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PubMed2000

Random effects modeling of multiple binomial responses using the multivariate binomial logit-normal distribution.

The multivariate binomial logit-normal distribution is a mixture distribution for which, (i) conditional on a set of success probabilities and sample size indices, a vector of counts is independent binomial variates, and (ii) the vector of logits of the parameters has a multivariate normal distribution. We use this distribution to model multivariate binomial-type responses using a vector of random effects. The vector of logits of parameters has a mean that is a linear function of explanatory variables and has an unspecified or partly specified covariance matrix. The model generalizes and provides greater flexibility than the univariate model that uses a normal random effect to account for positive correlations in clustered data. The multivariate model is useful when different elements of the response vector refer to different characteristics, each of which may naturally have its own random effect. It is also useful for repeated binary measurement of a single response when there is a nonexchangeable association structure, such as one often expects with longitudinal data or when negative association exists for at least one pair of responses. We apply the model to an influenza study with repeated responses in which some pairs are negatively associated and to a developmental toxicity study with continuation-ratio logits applied to an ordinal response with clustered observations.

Abnormalities, Drug-Induced↗

PubMed1992

Evidence that dyslexia may represent the lower tail of a normal distribution of reading ability.

BACKGROUND: Dyslexia is now widely believed to be a biologically based disorder that is distinct from other, less specific reading problems. According to this view, reading ability is considered to follow a bimodal distribution, with dyslexia as the lower mode. We hypothesized that, instead, reading ability follows a normal distribution, with dyslexia at the lower end of the continuum. METHODS AND RESULTS: We used data from the Connecticut Longitudinal Study, a sample survey of 414 Connecticut children who entered kindergarten in 1983 and were followed as a longitudinal cohort. Dyslexia was defined in terms of a discrepancy score, which represents the difference between actual reading achievement and achievement predicted on the basis of measures of intelligence. Data were available from intelligence tests administered in grades 1, 3, and 5 and achievement tests administered yearly in grades 1 through 6. For each child there were 108 possible discrepancy scores ([3 x 3 years] x [2 x 6 years]) based on combinations of the ability scores (full-scale, verbal, and performance IQ) in each of three years and two achievement scores (reading and mathematics) in each of six years. We demonstrated that each of the discrepancy scores followed a univariate normal distribution and that the interrelation of two different discrepancy scores followed a bivariate normal distribution. At most, only 9 of 108 discrepancy scores (8.3 percent) and 171 of 3402 pairs of discrepancy scores (5.0 percent) were significantly different (at the 5 percent level) from the expected scores--well within the expected values for data with univariate and bivariate normal distributions, respectively. We also examined the stability of dyslexia over time. The normal-distribution model predicted (and the data indicated) that only 7 of the 25 children (28 percent) classified as having dyslexia in grade 1 would also be classified as having dyslexia in grade 3. CONCLUSIONS: Reading difficulties, including dyslexia, occur as part of a continuum that also includes normal reading ability. Dyslexia is not an all-or-none phenomenon, but like hypertension, occurs in degrees. The variability inherent in the diagnosis of dyslexia can be both quantified and predicted with use of the normal-distribution model.

Child↗

PubMed1998

Method to assist in the scheduling of add-on surgical cases--upper prediction bounds for surgical case durations based on the log-normal distribution.

BACKGROUND: A problem that operating room (OR) managers face in running an OR suite on the day of surgery is to identify "holes" in the OR schedule in which to assign "add-on" cases. This process necessitates knowing the typical and maximum amounts of time that the case is likely to require. The OR manager may know previous case durations for the particular surgeon performing a particular scheduled procedure. The "upper prediction bound" specifies with a certain probability that the duration of the surgeon's next case will be less than or equal to the bound. METHODS: Prediction bounds were calculated by using methods that (1) do not assume that case durations follow a specific statistical distribution or (2) assume that case durations follow a log-normal distribution. These bounds were tested using durations of 48,847 cases based on 15,574 combinations of scheduled surgeon and procedure. RESULTS: Despite having 3 yr of data, 80 or 90% prediction bounds would not be able to be calculated using the distribution-free method for 35 or 49% of future cases versus 22 or 22% for the log-normal method, respectively. Prediction bounds based on the log-normal distribution overestimated the desired value less often than did the distribution-free method. The chance that the duration of the next case would be less than or equal to its 90% bound based on the log-normal distribution was within 2% of the expected rate. CONCLUSIONS: Prediction bounds classified by scheduled surgeon and procedure can be accurately calculated using a method that assumes that case durations follow a log-normal distribution.

