Small Sample Properties of the Mantel-Haenszel Test Statistic for Density Follow-Up Studies

For the analysis of combinations of 2×2 non-contingency tables as obtained from density follow-up studies (relating a number of events to a number of person-years of follow-up) an analogue of the Mantel-Haenszel test for 2×2 contingency tables is widely used. In this paper the small sample properties of this test, both with and without continuity correction, are evaluated. Also the improvement of the test-statistic by using the first four cumulants via the Edgeworth expansion was studied. Results on continuity correction agree with similar studies on the Mantel-Haenszel statistic for 2×2 contingency tables: Continuity correction gives a p-value which approximates the exact p-value better than the p-value obtained without this correction; both the exact test and its approximations show considerable conservatism in small samples; the uncorrected Mantel-Haenszel test statistic gives a p-value that agrees more with the nominal significance level, but can be anti-conservative. The p-value based on the first four cumulants gives a better approximation of the exact p-value than the continuity corrected test, especially when the distribution has marked skewness.     

The Discrete Distribution Used for the Log-rank Test Can Be Inaccurate

When the number of tumors is small, a significance level for the Cox-Mantel (log-rank) test Z is often computed using a discrete approximation to the permutation distribution. For j = 0,…, J let N_j(t) be the number of animals in group j alive and tumor-free at the start of time t. Make a 2 × (1+J) table for each time t of the number of animals R_j(t) with newly palpated tumor out of the total N_j(t) at risk. There are a total of say K tables, one for each distinct time t with observed death or newly palpated tumor. The usual discrete approximation to the permutation distribution of Z is defined by taking tables to be independent with fixed margins N_j(t) and ΣR_j(t) for all t. However, the N_j(t) are random variables for the actual permutation distribution of Z, resulting in dependence among the tables. Calculations for the exact permutation distribution are explained, and examples are given where the exact significance level differs substantially from the usual discrete approximation. The discrepancy arisis primarily because permutations with different Z-scores under the exact distribution can be equal for the discrete approximation, inflating the approximate P-value.     

Asymptotical Tests on the Equivalence,Substantial Difference and Non‐inferiority Problems with Two Proportions

Let d = p₂ ? p₁ be the difference between two binomial proportions obtained from two independent trials. For parameter d, three pairs of hypothesis may be of interest: H₁: d ≤ δ vs. K₁: d > δ; H₂: d ? (δ₁, δ₂) vs. K₂: d ∈ (δ₁, δ₂); and H₃: d ∈ [δ₁, δ₂] vs. K₃: d ? [δ₁, δ₂], where H_i is the null hypothesis and K_i is the alternative hypothesis. These tests are useful in clinical trials, pharmacological and vaccine studies and in statistics generally. The three problems may be investigated by exact unconditional tests when the sample sizes are moderate. Otherwise, one should use approximate (or asymptotical) tests generally based on a Z‐statistics like those suggested in the paper. The article defines a new procedure for testing H₂ or H₃, demonstrates that this is more powerful than tests based on confidence intervals (the classic TOST – two one sided tests – test), defines two corrections for continuity which reduce the liberality of the three tests, and selects the one that behaves better. The programs for executing the unconditional exact and asymptotic tests described in the paper can be loaded at http://www.ugr.es/~bioest/software.htm. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlocal nature of the electron energy spectrum in a glow discharge in pure O2: I. Nonlocal character of the electron distribution function

Abstrac A study is made of the nonlocal nature of the electron energy distribution function in the positive column of a glow discharge in a tube filled with pure oxygen. The distribution function and the axial (E _z) and radial (E _r) electric fields as functions of radius are measured using an array of mobile probes. The experimentally obtained spatial profiles of the distribution function are used to test the applicability of the two-term approximation to the distribution function of the electrons with a nonlocal energy spectrum. The distribution function in a specified electric field E=E _z +E _r (where E _z ⊥ E _r) is calculated by solving the coordinate-dependent Boltzmann equation in the two-term approximation and by directly integrating the equations of electron motion using the Monte Carlo method. A comparison between the experimental data and the results of simulations carried out for a broad parameter range shows that, in the case of a highly nonlocal electron energy spectrum, the two-term approximation makes it possible to calculate the electron distribution function with a fairly good accuracy, in which case, however, in imposing the boundary conditions, the electron losses at the plasma surface should be treated in the kinetic approximation. It is shown that using the reflection coefficient of the plasma surface for electrons instead of the loss cone in space makes it possible to accurately calculate the electron energy distribution function over the entire parameter range under consideration, including the transient region in which the electron-energy relaxation length is comparable to the characteristic plasma dimension. __________ Translated from Fizika Plazmy, Vol. 26, No. 11, 2000, pp. 1038–1045. Original Russian Text Copyright ? 2000 by Ivanov, Klopovskii, Lopaev, Rakhimov, Rakhimova.     

