Detection of two-component mixtures of lognormal distributions in grouped, doubly truncated data: analysis of red blood cell volume distributions

We have examined the statistical requirements for the detection of mixtures of two lognormal distributions in doubly truncated data when the sample size is large. The expectation-maximization algorithm was used for parameter estimation. A bootstrap approach was used to test for a mixture of distributions using the likelihood ratio statistic. Analysis of computer simulated mixtures showed that as the ratio of the difference between the means to the minimum standard deviation increases, the power for detection also increases and the accuracy of parameter estimates improves. These procedures were used to examine the distribution of red blood cell volume in blood samples. Each distribution was doubly truncated to eliminate artifactual frequency counts and tested for best fit to a single lognormal distribution or a mixture of two lognormal distributions. A single population was found in samples obtained from 60 healthy individuals. Two subpopulations of cells were detected in 25 of 27 mixtures of blood prepared in vitro. Analyses of mixtures of blood from 40 patients treated for iron-deficiency anemia showed that subpopulations could be detected in all by 6 weeks after onset of treatment. To determine if two-component mixtures could be detected, distributions were examined from untransfused patients with refractory anemia. In two patients with inherited sideroblastic anemia a mixture of microcytic and normocytic cells was found, while in the third patient a single population of microcytic cells was identified. In two family members previously identified as carriers of inherited sideroblastic anemia, mixtures of microcytic and normocytic subpopulations were found. Twenty-five patients with acquired myelodysplastic anemia were examined. A good fit to a mixture of subpopulations containing abnormal microcytic or macrocytic cells was found in two. We have demonstrated that with large sample sizes, mixtures of distributions can be detected even when distributions appear to be unimodal. These statistical techniques provide a means to characterize and quantify alterations in erythrocyte subpopulations in anemia but could also be applied to any set of grouped, doubly truncated data to test for the presence of a mixture of two lognormal distributions.     

Nested Sampling for Bayesian Model Comparison in the Context of Salmonella Disease Dynamics

Understanding the mechanisms underlying the observed dynamics of complex biological systems requires the statistical assessment and comparison of multiple alternative models. Although this has traditionally been done using maximum likelihood-based methods such as Akaike''s Information Criterion (AIC), Bayesian methods have gained in popularity because they provide more informative output in the form of posterior probability distributions. However, comparison between multiple models in a Bayesian framework is made difficult by the computational cost of numerical integration over large parameter spaces. A new, efficient method for the computation of posterior probabilities has recently been proposed and applied to complex problems from the physical sciences. Here we demonstrate how nested sampling can be used for inference and model comparison in biological sciences. We present a reanalysis of data from experimental infection of mice with Salmonella enterica showing the distribution of bacteria in liver cells. In addition to confirming the main finding of the original analysis, which relied on AIC, our approach provides: (a) integration across the parameter space, (b) estimation of the posterior parameter distributions (with visualisations of parameter correlations), and (c) estimation of the posterior predictive distributions for goodness-of-fit assessments of the models. The goodness-of-fit results suggest that alternative mechanistic models and a relaxation of the quasi-stationary assumption should be considered.     

Modeling the response of a population of olfactory receptor neurons to an odorant

We modeled the firing rate of populations of olfactory receptor neurons (ORNs) responding to an odorant at different concentrations. Two cases were considered: a population of ORNs that all express the same olfactory receptor (OR), and a population that expresses many different ORs. To take into account ORN variability, we replaced single parameter values in a biophysical ORN model with values drawn from statistical distributions, chosen to correspond to experimental data. For ORNs expressing the same OR, we found that the distributions of firing frequencies are Gaussian at all concentrations, with larger mean and standard deviation at higher concentrations. For a population expressing different ORs, the distribution of firing frequencies can be described as the superposition of a Gaussian distribution and a lognormal distribution. Distributions of maximum value and dynamic range of spiking frequencies in the simulated ORN population were similar to experimental results.     

