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PubMed1992

Estimation, reference ranges and goodness of fit for the three-parameter log-normal distribution.

The three-parameter log-normal distribution (3PL) is an appropriate model for many of the continuous variables encountered in medicine. It is shown how to obtain different types of estimate and approximate (sometimes conservative) confidence intervals for the parameters of the 3PL and for certain functions of them, particularly in the calculation of reference ranges of clinical measurements. A simple non-iterative estimate of the shift parameter is described. The Shapiro-Wilk test of non-normality is modified to allow it to be used for testing for departure from the 3PL. Its power is compared with that of other well-known tests. The methods are illustrated using several data sets.

Algorithms↗
PubMed1983

Static response of infrared neurons of crotaline snakes--normal distribution of interspike intervals.

Background discharges (static responses) of warm fibers in the pit organs (infrared receptive organs) of two species of crotaline snakes were recorded at various temperatures (water, 18-33 degrees C; air, 19-28 degrees C). Mean interspike intervals (means), standard deviations (SD), and coefficients of variation (CV) were calculated, and the goodness of fit of interspike interval histograms to a corresponding normal distribution (i.e., one having the same mean and SD) were tested. Means, SD, and CV were smallest at a certain temperature, which might be the optimum receptor temperature for the species. More than half of the histograms (22/42 for water, 7/10 for air) showed a normal distribution at a significance level of 0.01. This suggests that the spike intervals generated at the spike initiation site are constant, with some random error. Background discharges of three pure infrared secondary neurons from the lateral descending nucleus were analyzed in the same way and compared to the peripheral discharges. There were no histograms with a normal distribution in these central neurons, which might indicate that the constant interspike intervals which appear in the primary afferent fibers are not utilized for information processing at this level but occur only as part of a receptor mechanism which is still unknown. The discharge patterns of primary afferent fibers are also discussed in relation to the known discharge patterns of cold fibers in other animals.

Action Potentials↗
PubMed2003

A normal distribution for tensor-valued random variables: applications to diffusion tensor MRI.

Diffusion tensor magnetic resonance imaging (DT-MRI) provides a statistical estimate of a symmetric, second-order diffusion tensor of water, D, in each voxel within an imaging volume. We propose a new normal distribution, p(D) alpha exp(-1/2 D: A: D), which describes the variability of D in an ideal DT-MRI experiment. The scalar invariant, D : A : D, is the contraction of a positive definite symmetric, fourth-order precision tensor, A, and D. A correspondence is established between D: A: D and the elastic strain energy density function in continuum mechanics--specifically between D and the second-order infinitesimal strain tensor, and between A and the fourth-order tensor of elastic coefficients. We show that A can be further classified according to different classical elastic symmetries (i.e., isotropy, transverse isotropy, orthotropy, planar symmetry, and anisotropy). When A is an isotropic fourth-order tensor, we derive an explicit analytic expression for p(D), and for the distribution of the three eigenvalues of D, p(gamma1, gamma2, gamma3), which are confirmed by Monte Carlo simulations. We show how A can be estimated from either real or synthetic DT-MRI data for any given experimental design. Here we propose a new criterion for an optimal experimental design: that A be an isotropic fourth-order tensor. This condition ensures that the statistical properties of D (and quantities derived from it) are rotationally invariant. We also investigate the degree of isotropy of several DT-MRI experimental designs. Finally, we show that the univariate and multivariate distributions are special cases of the more general tensor-variate normal distribution, and suggest how to generalize p(D) to treat normal random tensor variables that are of third- (or higher) order. We expect that this new distribution, p(D), should be useful in feature extraction; in developing a hypothesis testing framework for segmenting and classifying noisy, discrete tensor data; and in designing experiments to measure tensor quantities.

Algorithms↗
PubMed1975

Log normal distribution of gonadotropins and ovarian steroid values in the normal menstrual cycle.

In the statistical analysis of the values of luteinizing hormone, follicle-stimulating hormone, estradiol, and progesterone obtained from normal menstrual cycles, a depature from normality was noted. Chi square, W test, and linear transformation were used to check the normality of the distributions. The results of this investigation showed that the distributions were not normal (Gaussian) but log-normal. By plotting the probit of the percentages of cumulative frequency on a log scale (probit-log), linearity of the data was obtained. This resulted in direct graphical estimations of values with a useful clinical range, which included the mean and the 95 per cent confidence interval.

Adult↗
PubMed1995

EM mixed model analysis of data from informatively censored normal distributions.