Approximating Exact Probabilities by χ2 and Continuity-Corrected χ2 in 2×2 Tables

Computations have been performed to find an adequate definition of exact two-sided probabilities in 2times2 contingency tables. It turns out, that both uncorrected χ² and Yates' correction for continuity give only unsatisfactory approximations to the exact probabilities of the hypergeometric distribution. The latter are therefore recommended for general use.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Validating the historical record: a relative distance test and correction formula for selection bias in presettlement land surveys

Presettlement land surveys have been used throughout North America to reconstruct forest characteristics prior to major Euro‐American settlement. The bearing tree record derived from these surveys is an example of a distance‐based ecological inventory lacking strict selection rules. Such inventories pose a problem of potential selection bias if the nearest individuals are not always selected. The possibility of bias presents a major impediment to compositional analysis from bearing tree proportions. This article presents a modification to distance‐based tests and correction formulas that utilize the corner‐to‐tree distances recorded in the General Land Office (GLO) and similar surveys. The proposed modification consists of replacing absolute with relative corner‐to‐tree distances to remove the effects of density variation. Monte Carlo simulation is conducted to assess the validity, power and effectiveness of the modified test and correction formula. The modified test is found to be robust in most forests that vary in density and aggregation pattern, but exhibits some statistical bias when density and composition vary simultaneously at local scales. The correction formulas accurately reflect the direction of surveyor bias, and adjusted estimates are consistently closer to the true values than unadjusted estimates. Based on a range of simulation results, upper bound limits on the effects of selection bias can be estimated. Application to the GLO bearing tree records for the state of Minnesota indicates that Minnesota surveyors favored Pinus resinosa, P. strobus and Quercus spp. and avoided five taxa including Salix spp. and Alnus spp. The magnitude of bias appears to be small, however, with an estimated upper bound of 5–6% dissimilarity between biased and unbiased bearing tree selection, some of which may be explained by size differences among taxa.     

Enzyme kinetics at high enzyme concentration

We re-visit previous analyses of the classical Michaelis-Menten substrate-enzyme reaction and, with the aid of the reverse quasi-steady-state assumption, we challenge the approximation d[C]/dt ≈ 0 for the basic enzyme reaction at high enzyme concentration. For the first time, an approximate solution for the concentrations of the reactants uniformly valid in time is reported. Numerical simulations are presented to verify this solution. We show that an analytical approximation can be found for the reactants for each initial condition using the appropriate quasi-steady-state assumption. An advantage of the present formalism is that it provides a new procedure for fitting experimental data to determine reaction constants. Finally, a new necessary criterion is found that ensures the validity of the reverse quasi-steady-state assumption. This is verified numerically.     

Impacts of the pesticides imidacloprid,propargite, and pymetrozine on <Emphasis Type="Italic">Chrysoperla carnea</Emphasis> (Stephens) (Neuroptera: Chrysopidae): IOBC and life table assays

The effects of imidacloprid, propargite, and pymetrozine on the common green lacewing, Chrysoperla carnea (Stephens) were investigated in the laboratory, using the IOBC-system and the life table response experiment. Residual glass plate bioassays were carried out using two-day-old larvae at the Iranian maximum field recommended rate of each commonly used pesticide. All three tested pesticides produced significant adverse effects on pre-imaginal survival (p < 0.01). Imidacloprid had no significant effect on fecundity, but propargite and pymetrozin caused significant reductions (p < 0.05). According to the IOBC classification, imidacloprid was found to be harmless (E = 27.44%); propargite (E = 49.78%) and pymetrozine (E = 66.9%) were slightly harmful. Life table assays revealed that imidacloprid and propargite had no significant effects on the intrinsic rate of natural increase, while pymetrozine caused a 34% reduction in r _m value (p < 0.05). Propargite was non-toxic to C. carnea under the tested conditions. In the case of imidacloprid, using a glass plate as test substrate led to␣underestimation of its effects. The life table assay showed more adverse effects of pymetrozine than the IOBC method.     