Algorithms↗

PubMed1981

The inadequacy of normal distribution models for the establishment of assigned values in control sera. The establishment of assigned values in control sera, II.

We compare 4 statistical models for the establishment of assigned values in a control serum which are based on the assumption of a normal distribution. The first model results in mean +/- 2s, whereas each of the following 3 models are based on a special analysis of variance. We studied by means of appropriate statistical tests the distributional properties of the data of the study described in part 1 of this series (Passing, H. et al. (1981) this j. 19, 1137-1144). Many model assumptions are violated: The totality of data of each method was never normally distributed, normal distribution within laboratories was not given in 27 out of 67 cases, and precision and accuracy varied from reference laboratory to reference laboratory. Moreover, assigned values and uncertainty intervals calculated by means of these methods can be misleading to the customer. Therefore, these models cannot be applied, and a distribution-free procedure has to be used instead.

Blood Chemical Analysis↗

PubMed1991

A new, sensitive graphical method for detecting deviations from the normal distribution of drug responses: the NTV plot.

1. A new graphical method was developed for the detection of deviations from the normal distribution. The approach took advantage of the similarity of graphical features of a graded dose-response relationship and a cumulative normal distribution. 2. The behaviour of the new normal test variable (NTV) plot was evaluated, in comparison with that of the probit plot and probability density functions (the generalization of histograms), for various assumed distributions. These included skewed distributions and composites of normal distributions with a variety of separations, ratios of peak sizes and widths. 3. The NTV approach generally detected deviations from the normal distribution more sensitively than the probit plot. 4. The NTV and probit plots may be able to identify biomadality by complementary approaches. 5. The characteristics of the three graphical representations were illustrated by a simulated sample from a composite of normal distributions and by an example of sparteine metabolism in 142 Cuna Amerindians.

Dose-Response Relationship, Drug↗

PubMed2002

Log-normal distribution of the trace element data results from a mixture of stocahstic input and deterministic internal dynamics.

In previous article, we showed a log-normal distribution of boron and lithium in human urine. This type of distribution is common in both biological and nonbiological applications. It can be observed when the effects of many independent variables are combined, each of which having any underlying distribution. Although elemental excretion depends on many variables, the one-compartment open model following a first-order process can be used to explain the elimination of elements. The rate of excretion is proportional to the amount present of any given element; that is, the same percentage of an existing element is eliminated per unit time, and the element concentration is represented by a deterministic negative power function of time in the elimination time-course. Sampling is of a stochastic nature, so the dataset of time variables in the elimination phase when the sample was obtained is expected to show Normal distribution. The time variable appears as an exponent of the power function, so a concentration histogram is that of an exponential transformation of Normally distributed time. This is the reason why the element concentration shows a log-normal distribution. The distribution is determined not by the element concentration itself, but by the time variable that defines the pharmacokinetic equation.

Adult↗

PubMed1993

Log normal distribution of bronchial responsiveness to methacholine in normal young adults.