Maximum likelihood techniques using the EM algorithm are applied to correlated normally distributed survival data from a placebo-controlled, double-blind, dose-ranging crossover study to assess the short-term efficacy of an antianginal drug in patients with chronic stable angina. Censoring was informative and nonterminal and was not due to death or withdrawal from the study. Unlike previous approaches these techniques are mathematically and computationally tractable, do not require computations of high-dimensional integrals, and do not require the inversion of large matrices.

Algorithms↗
PubMed1990

Disease entities, mixed multi-normal distributions, and the role of the hyperkinetic state in the pathogenesis of hypertension.

This paper concerns the theory and relevance of finite mixtures of univariate and multivariate normal distributions in medical research and suggests that multivariate normal mixture analysis, hitherto not extensively explored, is an appealing approach to the investigation of etiologically obscure, multifactorial diseases such as hypertension. We elaborate a statistical strategy to resolve and test for a normal mixture distribution in a seemingly heterogenous population actually comprising homogenous subpopulations. We use this strategy to validate the hypothesis that in the population at large there is a subgroup of individuals with the characteristic of a hyperkinetic circulatory state, defined as the association of an elevated cardiac index or heart rate with high blood pressure. This subgroup may have implications for the pathogenesis of hypertension. We discuss directions and implications for future research into the pathogenesis of hypertension.

Cluster Analysis↗
PubMed2005

Modeling errors in NOE data with a log-normal distribution improves the quality of NMR structures.

The distribution of the deviation of calculated from measured nuclear Overhauser effect (NOE) intensities is a priori unknown. The use of a log-normal distribution to describe these deviations permits the direct calculation of a structure from the measured intensities without first converting them into distance bounds. We show that the log-normal distribution is a natural choice for describing errors in NOE data and that it improves the accuracy, precision, and quality of the calculated structures compared to the usual bounds representation.

Magnetic Resonance Spectroscopy↗
PubMed1985

Miniature endplate potential amplitudes corrected for spatial decay are not normally distributed.

Miniature endplate potentials (MEPPs) were recorded simultaneously with two intracellular electrodes placed in the muscle fiber near the ends of the frog neuromuscular junction (nmj). The MEPP amplitudes are different in each electrode and are not normally distributed. A method is proposed to correct MEPP amplitudes for spatial decay. The MEPP amplitudes corrected for spatial decay are also not normally distributed. When the class interval of the distribution is small, multiple peaks are observed.

Action Potentials↗
PubMed1990

Testing nonlinear regression parameters under heteroscedastic, normally distributed errors.

Likelihood ratio tests for parameters estimated assuming normally distributed errors are examined under a variety of homoscedastic and heteroscedastic variance assumptions. It is assumed that gamma ij, the jth observation from the ith population, is distributed as N[mu(chi ij, beta i), (sigma i mu(chi ij, beta i)theta i)2]. By varying sigma 2i and theta i, this model contains the most obvious examples of homoscedastic and heteroscedastic variability. Under this model, the null hypothesis of equal beta i's can easily be tested using the likelihood ratio criterion. Also considered is the two-sample chi-square statistic, chi 2t = U'S-1 U, where U is the difference vector of nonlinear least squares parameter estimates and S is an estimate of the covariance matrix of U. Monte Carlo simulation using the von Bertalanffy growth curve as an example is used to evaluate several test statistics for Type I error rates under different sampling assumptions.

Biometry↗
PubMed2003

Application of the complex multivariate normal distribution to crystallographic methods with insights into multiple isomorphous replacement phasing.

Probabilistic methods involving maximum-likelihood parameter estimation have become a powerful tool in computational crystallography. At the centre of these methods are the relevant probability distributions. Here, equations are developed based on the complex multivariate normal distribution that generalize the distributions currently used in maximum-likelihood model and heavy-atom refinement. In this treatment, the effects of various sources of error in the experiment are considered separately and allowance is made for correlations among sources of error. The multivariate distributions presented are closely related to the distributions previously derived in ab initio phasing and can be applied to many different aspects of a crystallographic structure-determination process including model refinement, density modification, heavy-atom phasing and refinement or combinations of them. The underlying probability distributions for multiple isomorphous replacement are re-examined using these techniques. The re-analysis requires the underlying assumptions to be made explicitly and results in a variance term that, unlike those previously used for maximum-likelihood multiple isomorphous replacement phasing, is expressed explicitly in terms of structure-factor covariances. Test cases presented show that the newly derived multiple isomorphous replacement likelihood functions perform satisfactorily compared with currently used programs.