Distributions of Hardy-Weinberg equilibrium test statistics

It is well established that test statistics and P-values derived from discrete data, such as genetic markers, are also discrete. In most genetic applications, the null distribution for a discrete test statistic is approximated with a continuous distribution, but this approximation may not be reasonable. In some cases using the continuous approximation for the expected null distribution may cause truly null test statistics to appear nonnull. We explore the implications of using continuous distributions to approximate the discrete distributions of Hardy–Weinberg equilibrium test statistics and P-values. We derive exact P-value distributions under the null and alternative hypotheses, enabling a more accurate analysis than is possible with continuous approximations. We apply these methods to biological data and find that using continuous distribution theory with exact tests may underestimate the extent of Hardy–Weinberg disequilibrium in a sample. The implications may be most important for the widespread use of whole-genome case–control association studies and Hardy–Weinberg equilibrium (HWE) testing for data quality control.     

Approximating the Non-Null Distribution of the Likelihood Ratio Test Statistic

This paper is concerned with investigation into the behavior of the likelihood ratio test statistic G² when the alternative hypothesis M (QQ) is the true model. Exact moments of G² are computed empirically and three approximations are considered for approximating the non-null distribution of G². Our results show that the two parameter gamma distribution provides a closer approximation to the exact powers of G². A randomized procedure was employed to obtain critical values based on 1000 simulated samples.     

Michaelis-Menten kinetics at high enzyme concentrations

The total quasi-steady state approximation (tQSSA) for the irreversible Michaelis-Menten scheme is derived in a consistent manner. It is found that self-consistency of the initial transient guarantees the uniform validity of the tQSSA, but does not guarantee the validity of the linearization in the original derivation of Borghans et al. (1996, Bull. Math. Biol., 58, 43–63). Moreover, the present rederivation yielded the noteworthy result that the tQSSA is at least roughly valid for any substrate and enzyme concentrations. This reinforces and extends the original assertion that the parameter domain for which the tQSSA is valid overlaps the domain of validity of the standard quasi-steady state approximation and includes the limit of high enzyme concentrations. The criteria for the uniform validity of the original (linearized) tQSSA are corrected, and are used to derive approximate solutions that are uniformly valid in time. These approximations overlap and extend the domains of validity of the standard and reverse quasi-steady state approximations.     

The Probability of a Negative Linear Combination of Independent Mean Squares

SATTERTHWAITE'S (1941) approximation of the distribution of a linear combination, of independent mean squares is a commonly used technique in the analysis of variance. Confidence intervals and test statistics based on this approximation require that be positive. In this article, the probability that will be negative is considered in situations in which the mean squares are associated with a general balanced mixed model. Expressions are given for exact and approximate values of this probability in terms of the expected values and degrees of freedom of the mean squares. An example is presented to illustrate the implementation of the proposed methodology.     

Unconditional Confidence Interval for the Difference between Two Proportions

In applied statistics it is customary to have to obtain a one‐ or two‐tail confidence interval for the difference d = p₂ – p₁ between two independent binomial proportions. Traditionally, one is looking for a classic and non‐symmetric interval (with respect to zero) of the type d ∈ [δ_L;δ_U], d ≤ δ₀ or d ≥ δ₀. However, in clinical trials, equivalence studies, vaccination efficacy studies, etc., and even after performing the classic homogeneity test, intervals of the type |d| ≤ Δ₀ or |d| ≥ Δ₀, where Δ₀ > 0, may be necessary. In all these cases it is advisable to obtain the interval by inverting the appropriate test. The advantage of this procedure is that the conclusions obtained using the test are compatible with those obtained using the interval. The article shows how this is done using the new exact and asymptotic unconditional tests published. The programs for performing these tests may be obtained at URL http://www.ugr.es/~bioest/software.htm.     

