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The mathematical formulations of Neyman-Pearson hypothesis testing and the Fisherian concept of significance tests are examined as alternative ways of applying statistical models to data from clinical trials.
We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.
OBJECTIVES: To assume that a cause of relative infertility will decrease the monthly chance of conception (fecundability) in a dose-dependent manner and, by a mathematical model, to identify common clinical observations and paradoxes that are explainable within this hypothesis. DESIGN: An empirically based assumption of a population-mean fecundability of 0.2 and the accumulating probability of pregnancy equations and projections were used to examine over time the effects of diminishing such fecundability to one half, one fifth, and one twentieth of normal, and then reversing this effect with ideal treatment at points of 2 years and 5 years. RESULTS: [1] The duration of infertility is an important and powerful covariate in determining residual fecundability and the chance of pregnancy, with or without treatment. [2] The more substantial the pathology is, the greater should be the likelihood of pregnancy after its effective treatment. [3] Provided no harm is done by treatment, an increase in subsequent fecundability will result whatever the "dose" of the reproductive disturbance, but this will not always mean that pregnancy is probable. [4] The presence of a second infertility factor should compound dramatically the deleterious effects attributable to the first and make it more likely for either factor to be diagnosed. CONCLUSIONS: Duration of infertility is generally more important than the dose of an infertility factor as a covariate in clinical studies, and more emphasis should be placed on controlling for it. Discouraging clinical reports on the low success of treating certain conditions associated with infertility do not necessarily justify rejecting a hypothesis that such a condition decreases fertility in a dose-dependent manner.
Cell adhesion is mediated by specific interaction between receptors and ligands. Such interaction provides not only physical linkage but also communication between the cell and its environment. The kinetics and mechanics of cell adhesion are coupled, because force can influence the formation and dissociation of receptor-ligand bonds. The kinetic rates and their force dependence determine how likely, how rapidly and how strongly cells bind as well as how long they remain bound. Since adhesion molecules are linked to apposing cellular membranes, their interaction is governed by two-dimensional (2D) kinetics. This is in contrast to the three-dimensional (3D) binding of soluble ligands to cell surface receptors. Unlike the 3D case in which many methods are available for measuring kinetic rates, not until recently have the 2D kinetic rates become experimentally measurable. In this review, I will discuss the recent progress in the experimental methods that enable quantification of the relevant kinetic and mechanical parameters, the fundamental concepts that underlie the physics of the biological phenomena, and the mathematical models that relate functions to the intrinsic properties of the adhesion molecules.
The Doppler determination of the mitral pressure half-time has gained widespread acceptance as a reliable estimate for mitral valve area, despite little theoretical basis for its "independence" of other hemodynamic variables. A simple model of the left atrium and mitral valve has been developed and a governing equation derived from fluid dynamics fundamentals. Solution of this equation indicates that the pressure half-time should vary inversely with mitral valve area, but also proportionally to net left atrial and ventricular compliance and to the square root of the peak transmitral gradient. This complex relation is apparently masked in the typical clinical situation because pressure and compliance tend to change in opposite directions, thereby partly offsetting each other. In several clinical settings, such as balloon mitral valvotomy, left ventricular hypertrophy and aortic regurgitation, changes in initial pressure and compliance may be large enough to alter the relation between mitral area and pressure half-time. This study reviews the development of the pressure half-time concept, presents an overall method for studying mitral valve flow using mathematical modeling and describes the effects of factors other than mitral valve area on pressure half-time.
Laboratory-scale experiments were conducted to investigate the impact of ozonation on the degradability of wine distillery wastewaters, usually called vinasses, with the goal of developing combined chemical-biological methods for their treatment. Chemical oxygen demand (COD), biological oxygen demand (BOD), total organic carbon (TOC), and ultraviolet absorbance at 254 nm (UV(254)) were taken as reference parameters to follow the pollution level. The vinasses were treated both directly and after being mixed with domestic sewage. Ozonation of pure vinasses required high ozone doses to achieve a significant efficiency for removing the organic matter. Mixing vinasses with domestic sewage allowed higher degradation rates with ozone. Ozone was also demonstrated to be an appropriate oxidizing agent to improve vinasses's biodegradability and organic matter removal. A mathematical model of the ozonation kinetics based on the film theory concept is also presented.
