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At least 19 recordsLinked to original sources
PubMed2005

Variable Unstressed Volume Keeps Normal Distributions of Canine Left Ventricular Contractility and Total Mechanical Energy under Atrial Fibrillation.

We have reported that the contractility index (E(max)) and the total mechanical energy (PVA) of arrhythmic beats of the left ventricle (LV) distribute normally in canine hearts under electrically induced atrial fibrillation (AF). Here, E(max) is the ventricular elastance as the slope of the end-systolic (ES) pressure-volume (P-V) relation (ESPVR), and PVA is the systolic P-V area as the sum of the external mechanical work within the P-V loop and the elastic potential energy under the ESPVR. To obtain E(max) and PVA, we had to assume the systolic unstressed volume (V(o)) as the V-axis intercept of the ESPVR to be constant despite the varying E(max), since there was no method to obtain V(o) directly in each arrhythmic beat. However, we know that in regular stable beats V(o) decreases by approximately 7 ml/100 g LV with approximately 100 times the increases in E(max) from ~0.2 mmHg/(ml/100 g LV) of almost arresting weak beats to approximately 20 mmHg/(ml/100 g LV) of strong beats with a highly enhanced contractility. In the present study, we investigated whether E(max) and PVA under AF could still distribute normally, despite such E(max)-dependent V(o) changes. The present analyses showed that the E(max) changes were only approximately 3 times at most from the weakest to the strongest arrhythmic beat under AF. These changes were not large enough to affect V(o) enough to distort the frequency distributions of E(max) and PVA from normality. We conclude that one could practically ignore the slight E(max) and PVA changes with the Emax-dependent V(o) changes under AF.

Animals↗
PubMed1989

Estimation of parameters and missing values under a regression model with non-normally distributed and non-randomly incomplete data.

We carried out a simulation study to compare the performance of three algorithms (complete cases, ALLVALUE, and expectation maximization, EM) in estimating regression parameters and missing values for situations that have varying amounts of missing data, distributions (normal, mixture of normals and lognormal), patterns of incomplete data (random, related and censored), and degrees of correlational structure among the dependent and independent variables. We found that the EM and complete cases algorithms performed equally well regardless of the correlational structure, when the percentage of incomplete data was only 5 per cent. When this percentage increased to 25 per cent, the EM algorithm was generally best for estimation, but the complete cases algorithm was safe and conservative. This finding may be attributed to the study design, which required that the slopes be the same in the population of all cases, and in the population of complete cases. In addition, the one-step imputing method (ALLVALUE) was competitive only for situations with weak correlational structure and/or little missing data. In that situation the bias caused with use of all available information was less than that caused with use of only complete cases. On the other hand, for imputation, the EM algorithm performed optimally, even in situations of censored or log-normally distributed data.

Algorithms↗
PubMed1993

Normal distribution of hand preference and its bimodality.

The distribution of hand preference was studied in human subjects by means of The Waterloo Handedness Questionnaire and Edinburgh Handedness Inventory. The Waterloo Handedness scores (WS) and Geschwind Scores (GS) were estimated. Left-handers (WS < 0; GS < 0) and right-handers (WS > 0; GS > 0) constituted separate groups (dichotomy). The WSs, and to a lesser extent GSs, were distributed normally when only the right-handers were considered. There was a significant positive-linear correlation between WSs and GSs. These results suggested that a detailed questionnaire such as the Waterloo Handedness Questionnaire would be more appropriate than concise questionnaires in assessing hand preference. A normally distributed right-hand preference and a widely dispersed left-hand preference also supports the concept of anomalous dominance for left-handedness.

Adolescent↗
PubMed2007

Transformation of arbitrary distributions to the normal distribution with application to EEG test-retest reliability.

Many variables in the social, physical, and biosciences, including neuroscience, are non-normally distributed. To improve the statistical properties of such data, or to allow parametric testing, logarithmic or logit transformations are often used. Box-Cox transformations or ad hoc methods are sometimes used for parameters for which no transformation is known to approximate normality. However, these methods do not always give good agreement with the Gaussian. A transformation is discussed that maps probability distributions as closely as possible to the normal distribution, with exact agreement for continuous distributions. To illustrate, the transformation is applied to a theoretical distribution, and to quantitative electroencephalographic (qEEG) measures from repeat recordings of 32 subjects which are highly non-normal. Agreement with the Gaussian was better than using logarithmic, logit, or Box-Cox transformations. Since normal data have previously been shown to have better test-retest reliability than non-normal data under fairly general circumstances, the implications of our transformation for the test-retest reliability of parameters were investigated. Reliability was shown to improve with the transformation, where the improvement was comparable to that using Box-Cox. An advantage of the general transformation is that it does not require laborious optimization over a range of parameters or a case-specific choice of form.

