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PubMed2003

Geometry of escort distributions.

Given an original distribution, its statistical and probabilistic attributes may be scanned using the associated escort distribution introduced by Beck and Schlögl and employed in the formulation of nonextensive statistical mechanics. Here, the geometric structure of the one-parameter family of the escort distributions is studied based on the Kullback-Leibler divergence and the relevant Fisher metric. It is shown that the Fisher metric is given in terms of the generalized bit variance, which measures fluctuations of the crowding index of a multifractal. The Cramér-Rao inequality leads to a fundamental limit for the precision of the statistical estimate of the order of the escort distribution. We also show quantitatively that it is inappropriate to use the original distribution instead of the escort distribution for calculating the expectation values of physical quantities in nonextensive statistical mechanics.

Journal Article↗
PubMed2004

Statistical analysis of the spatial distribution of operons in the transcriptional regulation network of Escherichia coli.

We have performed a statistical analysis of the spatial distribution of operons along the DNA in the transcriptional regulation network of Escherichia coli. The analysis reveals that pairs of operons that regulate each other and those that are co-regulated tend to lie much closer to one another than would be expected for a random network. Moreover, these pairs of operons tend to be transcribed in diverging directions. This spatial arrangement of operons allows the upstream regulatory domains to overlap and interfere with each other and our analysis also demonstrates the statistical significance of this motif of overlapping operons. Overlapping operons afford additional regulatory control, such as the correlated or anticorrelated expression of operons. We show by a mean-field analysis of a feed-forward loop that overlapping operons can drastically enhance the performance of gene regulatory networks. Our results suggest that regulatory control can provide a selective pressure that drives operons together in the course of evolution.

DNA, Bacterial↗
PubMed2004

Wilcoxon test after Levene's transformation can have an inflated Type I error rate.

It is shown that the procedure of applying the Wilcoxon test after Levene's transformation can have an inflated Type I error rate when distributions are skewed. Thus, when the data may come from an asymmetric distribution, the Wilcoxon test should not be applied as a test for homogeneity of variances after Levene's transformation.

Humans↗
PubMed1998

Alternative criteria for optimal designs of limiting dilution assays.

It is well known that criteria for optimal non-linear designs usually depend on the unknown value of parameters. An approximate Bayesian approach imposes a prior on these values and optimizes the expectation of the criterion over this distribution. While this method produces designs that perform well on average, the design may perform badly in some parts of the parameter space, and if the true parameter appears to fall in one of these regions, then the good average performance will be little compensation. Several alternative criteria are introduced in the context of deriving designs for limiting dilution assays. These include constrained optimization of a familiar criterion, a minimax criterion and designs to optimize prespecified centiles of the variance under the prior distribution. The latter are shown to offer a useful compromise between good overall performance and possible poor performance.

Analysis of Variance↗
PubMed1987

An approximation for the distribution of the scan statistic.

The scan statistic evaluates whether an apparent cluster of disease in time is due to chance. The statistic employs a 'moving window' of length w and finds the maximum number of cases revealed through the window as it scans or slides over the entire time period T. Computation of the probability of observing a certain size cluster, under the hypothesis of a uniform distribution, is infeasible when N, the total number of events, is large, and w is of moderate or small size relative to T. We give an approximation that is an asymptotic upper bound, easy to compute, and, for the purposes of hypothesis testing, more accurate than other approximations presented in the literature. The approximation applies both when N is fixed, and when N has a Poisson distribution. We illustrate the procedure on a data set of trisomic spontaneous abortions observed in a two year period in New York City.

Abortion, Spontaneous↗
PubMed2000

Characteristics of the myopic patient population applying for refractive surgery.