Landscape‐scale spatial abundance distributions discriminate core from random components of boreal lake bacterioplankton

Aquatic bacterial communities harbour thousands of coexisting taxa. To meet the challenge of discriminating between a ‘core’ and a sporadically occurring ‘random’ component of these communities, we explored the spatial abundance distribution of individual bacterioplankton taxa across 198 boreal lakes and their associated fluvial networks (188 rivers). We found that all taxa could be grouped into four distinct categories based on model statistical distributions (normal like, bimodal, logistic and lognormal). The distribution patterns across lakes and their associated river networks showed that lake communities are composed of a core of taxa whose distribution appears to be linked to in‐lake environmental sorting (normal‐like and bimodal categories), and a large fraction of mostly rare bacteria (94% of all taxa) whose presence appears to be largely random and linked to downstream transport in aquatic networks (logistic and lognormal categories). These rare taxa are thus likely to reflect species sorting at upstream locations, providing a perspective of the conditions prevailing in entire aquatic networks rather than only in lakes.     

Fitting mixture models to grouped and truncated data via the EM algorithm

The fitting of finite mixture models via the EM algorithm is considered for data which are available only in grouped form and which may also be truncated. A practical example is presented where a mixture of two doubly truncated log-normal distributions is adopted to model the distribution of the volume of red blood cells in cows during recovery from anemia.     

Dealing with Uncertainty in Risk Assessment

This paper is a commentary on Hattis’ three laws of risk assessment. The first law, that “application of standard statistical techniques to a single data set will nearly always reveal only a trivial proportion of the overall uncertainty in the parameter value” is illustrated both by examining the relevance of animal models to man and by a retrospective view of exposure conditions whose importance has only recently been recognized to be important. The second law, that “any estimate of the uncertainty of a parameter value will always itself be more uncertain than the estimate of the parameter value,” is examined in terms of a model addressing multiple levels of uncertainty, e.g., the “uncertainty in the uncertainty”. A argument is made that the number of terms needed for convergence of this uncertainty hierarchy depends on how far from the central tendency of the risk distribution one goes. The further out the “tail” of the distribution, the more terms in the uncertainty hierarchy are needed for convergence. The third law, that “nearly all parameter distribu tions look lognormal, as long as you don't look too closely,” is illustrated with a number of examples. Several reasons are put forward as to why risk variables appear so frequently to be lognormal. Recognition of the lognormal character of variable distributions can provide insight into the proper form for the associated uncertainty distributions.     

Effects of Distribution Choice on the Modeling of Life Cycle Inventory Uncertainty: An Assessment on the Ecoinvent v2.2 Database

Life cycle inventory data have multiple sources of uncertainty. These data uncertainties are often modeled using probability density functions, and in the ecoinvent database the lognormal distribution is used by default to model exchange uncertainty values. The aim of this article is to systematically measure the effect of this default distribution by changing from the lognormal to several other distribution functions and examining how this change affects the uncertainty of life cycle assessment results. Using the ecoinvent 2.2 inventory database, data uncertainty distributions are switched from the lognormal distribution to the normal, triangular, and gamma distributions. The effect of the distribution switching is assessed for both impact assessment results of individual products system, as well as comparisons between product systems. Impact assessment results are generated using 5,000 Monte Carlo iterations for each product system, using the Intergovernmental Panel on Climate Change (IPCC) 2001 (100‐year time frame) method. When comparing the lognormal distribution to the alternative default distributions, the difference in the resulting median and standard deviation values range from slight to significant, depending on the distributions used by default. However, the switch shows practically no effect on product system comparisons. Yet, impact assessment results are sensitive to how the data uncertainties are defined. In this article, we followed what we believe to be ecoinvent standard practice and preserved the “most representative” value. Practitioners should recognize that the most representative value can depart from the average of a probability distribution. Consistent default distribution choices are necessary when performing product system comparisons.     