The celebrated inequalities of Bell are based on the assumption that local hidden parameters exist. When combined with conflicting experimental results, these inequalities appear to prove that local hidden parameters cannot exist. This contradiction suggests to many that only instantaneous action at a distance can explain the Einstein, Podolsky, and Rosen type of experiments. We show that, in addition to the assumption that hidden parameters exist, Bell tacitly makes a variety of other assumptions that contribute to his being able to obtain the desired contradiction. For instance, Bell assumes that the hidden parameters do not depend on time and are governed by a single probability measure independent of the analyzer settings. We argue that the exclusion of time has neither a physical nor a mathematical basis but is based on Bell's translation of the concept of Einstein locality into the language of probability theory. Our additional set of local hidden variables includes time-like correlated parameters and a generalized probability density. We prove that our extended space of local hidden variables does not permit Bell-type proofs to go forward.
We present a new class discovery method for microarray gene expression data. Based on a collection of gene expression profiles from different tissue samples, the method searches for binary class distinctions in the set of samples that show clear separation in the expression levels of specific subsets of genes. Several mutually independent class distinctions may be found, which is difficult to obtain from most commonly used clustering algorithms. Each class distinction can be biologically interpreted in terms of its supporting genes. The mathematical characterization of the favored class distinctions is based on statistical concepts. By analyzing three data sets from cancer gene expression studies, we demonstrate that our method is able to detect biologically relevant structures, for example cancer subtypes, in an unsupervised fashion.
Many current quality assurance programs aim at maximizing the quality of health services instead of optimizing it, because they rest on the erroneous "best to all" approach, which overemphasizes the scientific and technical aspects of quality. The result is prohibitively expensive "Cadillac care." It is suggested that the optimal qualitative level can be identified by answering a chain of hierarchical questions beginning with the relevance of medical care to the solution of the problem at hand and ending with such detailed questions as what diagnostic procedures should be performed by whom, and with what technique. Using the functional definition of quality applied in industrial quality control, a mathematical model is developed to illustrate the relative nature of the concept of quality and its implications for quality assurance programs. Special attention is paid to logical quality, i.e., the efficacy with which information is used in arriving at decisions.
The concept of the density of modes has been lacking a precise mathematical definition for a finite-size structure. With the explosive growth in the fabrication of photonic crystals and nanostructures, which are inherently finite in size, a workable definition is imperative. We give a simple and physically intuitive definition of the electromagnetic density of modes based on the Green's function for a generic three-dimensional open cavity filled with a linear, isotropic, dielectric material.
We demonstrate that kinematic simulation (KS) of three-dimensional homogeneous turbulence produces fluid element pair statistics in agreement with the predictions of L F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)] even though KS lacks explicit modeling of turbulent sweeping of small eddies by large ones. This scaling is most clearly evident in the turbulent diffusivity's dependence on rms pair separation and, to a lesser extent, on the pair's travel time statistics. It is also shown that kinematic simulation generates a probability density function of pair separation which is in good agreement with recent theory [S. Goto and J. C. Vassilicos, New J. Phys. 6, 65 (2004)] and with the scaling of the rms pair separation predicted by L. F. Richardson [Proc. R. Soc. London, Ser. A 110, 709 (1926)]. Finally, the statistical persistence hypothesis (SPH) is formulated mathematically and its validity tested in KS. This formulation introduces the concept of stagnation point velocities and relates these to fluid accelerations. The scaling of accelerations found in kinematic simulation supports the SPH, even though KS does not generate a Kolmogorov scaling for the acceleration variance (except for a specific case and a limited range of outer to inner length-scale ratios). An argument is then presented that suggests that the stagnation points in homogeneous isotropic turbulence are on average long-lived.