Algorithms↗
PubMed1985

Explorative studies on karyometric frequency distributions. II. Approximation using normally distributed curves versus median interpolation.

An approximation using normally distributed curves and a median interpolation were applied to frequency distributions of nuclear size with one maximum. Both methods gave the same results as far as the positions of the curves on the axis of nuclear volume classes were concerned. Further detailed information on the course of the curves by the median interpolation, however, may be falsified by artificial differences due to the interpolation.

Animals↗
PubMed2000

A computer program for pharmacokinetics based on maximum likelihood estimation using the gamma distribution with a probability density function: comparison with the normal distribution.

A computer program is described for maximum likelihood estimation within the gamma or normal distribution which can be used to estimate pharmacokinetic parameters. Pharmacokinetic analysis using this proposed program was investigated by the Monte Carlo method. The assumed pharmacokinetic models were a one-compartment intravenous model and an oral model. The simulated drug concentrations were generated using a 10% S.D. based on the gamma or normal distribution. The gamma or normal distribution was adopted as the probability density function (p.d.f.) to estimate model parameters. The Powell method was used to maximize the logarithmic likelihood. There were no differences in the estimated parameters in terms of statistical and frequency distributions between the gamma and normal distributions using the generated data and the p.d.f. distributions. However, the number of failures to calculate the parameters using the p.d.f. with the normal distribution was more than five times that using the gamma distribution. This result suggests that it may be necessary to evaluate the validity of results computed using the maximum likelihood estimation based on a normal distribution as a data error distribution and p.d.f.

Algorithms↗
PubMed2005

Is the diaphragm motion probability density function normally distributed?

During radiotherapy treatment planning, the margins given to the clinical target volume to form the planning target volume accounts for internal motion and set-up error. Most margin formulas assume that the underlying distributions are independent and normal. Clinical data suggests that the set-up error probability density function (pdf) can be considered to have an approximately normal distribution. However, there is evidence that internal motion does not have a normal distribution. Thus, in general, a convolution of the two pdfs should be performed to determine the total geometric error. The goals of this article were to (1) determine if the internal motion pdf due to respiration can be characterized using a normal distribution, and (2) if not, determine if the total geometric uncertainty for combining internal motion and set-up error can be characterized by a normal distribution. Sixty fluoroscopy diaphragm motion data sets were obtained using three breathing training types: free breathing, audio instruction, and visual feedback. Diaphragm motion was used as a surrogate for liver and lung cancer motion. The data were analyzed with normality tests in the following groups: (1) single motion measurements, (2) combined motion measurements for each patient, and (3) combined motion measurements for all patients. Following this analysis, the diaphragm motion pdfs were convolved with a set-up error pdf, and the standard deviation of the set-up error pdf at which the total geometric error pdf became normal was determined. At set-up error standard deviation values of at least 0.27 and 0.1 cm for free breathing, 0.57 and 0.42 cm for audio instruction, and 0.55 and 0 cm for visual feedback, for single motion measurements and combined motion measurements for each patient, respectively, total geometric error pdfs became approximately normal. When the motion measurements for all the patients were combined, diaphragm motion pdfs were approximately normal for all feedback types. Therefore, for treatment planning purposes in the absence of individual patient measurements, the diaphragm motion pdf can be considered an approximately normal distribution. However, care should be taken when determining a margin based on individual patients measurements as the total geometric error will, in general, not be normally distributed.

Artifacts↗
PubMed1989

Distribution of in vivo insulin action in Pima Indians as mixture of three normal distributions.

If a single gene produced insulin resistance, with environmental effects creating some additional variance, insulin action might be distributed as a mixture of two normal distributions if the gene is dominant or recessive or as a mixture of three normal distributions if the gene is codominant. To estimate maximal insulin-stimulated glucose uptake rates (MaxMs), hyperinsulinemic-euglycemic clamps were performed on 245 nondiabetic Pima Indians (126 men, 119 women). Five models (for 1, 2, 3, 4, or 5 components each, normally distributed with a common variance) were fitted to the frequency distribution of MaxM by iterative maximum-likelihood estimation. The three-component model fit the data significantly better than a single normal distribution (chi 2 = 14.3 with 4 df P less than .01) or a mixture of two normal distributions (chi 2 = 9.9 with 2 df, P less than .01). Mixtures of four or five normal distributions did not fit the data significantly better than a mixture of three normal distributions. The first component of the distribution comprised 23%, the second 48%, and the third 29% of the total distribution. Similarly, the frequency distributions of fasting plasma insulin concentrations and a principal component score derived from MaxM and fasting insulin were best fitted by a mixture of three normal distributions. These results are consistent with the hypothesis that among Pimas, insulin resistance is determined by a single gene with a codominant mode of inheritance. Segregation analyses of studies performed in pedigrees are indicated to prove or disprove this genetic hypothesis.