It has not been well known whether or not the distribution of non-specific bronchial responsiveness is log normal in normal subjects. We measured bronchial responsiveness to methacholine using partial flow-volume curves. The expiratory flow at 40% (above RV) from the full forced vital capacity (MEF40) was obtained and used as a reference volume to compare the isovolume flow from the partial flow-volume curve (PEF40). Provocative concentrations of methacholine causing a 20% fall in FEV1 (PC20-FEV1) and a 35% fall in MEF40 (PC35-MEF40) and PEF40 (PC35-PEF40) were measured as indices of bronchial responsiveness in 93 Japanese college students including 36 atopic and 57 non-atopic normal subjects. A 35% or more fall in PEF40 was produced by 160 mg/ml or less of methacholine in all the subjects whereas 20% or more fall in FEV1 did not occur in 4 atopic and 10 non-atopic subjects, and 35% or more fall in MEF40 did not occur in 4 non-atopic subjects at the final concentration of methacholine (160 mg/ml). The geometric mean value of PC35-PEF40 was 2.41 (GSEM, 1.21), 3.74 (GSEM, 1.16), and 3.16 (GSEM, 1.12) mg/ml in atopic, non-atopic, and both subjects, respectively. The distribution of log PC35-PEF40 could not be differentiated at the 5% level from a normal distribution using chi-square test to examine the goodness of fit in each group of subjects and in the whole subjects. These results directly confirm the log normal distribution of non-specific bronchial responsiveness in normal humans.

Adult↗

PubMed2006

Log normal distribution of cellular uptake of radioactivity: implications for biologic responses to radiopharmaceuticals.

UNLABELLED: It is widely recognized that radiopharmaceuticals are generally distributed nonuniformly in tissues. Such nonuniformities are observed over the entire range of spatial levels, ranging from organ to subcellular levels. The implications of nonuniform distributions of radioactivity for dosimetry, and ultimately for the biologic response of tissues containing radioactivity, have been investigated extensively. However, there is a paucity of experimental data on the distribution of cellular activity within a population of cells. In the present study, the distribution of activity per cell is experimentally determined and its implications for predicting biologic response are examined. METHODS: Chinese hamster V79 cells were exposed to different concentrations of (210)Po-citrate. The radiolabeled cells were washed, seeded into culture dishes or glass slides, covered with photographic emulsion, and stored in an opaque container. Subsequently, the emulsion was developed, thereby resulting in observable alpha-particle tracks that were scored. RESULTS: The distribution of activity per cell was found to be well described by a log normal distribution function. Theoretic modeling of cell survival as a function of mean activity per cell showed that survival curves differed substantially when the activity per cell was log normally distributed versus when it was assumed conventionally that every cell in the population contained the mean activity. CONCLUSION: The present study provides experimental evidence of log normal cellular uptake of radioactivity. Theoretic calculations show that a log normal distribution of cellular activity can have a substantial impact on modeling the biologic response of cell populations.

Animals↗

PubMed1977

[Log-normal distributed plasma cortisol levels of four daily measurements in normal subjects. A comparison of the fluorescence method after de Moor with a radioimmuno assay of plasma cortisol (author's transl)].

The present paper describes the plasma cortisol concentrations on four daily measurements in normal subjects assayed with the fluorescence method after de Moor and with a radioimmuno assay. A comparison of these two methods shows that the determinations of plasma cortisol with the fluorescence method always yielded higher values. Both methods are sensitive enough to detect the circadian rhythm of cortisol secretion. For the purpose of routine diagnostics it is sufficient to measure the maximal and minimal concentrations of plasma cortisol in blood samples taken at 8 a.m. and 8 p.m., respectively. The plasma cortisol measurements should only be attempted after allowing a few days of adaption to the hospital situation. Table 2 shows that the normal plasma cortisol levels are represented only by log-normal distributed values. These concentrations are lower than the linear distributed levels reported so far.

Adult↗

PubMed2007

Bivariate random effect model using skew-normal distribution with application to HIV-RNA.

Correlated data arise in a longitudinal studies from epidemiological and clinical research. Random effects models are commonly used to model correlated data. Mostly in the longitudinal data setting we assume that the random effects and within subject errors are normally distributed. However, the normality assumption may not always give robust results, particularly if the data exhibit skewness. In this paper, we develop a Bayesian approach to bivariate mixed model and relax the normality assumption by using a multivariate skew-normal distribution. Specifically, we compare various potential models and illustrate the procedure using a real data set from HIV study.

Bayes Theorem↗

PubMed2000

Evaluating group distributional characteristics: why psychophysiologists should be interested in qualitative departures from the normal distribution.