Notes on Testing Equality in Dichotomous Data with Matched Pairs

In attempting to improve the efficiency of McNemar's test statistic, we develop two test procedures that account for the information on both the discordant and concordant pairs for testing equality between two comparison groups in dichotomous data with matched pairs. Furthermore, we derive a test procedure derived from one of the most commonly‐used interval estimators for odds ratio. We compare these procedures with those using McNemar's test, McNemar's test with the continuity correction, and the exact test with respect to type I error and power in a variety of situations. We note that the test procedures using McNemar's test with the continuity correction and the exact test can be quite conservative and hence lose much efficiency, while the test procedure using McNemar's test can actually perform well even when the expected number of discordant pairs is small. We also find that the two test procedures, which incorporate the information on all matched pairs into hypothesis testing, may slightly improve the power of using McNemar's test without essentially losing the precision of type I error. On the other hand, the test procedure derived from an interval estimator of adds ratio with use of the logarithmic transformation may have type I error much larger than the nominal α‐level when the expected number of discordant pairs is not large and therefore, is not recommended for general use.     

The translational friction of toroids

An approximate analytic expression for the translational friction coefficient of a toroid modeled as a continuous shell of frictional elements is derived using the Kirkwood approximation. The accuracy of this expression was determined by comparing the friction coefficients predicted by it to those predicted by extrapolated shell-model calculations using the modified Oseen tensor. To show that these calculations do indeed yield the correct friction coefficients, actual translational friction coefficients were determined by observing settling rates of macroscopic model rings or toroids in a high-viscosity silicone fluid. Our conclusion is that the approximate expression yields friction coefficients that are about 1.5–3% low for finite rings. For thin rings, a comparison is also made with the exact result of Yamakawa and Yamaki [J. Chem. Phys. 57 , 1572 (1972); 58 , 2049 (1973)] for the translational friction of plane polygonal rings. This comparison shows that the approximate expression yields results which are low by 2–3% unless the rings are extremely thin, in which case the error is larger. In the limit of an infinitely thin ring the approximate expression reduces to the Kirkwood result [J. Polym. Sci. 12 , 1 (1954)], which is low by 8.3%. We discuss briefly how this work may be useful in determining the structure of DNA compacted by various solvent–electrolyte systems and polyamines.     

Stratified Fisher's exact test and its sample size calculation

Chi‐squared test has been a popular approach to the analysis of a 2 × 2 table when the sample sizes for the four cells are large. When the large sample assumption does not hold, however, we need an exact testing method such as Fisher's test. When the study population is heterogeneous, we often partition the subjects into multiple strata, so that each stratum consists of homogeneous subjects and hence the stratified analysis has an improved testing power. While Mantel–Haenszel test has been widely used as an extension of the chi‐squared test to test on stratified 2 × 2 tables with a large‐sample approximation, we have been lacking an extension of Fisher's test for stratified exact testing. In this paper, we discuss an exact testing method for stratified 2 × 2 tables that is simplified to the standard Fisher's test in single 2 × 2 table cases, and propose its sample size calculation method that can be useful for designing a study with rare cell frequencies.     

An Alternative Approach for the Assessment of Bioequivalence Between Two Formulations of a Drug

The problem of the assessment of bioequivalence between a test formulation (T) and a reference formulation (R) of a drug using a two-way crossover experiment is considered. To claim bioequivalence between two formulations, it is required by the United States Food and Drug Administration (FDA) to demonstrate that the true ratio of means μT/μR of pharmacokinetic parameters of concern falls within some reasonable limits (e.g., (80%, 120%)) with certain assurance. A commonly used approach is to construct an approximate 90% confidence interval for μT/μR and compare it with (80%, 120%). In this paper, an exact approach according to the FDA's criteria is proposed. The proposed procedure is derived by constructing an exact confidence region (an ellipse) for (μR, μT) and comparing it with the region bounded by μT = 0.8 μR and μT = 1.2 μR. Bioequivalence is concluded if the ellipse is within the critical region.     

Optimal design for the dose-response screening of tight-binding enzyme inhibitors

Optimal experimental designs for the dose–response screening of enzyme inhibitors were studied within the framework of the Box–Lucas theory. If the enzyme concentration E is considered as a fixed constant, an exact two-point D-optimal design consists of a pair of inhibitor concentrations equal to I₁=0 and I₂=E+K, where K is the apparent inhibition constant. If the enzyme concentration is treated as an adjustable parameter, an empirical three-point D-optimal design consists of three inhibitor concentrations equal to I₁=0, I₂=E+3K, and I₃=0.7E. These results were applied to design optimized, irregularly spaced concentration series for routine inhibitor screening. A heuristic Monte Carlo simulation study confirmed that the optimized dilution series is significantly more efficient than the classic series characterized by a constant dilution ratio. An online calculator to create optimized dilution series is freely available at http://www.biokin.com/design/.