Crystallography, X-Ray↗
PubMed1991

Neonatal brain parameters: outstanding tails of normal distribution.

Previous studies were concerned with distribution curves of values of brain parameters (weight, DNA or cell number, and protein) in a population of 720 neonatal rats. These values followed normal distribution curves, and the tails of such curves, with values in excess of two standard deviations (SD) above (right hand) or below (left hand) the mean (X), were statistically significant. In the present work a larger sample (1,948) of neonatal rats, with particular reference to those whose brain parameters exceeded three SD above or below X, respectively, was examined. Such occurrences are rare; they are somewhat skewed with respect to numbers expected in the tails of a normal distribution curve. For DNA the highest deviations from X were for 'maximal' brain, +3.62 SD (right tail); and for 'minimal' brain, -4.33 SD (left tail). Further study suggested that one of the factors increasing the maximal values might be a lower number in the litter, and one of the factors decreasing the minimal values was a lower maternal weight at conception. Thus the factors affecting maximal and minimal brains are not the same. The occurrence of such individuals with maximal or minimal values of neonatal brain parameters is of particular interest if these parameters are correlated with behavioural performance.

Animals↗
PubMed2004

A log-normal distribution model of the effect of bacteria and ear fenestration on hearing loss: a Bayesian approach.

Chronic ear infection is a potentially life-threatening illness that medical doctors typically treat with ear surgery. Despite the success of this treatment, complications can occur due to bacteria infection. Surgeons believe that this infection causes the patient to have clinically significant hearing damage. In order to understand such complications, surgeons must quantify the effect of bacteria, their toxins and ear surgery on hearing loss. To this end, the other two authors of this paper performed two experiments on guinea pigs to measure hearing thresholds following a bacterial infection and surgery of the inner ear. The response variable in these experiments is hearing thresholds measured in decibels (dB). The problem in analysing such experiments is that the hearing threshold observations often suffer from missing data and censoring mechanisms of various types. Additionally, the distribution of hearing thresholds has heavy tails and is peaked. In order to account for the above statistical issues, we present a Bayesian method with a location-shifted log-normal distribution. The method accounts for the uncertainty in the data collection mechanism and the parameters associated with a location-shifted log-normal distribution. We refer to one of the parameters as the "location-shift" parameter. The Bayesian approach provides a posterior distribution of the location-shift parameter that we compare with values estimated in previously published studies. The immediate goal of our proposed method was to quantify the effects of ear surgery and bacteria infection on hearing loss. Thus, we present the merits of the method in the form of a case study, and report posterior distributions of mean hearing loss, probability of clinically significant hearing loss and relative risk. The results show that surgeon 2, using the surgical procedure "oval window", poses a greater than 40 per cent chance of a 15dB hearing loss regardless of injection of bacteria or not. However, surgeon 1, using the surgical procedure "semicircular canal", does not pose a significantly greater than 40 per cent chance of a 15dB hearing loss unless there is a Pseudomonas aeruginosa-induced infection.

Aged↗
PubMed2006

Interstitial cells of Cajal could regenerate and restore their normal distribution after disrupted by intestinal transection and anastomosis in the adult guinea pigs.

Surgical manipulations of the gastrointestinal (GI) tract usually lead to loss of interstitial cells of Cajal (ICCs). The present study prepared to investigate whether ICCs can regenerate and restore their normal distribution up to 5 months after semitransection and end-to-end anastomosis of small intestines of adult guinea pigs. The segments of anastomosis were studied by immunohistochemistry with anti-KIT, 5-bromo-2'-deoxyuridine (BrdU), stem cell factor (SCF), and neurofilament 200 antibodies and also by transmission electron microscopy (TEM). At early stage, intestinal surgery led to intestinal wall impairment and ICCs loss, and ICCs near the site of anastomosis gradually increased in numbers. About 150 days postoperation, the distribution of ICCs and the microstructure of intestinal wall appeared to be similar with those of the control. By double immunostaining with BrdU and KIT antibodies, a number of proliferated ICCs were seen near the site of transection/anastomosis. Furthermore, KIT ligand, SCF, was mainly observed in the smooth muscle cells (SMCs), which are located close to ICCs. TEM observation revealed a number of immature and mature ICCs in this region. Our results indicated that ICCs could regenerate and restore their normal distribution after intestinal surgery and SMCs might be involved in the regenerated events of ICCs in the adult guinea pig GI tract.