PURPOSE: Assessment of the characteristics of the myopic patient population applying for refractive surgery in order to determine the potential market for myopic refractive surgery. METHODS: Records of consecutive patients who responded to an advertisement for refractive surgery to correct myopia were evaluated retrospectively with regards to patient demographics and the amount and distribution of the refractive error. Data were compared to that available from population-based statistics for distribution of myopia in the general population. For statistical analysis, one sample Student's t-test and two tailed Student's t-test were utilized. RESULTS: Two hundred fifty seven patients (140 women and 117 men) responded to an advertisement for refractive surgery during the six month period between January and June 1998. Mean spherical equivalent (SEQ) of the patient population was -4.59+/-2.54 D (min;-0.25 D, max;-15.75 D) right eye (OD) and -4.62+/-2.82 D (min;-0.25 D, max;-15.25 D) left eye (OS). Among the patients who had myopia with an astigmatism of at most 1.00 D (n=165), the distribution of refractive error was statistically significantly different from that obtained from population-based statistics, such that, although most of the myopic population (40%) had an SEQ of -1.00 to -2.25 D, the majority of our patients (54.8%) who applied for myopic refractive surgery had an SEQ of -2.50 to -5.00 D. Another striking difference was that, although patients with an SEQ more than -6.00 D were a minority(2%) in the population study, in our study group, they comprised 16.7% of the patients seeking refractive correction. The difference between the SEQ of the right and left eyes ranged from 0.00 D to 13.0 D (mean, 0.89+/-1.5 D), 47.1 % having a difference of at most +/-0.5 D between the two eyes. The mean cylindrical error in the patient population was 0.69+/-0.93 D (min: 0, max: -4.5) OD and 0.69+/-0.96 D (min: 0, max: -4.5) OS. There were no statistically significant differences between the distribution of SEQ or cylindrical refractive error between males and females. CONCLUSION: Although a population-based study reported that most of the myopic population (40%) had an SEQ of -1.00 to -2.25 D, the majority of our patients (54.8%) who applied for myopic refractive surgery had an SEQ of -2.50 to -5.00 D. On the other hand, while patients with an SEQ of -6.00 D and more constituted only about 2% of the general population, they accounted for 16.7% of our study population. Therefore, the refractive characteristics of the patient population applying for myopic refractive surgery may not necessarily parallel that of general population-based statistics. In order to establish a more effective refractive surgery practice, it is feasible to perform local studies and reevaluate the requirements of your practice accordingly.

Adolescent↗
PubMed1997

Saccadic reaction times: a statistical analysis of multimodal distributions.

The distributions of saccadic reaction times (SRT) often deviate from unimodal normal distributions. An excess-mass procedure was used to detect peaks in 963 data sets containing 90,927 reaction times from 170 subjects. About 55% showed one, 30% two, 12% three and 3% four peaks. According to their clustering along the reaction time scale the modes could be classified into express (90-120 msec), fast regular (135-170 msec) and slow regular (200-220 msec) modes. Among the unimodal distributions 29% had peaks in the range of the express mode and 46% had peaks in the range of the fast regular mode. Therefore, 87% of the data sets support the notion of saccadic reaction time distributions being the superposition of three modes. All experimental distributions were fitted by as many gamma distributions as determined by the excess-mass test. The significance of the multimodality for saccade generation processes is discussed.

Fixation, Ocular↗
PubMed2000

Bayesian mapping of quantitative trait loci for complex binary traits.

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.

Algorithms↗
PubMed2001

18O pattern and biosynthesis of natural plant products.

Oxygen atoms in plant products originate from CO(2), H(2)O and O(2), precursors with quite different delta18O values. Furthermore their incorporation by different reactions implies isotope effects. On this base the resulting non-statistical 18O distributions in natural compounds are discussed. The delta18O value of cellulose is correlated to that of the leaf water, and the observed 18O enrichment (approximately +27 per thousand) is generally attributed to an equilibrium isotope effect between carbonyl groups and water. However, as soluble and heterotrophically synthesised carbohydrates show other correlations, a non-statistical 18O distribution - originating from individual biosynthetic reactions - is postulated for carbohydrates. Similarly, the delta18O values of organic acids, carbonyl compounds, alcohols and esters indicate water-correlated, but individual 18O abundances (e.g. O from acyl groups approximately +19% above water), depending upon origin and biosyntheses. Alcoholic groups introduced by monooxygenase reactions, e.g. in sterols and phenols, show delta18O values near +5 per thousand, in agreement with an assumed isotope fractionation factor of approximately 1.02 on the reaction with atmospheric oxygen (delta18O=+23.5 per thousand). Correspondingly, a "thermodynamically ordered isotope distribution" is only observed for oxygen in some functional groups correlated to an origin from CO(2) and H(2)O, not from O(2). The individual isotopic increments of functional groups permit the prediction of global delta18O values of natural compounds on the basis of their biosynthesis.

Alcohols↗
PubMed1996

Combinatorial peptide synthesis: statistical evaluation of peptide distribution.

Peptide libraries are large collections of oligopeptides used as screening mixtures for the discovery of new pharmacological leads. Their synthesis is achieved on solid phase in a combinatorial way, one of the reacting components being either a mixture of resin-bound amino acids or peptides, or a mixture of amino acids in solution. In practice, the various peptides are not represented in equimolar amounts in the resulting libraries. Here, Jean Boutin and Alban Fauchère explain the statistical considerations important in the design of a peptide library, show to what extent one bead-one peptide libraries fulfill statistical criteria, and discuss resin bead quantities necessary to minimize the range of relative concentrations of the different peptides present in the library. This range is crucial in the correct interpretation of the results of biological tests.