Adult↗
PubMed1991

Cytosolic calcium is normally distributed in brain synaptosomes of normal and chronic renal failure rats.

Available data indicate that the basal levels of cytosolic calcium ([Ca2+]i) varies widely in polymorphonuclear leukocytes and in thymocytes. The present study examined whether a similar phenomenon exists in other cells. We measured the basal level of [Ca2+]i in brain synaptosomes of normal and chronic renal failure (CRF) rats. We find these levels (278-439 mM) vary widely in both normal rats and in CRF animals (350-495 mM) as well. However, despite these wide variations, the values followed normal distribution in both groups of animals. The data are consistent with the proposition that the values of [Ca2+]i in many cells display wide variations among a particular species and the levels are most likely predetermined genetically for each animal.

Animals↗
PubMed2006

Optimum parameters in a model for tumour control probability, including interpatient heterogeneity: evaluation of the log-normal distribution.

The heterogeneity of human tumour radiation response is well known. Researchers have used the normal distribution to describe interpatient tumour radiosensitivity. However, many natural phenomena show a log-normal distribution. Log-normal distributions are common when mean values are low, variances are large and values cannot be negative. These conditions apply to radiosensitivity. The aim of this work was to evaluate the log-normal distribution to predict clinical tumour control probability (TCP) data and to compare the results with the homogeneous (delta-function with single alpha-value) and normal distributions. The clinically derived TCP data for four tumour types-melanoma, breast, squamous cell carcinoma and nodes-were used to fit the TCP models. Three forms of interpatient tumour radiosensitivity were considered: the log-normal, normal and delta-function. The free parameters in the models were the radiosensitivity mean, standard deviation and clonogenic cell density. The evaluation metric was the deviance of the maximum likelihood estimation of the fit of the TCP calculated using the predicted parameters to the clinical data. We conclude that (1) the log-normal and normal distributions of interpatient tumour radiosensitivity heterogeneity more closely describe clinical TCP data than a single radiosensitivity value and (2) the log-normal distribution has some theoretical and practical advantages over the normal distribution. Further work is needed to test these models on higher quality clinical outcome datasets.

Breast Neoplasms↗
PubMed2001

The Normal Distribution Derived from Qualitative Conditions.

The normal distribution is characterized in a measurement theoretic framework. The qualitative conditions guarantee that representations can be regarded as random variables. Additional axioms, also qualitative in the measurement sense, yield the normal. One characterization draws on a limit theorem. The main result derives the normal distribution from conjoint measurement axioms. This approach consists of formulating properties of a linear model as a component structure with error as one component. The normal distribution of errors is shown to be a consequence of the measurement theoretic assumptions. The possible impact of these results on statistical models is discussed. Copyright 2001 Academic Press.

Journal Article↗
PubMed1993

The normal distribution in scaling subjective stimulus differences: less "normal" than we think?

The Gaussian, or "normal," distribution is routinely used to model distributions of subjective quantities. This practice rests on a trust in the central limit theorem, yet this theorem does not cover the case of a mixture of Gaussian distributions with different standard deviations, which yields a distribution that is heavier tailed than the Gaussian. In psychophysical judgment experiments, this may result, for example, from fluctuating or interindividually varying attention. Two candidates for describing this commonly encountered type of distribution are the logistic distribution and the t distribution with a small number of degrees of freedom. In reanalyses of experimental data on three-category loudness comparisons, as well as in a Monte Carlo simulation, t(4) was found to model the underlying mixed inter- and intraindividual distribution of subjective loudness differences quite satisfactorily.

Humans↗
PubMed2005

Analysis of quantitative data obtained from toxicity studies showing non-normal distribution.

The data obtained from toxicity studies are examined for homogeneity of variance, but, usually, they are not examined for normal distribution. In this study I examined the measured items of a carcinogenicity/chronic toxicity study with rats for both homogeneity of variance and normal distribution. It was observed that a lot of hematology and biochemistry items showed non-normal distribution. For testing normal distribution of the data obtained from toxicity studies, the data of the concurrent control group may be examined, and for the data that show a non-normal distribution, non-parametric tests with robustness may be applied.

Animals↗