phylogeographic summary statistics for testing simultaneous vicariance

Testing for simultaneous vicariance across comparative phylogeographic data sets is a notoriously difficult problem hindered by mutational variance, the coalescent variance, and variability across pairs of sister taxa in parameters that affect genetic divergence. We simulate vicariance to characterize the behaviour of several commonly used summary statistics across a range of divergence times, and to characterize this behaviour in comparative phylogeographic datasets having multiple taxon-pairs. We found Tajima's D to be relatively uncorrelated with other summary statistics across divergence times, and using simple hypothesis testing of simultaneous vicariance given variable population sizes, we counter-intuitively found that the variance across taxon pairs in Nei and Li's net nucleotide divergence (pi(net)), a common measure of population divergence, is often inferior to using the variance in Tajima's D across taxon pairs as a test statistic to distinguish ancient simultaneous vicariance from variable vicariance histories. The opposite and more intuitive pattern is found for testing more recent simultaneous vicariance, and overall we found that depending on the timing of vicariance, one of these two test statistics can achieve high statistical power for rejecting simultaneous vicariance, given a reasonable number of intron loci (> 5 loci, 400 bp) and a range of conditions. These results suggest that components of these two composite summary statistics should be used in future simulation-based methods which can simultaneously use a pool of summary statistics to test comparative the phylogeographic hypotheses we consider here.