Anastomosis, Surgical↗
PubMed2002

Modeling of human hepatic CYP3A4 enzyme kinetics, protein, and mRNA indicates deviation from log-normal distribution in CYP3A4 gene expression.

AIMS: To determine whether the variability in cytochrome P(450) (CYP)3A4 metabolic function is exhibited at both transcription and translation levels and to examine the population distribution of CYP3A4 enzyme kinetics, protein, and mRNA. METHODS: Enzyme kinetics of testosterone 6beta-hydroxylation, immunoblot CYP3A4 protein, and CYP3A4 mRNA were determined in a microsomal bank of human livers. The distribution of these determinations was analyzed using cumulative distribution (probit) plots and normality test variable (NTV) to detect deviation from normality. RESULTS: Mean hepatic CYP3A4 protein and relative CYP3A4 mRNA were 35+/-23 pmol/mg and 79+/-59 (CYP3A4/beta-actin), respectively. Kinetic parameter estimates of testosterone 6beta-hydroxylation were 611+/-684 pmol/mg/min for maximum rate of the reaction (V(max)) and 206+/-48 micro M for the Michaelis constant (K(m)). The CYP3A4 gene expression and its activity exhibited a relatively high degree of interindividual variability. Furthermore, significant correlation between CYP3A4 protein and V(max) of testosterone 6beta-hydroxylation (r=0.82, P<0.001) as well as CYP3A4 protein and its mRNA (r=0.52, P<0.01) was observed. Cumulative distribution plots and histograms for the CYP3A4 protein, its mRNA, and maximum testosterone 6beta-hydroxylation exhibited evidences of deviation from log-normal distribution. The minimum NTV value for the distribution of CYP3A4 protein corresponding to the inflection point in the probit occurred at approximately 10% cumulative frequency. The percentage of low CYP3A4 protein phenotype was consistent for CYP3A4 activity and its mRNA. In contrast, the distribution of K(m) of testosterone 6beta-hydroxylation does not show evidence of bimodality. CONCLUSIONS: The distribution of CYP3A4 metabolic function, protein, and mRNA is non-normal and may represent a regulatory polymorphism in hepatic CYP3A4 gene expression. In contrast, assessment of the possibility of a structural variant of CYP3A4 through evaluation of the standard deviation relative to the mean kinetic constant value suggests that structural mutation of CYP3A4 may not be a major factor affecting interindividual variation in CYP3A4 metabolic function.

Adolescent↗
PubMed1993

Derivative domain fitting: a new method for resolving a mixture of normal distributions in the presence of a contaminating background.

Derivative domain least squares analysis is a new method for resolving multiple peaks superimposed on a slowly varying continuum into separate normal (Gaussian) distributions without developing a functional approximation for the continuum. The method is based on fitting the first derivative of the data with the first derivative of the sum of a series of normal distributions. A functional approximation for the continuum is not necessary as long as the first derivative of the continuum is approximately zero (i.e., the continuum varies slowly compared to the normal distributions).

Background Radiation↗
PubMed2004

Normal distribution of ventricular pressure-volume area of arrhythmic beats under atrial fibrillation in canine heart.

We previously found the frequency distribution of the left ventricular (LV) effective afterload elastance (E(a)) of arrhythmic beats to be nonnormal or non-Gaussian in contrast to the normal distribution of the LV end-systolic elastance (E(max)) in canine in situ LVs during electrically induced atrial fibrillation (AF). These two mechanical variables determine the total mechanical energy [systolic pressure-volume area (PVA)] generated by LV contraction when the LV end-diastolic volume is given on a per-beat basis. PVA and E(max) are the two key determinants of the LV O(2) consumption per beat. In the present study, we analyzed the frequency distribution of PVA during AF by its chi(2), significance level, skewness, and kurtosis and compared them with those of other major cardiodynamic variables including E(a) and E(max). We assumed the volume intercept (V(0)) of the end-systolic pressure-volume relation needed for E(max) determination to be stable during arrhythmia. We found that PVA distributed much more normally than E(a) and slightly more so than E(max) during AF. We compared the chi(2), significance level, skewness, and kurtosis of all the complex terms of the PVA formula. We found that the complexity of the PVA formula attenuated the effect of the considerably nonnormal distribution of E(a) on the distribution of PVA along the central limit theorem. We conclude that mean (SD) of PVA can reliably characterize the distribution of PVA of arrhythmic beats during AF, at least in canine hearts.

Animals↗