Curve fitting approach for measurement of cellular osmotic properties by the electrical sensing zone method. II. Membrane water permeability

In a companion paper, we demonstrated that dynamic range limitations can confound measurement of the osmotically inactive volume using electrical sensing zone instruments (e.g., Coulter counters), and presented an improved parameter estimation method in which a lognormal function was fit to the cell volume distribution to allow extrapolation beyond the bounds of the data. Presently, we have investigated the effect of dynamic range limitations on measurement of the cell membrane water permeability (L_p), and adapted the lognormal extrapolation method for estimation of L_p from transient volume data. An alternative strategy (the volume limit adjustment method, in which the measured isotonic volume distribution is used to generate model predictions for curve fitting, and the bounds of the dynamic range are adjusted such that extrapolation is not required) was also developed. The performance of these new algorithms was compared to that of a conventional parameter estimation method. The best-fit L_p values from in vitro experiments with mouse insulinoma (MIN6) cells differed significantly for the different parameter estimation techniques (p < 0.001). Using in silico experiments, the volume limit adjustment method was shown to be the most accurate (relative error 0.4 ± 3.2%), whereas the conventional method underestimated L_p by 19 ± 2% for MIN6 cells. Parametric analysis revealed that the error associated with the conventional method was sensitive to the dynamic range and the width of the volume distribution. Our initial implementation of the lognormal extrapolation method also yielded significant errors, whereas accuracy of this algorithm improved after including a normalization scheme.     

Statistical analyses support power law distributions found in neuronal avalanches

The size distribution of neuronal avalanches in cortical networks has been reported to follow a power law distribution with exponent close to -1.5, which is a reflection of long-range spatial correlations in spontaneous neuronal activity. However, identifying power law scaling in empirical data can be difficult and sometimes controversial. In the present study, we tested the power law hypothesis for neuronal avalanches by using more stringent statistical analyses. In particular, we performed the following steps: (i) analysis of finite-size scaling to identify scale-free dynamics in neuronal avalanches, (ii) model parameter estimation to determine the specific exponent of the power law, and (iii) comparison of the power law to alternative model distributions. Consistent with critical state dynamics, avalanche size distributions exhibited robust scaling behavior in which the maximum avalanche size was limited only by the spatial extent of sampling ("finite size" effect). This scale-free dynamics suggests the power law as a model for the distribution of avalanche sizes. Using both the Kolmogorov-Smirnov statistic and a maximum likelihood approach, we found the slope to be close to -1.5, which is in line with previous reports. Finally, the power law model for neuronal avalanches was compared to the exponential and to various heavy-tail distributions based on the Kolmogorov-Smirnov distance and by using a log-likelihood ratio test. Both the power law distribution without and with exponential cut-off provided significantly better fits to the cluster size distributions in neuronal avalanches than the exponential, the lognormal and the gamma distribution. In summary, our findings strongly support the power law scaling in neuronal avalanches, providing further evidence for critical state dynamics in superficial layers of cortex.     

Model of Red Blood Cell Rotation in the Flow toward a Cell Sizing Orifice: Application to Volume Distribution

The rotation of human red blood cells (RBC) as they flow in the shear field established by a Coulter type orifice is modeled. This model, based on hydrodynamics of ellipsoid rotation in laminar creeping flow, is used to calculate the probability of the cells entering the orifice with a specific orientation. The electrical resistance change produced by a cell passing through the orifice of an electronic cell volume detector is the product of an orientation-dependent shape factor and the cell volume. This paper presents a method to calculate the shape factor probability distribution which can be used to predict its effect on the cell volume distribution. Experimental results confirm the theoretical prediction that the right skewness of resistance change distributions is in part a result of the nonspherical shape of red cells.     