Wall turnover was studied in Bacillus subtilis. The loss of radioactively labeled wall polymers was followed during exponential growth in batch and chemostat cultures. Turnover kinetics were identical under all growth conditions; pulse-labeled wall material was lost with first-order kinetics, but only after exponential growth for 1 generation time after its incorporation. Similarly, continuously labeled cells showed an accelerating decrease in wall-bound radioactivity starting immediately after removal of the labeled precursor and also reached first-order kinetics after 1 generation time. A mathematical description was derived for these turnover kinetics, which embraced the concept of "spreading" of old wall chains (H. M. Pooley, J. Bacteriol. 125:1127-1138, 1976). Using this description, we were able to calculate from our experimental data the rate of loss of wall polymers from cells and the fraction of the wall which was sensitive to turnover. We found that about 20% of the wall was lost per generation time and that this loss was affected by turnover activity located in the outer 20 to 45% of the wall; rather large variations were found with both quantities and also between duplicate cultures. These parameters were quite independent of the growth rate (the specific growth rate varied from 1.3 h-1 in broth cultures to 0.2 to 0.3 h-1 in chemostat cultures) and of the nature of the anionic polymer in the wall (which was teichoic acid in cultures with an excess of phosphate and teichuronic acid in phosphate-limited chemostat cultures). Some implications of the observed wall turnover kinetics for models of wall growth in B. subtilis are discussed.
In this article, I argue that the regression-based discrepancy method used in the diagnosis of learning disabilities is invalid because it is inconsistent with the underlying underachievement concept of which it is intended to be the operationalization. I mathematically demonstrate that the regression-based discrepancy method largely reflects achievement-specific determinants, thereby defeating its own object of describing aptitude-achievement discrepancies. The implications for research examining the role of intelligence in learning disabilities are outlined.
The premise that progress in document examination will depend on employing techniques useful in the more formal branches of science is not exactly logical. The correlation between the work of the document examiner and the behavioral sciences has been discussed by presenting some random thoughts which have occurred to the author over a period of years. The suggestion is made, by illustration and implication, that the unfortunate connotation of the word "behavior" with the word "graphology" has tended to direct the attention of document examiners away from a study of the behavioral sciences, a branch of science from which much can be learned. The fact that the subjective concepts of probability formed by the mature document examiner will approach mathematical expectation has been noted.
A schematic conception of staged development of chronic bronchitis and a clinical-mathematical model of the diseases staging have been proposed. The study involved two samples of population: 5204 males aged 40-59 from small towns of the Lithuanian SSR and 1012 residents of the industrial city of Kaunas. It was established that various stages of the disease among which stage II predominated had different incidence rates. Long-term results of the stage-differentiated prophylaxis have furnished evidence for its effectiveness. This principle of stage-distributed therapeutic measures can be adopted for management of other affections.
Obtaining a minimal amount of postoperative astigmatism following cataract surgery is becoming increasingly important. One aspect of the patient's surgery which should not be overlooked is the preoperative keratometry which provides a basis for preoperative planning of surgical technique to be used and a point of reference for determining the amount of change in astigmatism produced by the surgery. Analysis of the surgically induced change in astigmatism using the calculations described in this paper will allow the surgeon to evaluate his own techniques and to maximize his potential for obtaining consistently good postoperative astigmatic results without the need for suture removal. The method presented is based upon concepts in common use in surgical ophthalmology and requires only simple mathematical procedures, familiar to all with a background in algebra and trigonometry.
Digital filtering is a powerful mathematical technique in computer analysis of nuclear medicine studies. The basic concepts of object-domain and frequency-domain filtering are presented in simple, largely nonmathematical terms. Computational methods are described using both the Fourier transform and convolution techniques. The frequency response is described and used to represent the behavior of several classes of filters. These concepts are illustrated with examples drawn from a variety of important applications in nuclear medicine.
The concept of the ventricular gradient was conceived in the mind of Frank Wilson in the early 1930s. Wilson, a mathematical genius, believed that the calculation of the ventricular gradient yielded information that was not otherwise obtainable. The method of analysis was not utilized by clinicians at large because the concept was not easy to understand and because the method used to compute the direction of the ventricular gradient was so time consuming that clinicians could not use it. Grant utilized the concept to create vector electrocardiography, but he believed that if his method of analysis was used, it was not often necessary to compute the direction of the ventricular gradient. He did, however, describe an easy way to compute the direction of the ventricular gradient. The current major clinical use of the ventricular gradient is to identify primary and secondary T-wave abnormalities in an electrocardiogram showing left or right ventricular hypertrophy or left or right ventricular conduction abnormalities. In addition, the author uses the term ventricular time gradient instead of ventricular gradient in an effort to clarify the concept. Finally, the author discusses the possible clinical significance of a normally directed, but shorter than normal, ventricular time gradient, an attribute that has not been emphasized previously.