Real data often do not approximate the normal distribution. Under nonnormal conditions, psycho-physiologists who use parametric statistics may be testing with inadequate power and/or testing a measure of location (i.e., the mean) that does not represent the desired portion of the distribution for statistical comparison. The purpose of this paper is: first, to provide psycho-physiologists with a method to investigate group distributional characteristics; second, to evaluate the distributional characteristics of heart period and respiratory sinus arrhythmia in human adults and newborns; third, to demonstrate the increased statistical power that can accompany the selection of an alternative statistical analysis for nonnormal data. Suggestions are provided on how to analyze nonnormal data.

Adult↗

PubMed1978

Estimation of the mixing proportion in a mixture of two normal distributions from simple, rapid measurements.

A mixture of two or more normal distributions often provides an adequate model for the distribution of a population consisting of varying proportions of component subpopulations. We consider here the problem of estimating the mixing proportion in a mixture of two normal distributions, the parameters of which can be assumed known. Very large samples may be needed if reasonably precise estimates are to be obtained, thus bringing into consideration the cost or time involved in obtaining large numbers of exact measurements and computing the estimates from them. Simple estimators based on simple, rapidly obtained measurements may then be attractive alternatives provided efficiency losses are not too great. Three such estimators studied here are based on (a) the number of observations less than a fixed point r, (b) the nembers less than s and greater than t, and (c) the sample mean. Optimal choices of the points r, s and t are considered, and the efficiencies of the estimators relative to maximum likelihood estimators (MLE) using the full data are obtained. The simple estimators often perform sufficiently well to make the collection of full data not worthwhile in practice.

Genetics, Population↗

PubMed2005

Parametric versus non-parametric statistics in the analysis of randomized trials with non-normally distributed data.

BACKGROUND: It has generally been argued that parametric statistics should not be applied to data with non-normal distributions. Empirical research has demonstrated that Mann-Whitney generally has greater power than the t-test unless data are sampled from the normal. In the case of randomized trials, we are typically interested in how an endpoint, such as blood pressure or pain, changes following treatment. Such trials should be analyzed using ANCOVA, rather than t-test. The objectives of this study were: a) to compare the relative power of Mann-Whitney and ANCOVA; b) to determine whether ANCOVA provides an unbiased estimate for the difference between groups; c) to investigate the distribution of change scores between repeat assessments of a non-normally distributed variable. METHODS: Polynomials were developed to simulate five archetypal non-normal distributions for baseline and post-treatment scores in a randomized trial. Simulation studies compared the power of Mann-Whitney and ANCOVA for analyzing each distribution, varying sample size, correlation and type of treatment effect (ratio or shift). RESULTS: Change between skewed baseline and post-treatment data tended towards a normal distribution. ANCOVA was generally superior to Mann-Whitney in most situations, especially where log-transformed data were entered into the model. The estimate of the treatment effect from ANCOVA was not importantly biased. CONCLUSION: ANCOVA is the preferred method of analyzing randomized trials with baseline and post-treatment measures. In certain extreme cases, ANCOVA is less powerful than Mann-Whitney. Notably, in these cases, the estimate of treatment effect provided by ANCOVA is of questionable interpretability.

Analysis of Variance↗

PubMed2002

Likelihood analysis for the ratio of means of two independent log-normal distributions.

Existing methods for comparing the means of two independent skewed log-normal distributions do not perform well in a range of small-sample settings such as a small-sample bioavailability study. In this article, we propose two likelihood-based approaches-the signed log-likelihood ratio statistic and modified signed log-likelihood ratio statistic-for inference about the ratio of means of two independent log-normal distributions. More specifically, we focus on obtaining p-values for testing the equality of means and also constructing confidence intervals for the ratio of means. The performance of the proposed methods is assessed through simulation studies that show that the modified signed log-likelihood ratio statistic is nearly an exact approach even for very small samples. The methods are also applied to two real-life examples.