Evaluation Studies as Topic↗
PubMed1993

Tests for qualitative treatment-by-centre interaction using a 'pushback' procedure.

In multicentre clinical trials using a common protocol, the centres are usually regarded as being a fixed factor, thus allowing any treatment-by-centre interaction to be omitted from the error term for the effect of treatment. However, we feel it necessary to use the treatment-by-centre interaction as the error term if there is substantial evidence that the interaction with centres is qualitative instead of quantitative. To make allowance for the estimated uncertainties of the centre means, we propose choosing a reference value (for example, the median of the ordered array of centre means) and converting the individual centre results into standardized deviations from the reference value. The deviations are then reordered, and the results 'pushed back' by amounts appropriate for the corresponding order statistics in a sample from the relevant distribution. The pushed-back standardized deviations are then restored to the original scale. The appearance of opposite signs among the destandardized values for the various centres is then taken as 'substantial evidence' of qualitative interaction. Procedures are presented using, in any combination: (i) Gaussian, or Student's t-distribution; (ii) order-statistic medians or outward 90 per cent points of the corresponding order statistic distributions; (iii) pooling or grouping and pooling the internally estimated standard deviations of the centre means. The use of the least conservative combination--Student's t, outward 90 per cent points, grouping and pooling--is recommended.

Clinical Trials as Topic↗
PubMed2006

Coherence solution for bidirectional reflectance distributions of surfaces with wavelength-scale statistics.

The scalar bidirectional reflectance distribution function (BRDF) due to a perfectly conducting surface with roughness and autocorrelation width comparable with the illumination wavelength is derived from coherence theory on the assumption of a random reflective phase screen and an expansion valid for large effective roughness. A general quadratic expansion of the two-dimensional isotropic surface autocorrelation function near the origin yields representative Cauchy and Gaussian BRDF solutions and an intermediate general solution as the sum of an incoherent component and a nonspecular coherent component proportional to an integral of the plasma dispersion function in the complex plane. Plots illustrate agreement of the derived general solution with original bistatic BRDF data due to a machined aluminum surface, and comparisons are drawn with previously published data in the examination of variations with incident angle, roughness, illumination wavelength, and autocorrelation coefficients in the bistatic and monostatic geometries. The general quadratic autocorrelation expansion provides a BRDF solution that smoothly interpolates between the well-known results of the linear and parabolic approximations.

Journal Article↗
PubMed1991

Fixed-bin analysis for statistical evaluation of continuous distributions of allelic data from VNTR loci, for use in forensic comparisons.

The detection of DNA polymorphisms by RFLP analysis is having a major impact on identity testing in forensic science. At present, this approach is the best effort a forensic scientist can make to exclude an individual who has been falsely associated with an evidentiary sample found at a crime scene. When an analysis fails to exclude a suspect as a potential contributor of an evidentiary sample, a means should be provided to assess suitable weight to the putative match. Most important, the statistical analysis should not place undue weight on a genetic profile derived from an unknown sample that is attributed to an accused individual. The method must allow for limitations in conventional agarose-submarine-gel electrophoresis and Southern blotting procedure, limited sample population data, possible subpopulation differences, and potential sampling error. A conservative statistical method was developed based on arbitrarily defined fixed bins. This approach permits classification of continuous allelic data, provides for a simple and portable data-base system, and is unlikely to underestimate the frequency of occurrence of a set of alleles. This will help ensure that undue weight is not placed on a sample attributed to an accused individual.

Alleles↗
PubMed1988

Reliability of Bayesian probability analysis for predicting coronary artery disease in a veterans hospital.

To assess the accuracy of Bayesian probability analysis for the prediction of coronary artery disease, post-test probabilities were generated by the application of three Bayesian algorithms to the clinical and noninvasive test results of 199 patients undergoing angiography in a veterans' hospital. All assumed conditional independence but each used different pre-test and conditional probabilities. Two statistical approaches were employed: (1) Sorting of patients in ascending deciles of probability and comparing expected and observed probabilities in each decile. (2) Calculation of normally distributed reliability statistics which do not depend on probability subsets and the comparison of resulting probability distributions using these statistics. Both statistical approaches revealed that the Bayesian algorithms overestimated disease probability when it was high and underestimated it when low. Though all three algorithms were frequently incorrect, they differed significantly in their accuracies, suggesting that errors in Bayesian analysis are caused by factors other than the assumption of independence. The errors may be due to differences in sensitivity and specificity of tests applied in different institutions.

Adult↗