Biological Availability↗

PubMedDate not supplied

CAN'T MISS--conquer any number task by making important statistics simple. Part 2. Probability, populations, samples, and normal distributions.

Healthcare quality improvement professionals need to understand and use inferential statistics to interpret sample data from their organizations. In quality improvement and healthcare research studies all the data from a population often are not available, so investigators take samples and make inferences about the population by using inferential statistics. This three-part series will give readers an understanding of the concepts of inferential statistics as well as the specific tools for calculating confidence intervals for samples of data. This article, Part 2, describes probability, populations, and samples. The uses of descriptive and inferential statistics are outlined. The article also discusses the properties and probability of normal distributions, including the standard normal distribution.

Data Interpretation, Statistical↗

PubMed1998

Coregistration of FDG PET and MRI of the head and neck using normal distribution of FDG.

UNLABELLED: For better localization of head and neck structures by PET with 2-(18)F-2-deoxy-D-glucose (FDG), direct incorporation of anatomical information from MRI by the coregistration of FDG PET and MRI without external markers is proposed. METHODS: Seventeen patients with neoplasms and 16 normal subjects who had both FDG PET and MRI were studied. First, the three-dimensional normal distribution of FDG was evaluated, and then the structures of the head and neck regions with normal distribution patterns of FDG were used as internal markers for the coregistration of PET and MRI. The effectiveness of the coregistration was evaluated using focal neoplasms that were identified by both PET and MRI as fiducial internal markers. RESULTS: The normal structures selected as internal landmarks for coregistration were the tonsils, salivary glands, mucosal layers of the oral cavity and pharynx, spinal cord, inferior portion of the frontal lobe, cerebellum and nasal turbinates. These structures were more easily observed in sagittal or coronal sections than in transaxial sections. All primary neoplasms were delineated by PET, whereas 4 were missed by MRI. Thirteen primary tumors and 7 cervical lymph node metastases coregistered well, with a center-of-mass distance of <2 mm, whereas 10 lymph node metastases were slightly misregistered, with a center-of-mass distance of 7.8+/-6.5 mm (mean+/-s.d.), probably due to differences in neck positions. CONCLUSION: Normal distribution of FDG uptake in the head and neck regions delineated by multidirectional sections is important for effective coregistration of FDG PET with MRI.

Aged↗

PubMed1991

Heritability estimates of transformations of normally distributed records.

Milk yields were simulated for identical twin and parent-progeny pairs to determine effect of transformations on genetic variance. Use of identical twin pairs excludes variability due to Mendelian sampling and contribution by the other parent present in parent-progeny records. Phenotypic variances of 616,800; 1,439,200; and 2,261,600 and means from 2267 to 13,603 by increments of 2267 were used for linear scale records. Phenotypic variances of .01000, .01875, and .02750 and means equal to the natural logarithm of the means for linear scale were used for log scale records. Heritability varied from .05 to .95 by increments of .10. For each combination of parameters, 10 replicates of 10,000 pairs of relatives were created. Data normally distributed on a linear scale were log transformed. Log normally distributed data were exponentially transformed. Heritability estimates from correlations among relatives indicated heritability is larger for normally distributed data than for transformed data. For parent-progeny pairs, the difference increased as heritability increased, and for identical twin pairs, the differences were least at extremely large and small heritabilities. For both types of relative pairs, the difference increased as phenotypic variance increased. Absolute differences in heritability ranged from .0000 to .0089, a difference of little practical importance.

Animals↗

PubMed2002

Fracture statistics of brittle materials: Weibull or normal distribution.

The fit of fracture strength data of brittle materials (Si(3)N(4), SiC, and ZnO) to the Weibull and normal distributions is compared in terms of the Akaike information criterion. For Si(3)N(4), the Weibull distribution fits the data better than the normal distribution, but for ZnO the result is just the opposite. In the case of SiC, the difference is not large enough to make a clear distinction between the two distributions. There is not sufficient evidence to show that the Weibull distribution is always preferred to other distributions, and the uncritical use of the Weibull distribution for strength data is questioned.

Journal Article↗