Modelling rate distributions using character compatibility: implications for morphological evolution among fossil invertebrates

Rate distributions are important considerations when testing hypotheses about morphological evolution or phylogeny. They also have implications about general processes underlying character evolution. Molecular systematists often assume that rates are Poisson processes with gamma distributions. However, morphological change is the product of multiple probabilistic processes and should theoretically be affected by hierarchical integration of characters. Both factors predict lognormal rate distributions. Here, a simple inverse modelling approach assesses the best single-rate, gamma and lognormal models given observed character compatibility for 115 invertebrate groups. Tests reject the single-rate model for nearly all cases. Moreover, the lognormal outperforms the gamma for character change rates and (especially) state derivation rates. The latter in particular is consistent with integration affecting morphological character evolution.     

Perceptual dominance time distributions in multistable visual perception

Perceptual multistability, alternative perceptions of an unchanging stimulus, gives important clues to neural dynamics. The present study examined 56 perceptual dominance time series for a Necker cube stimulus, for ambiguous motion, and for binocular rivalry. We made histograms of the perceptual dominance times, based on from 307 to 2478 responses per time series (median=612), and compared these histograms to gamma, lognormal and Weibull fitted distributions using the Kolmogorov–Smirnov goodness-of-fit test. In 40 of the 56 tested cases a lognormal distribution provided an acceptable fit to the histogram (in 24 cases it was the only fit). In 16 cases a gamma distribution, and in 11 cases a Weibull distribution, were acceptable but never as the only fit in either case. Any of the three distributions were acceptable in three cases and none provided acceptable fits in 12 cases. Considering only the 16 cases in which a lognormal distribution was rejected (p<0.05) revealed that minor adjustments to the fourth-moment term of the lognormal characteristic function restored good fits. These findings suggest that random fractal theory might provide insight into the underlying mechanisms of multistable perceptions.     

Stage duration distributions in matrix population models

Matrix population models are a standard tool for studying stage‐structured populations, but they are not flexible in describing stage duration distributions. This study describes a method for modeling various such distributions in matrix models. The method uses a mixture of two negative binomial distributions (parametrized using a maximum likelihood method) to approximate a target (true) distribution. To examine the performance of the method, populations consisting of two life stages (juvenile and adult) were considered. The juvenile duration distribution followed a gamma distribution, lognormal distribution, or zero‐truncated (over‐dispersed) Poisson distribution, each of which represents a target distribution to be approximated by a mixture distribution. The true population growth rate based on a target distribution was obtained using an individual‐based model, and the extent to which matrix models can approximate the target dynamics was examined. The results show that the method generally works well for the examined target distributions, but is prone to biased predictions under some conditions. In addition, the method works uniformly better than an existing method whose performance was also examined for comparison. Other details regarding parameter estimation and model development are also discussed.     

Detection of inbreeding effects by the chi-square test on genotypic and phenotypic frequencies.

The problem of detection of inbreeding effects by the chi-square goodness-of-fit test is investigated. The noncentrality parameter of the limiting chi-square distribution is considered as a function of the inbreeding coefficient in two models: (1) a gene locus with m codominant alleles and (2) the generalized ABO model. Previous results by Ward and Sing and by Chakraborty and Rao were based on inadequate statistical reasoning. In the first model, the resulting noncentrality parameter is identical with that found by Ward and Sing. For the second model, the limiting noncentrality parameter is found to be zero.     

Analytical Quantile Solution for the S‐distribution,Random Number Generation and Statistical Data Modeling

The selection of a specific statistical distribution as a model for describing the population behavior of a given variable is seldom a simple problem. One strategy consists in testing different distributions (normal, lognormal, Weibull, etc.), and selecting the one providing the best fit to the observed data and being the most parsimonious. Alternatively, one can make a choice based on theoretical arguments and simply fit the corresponding parameters to the observed data. In either case, different distributions can give similar results and provide almost equivalent models for a given data set. Model selection can be more complicated when the goal is to describe a trend in the distribution of a given variable. In those cases, changes in shape and skewness are difficult to represent by a single distributional form. As an alternative to the use of complicated families of distributions as models for data, the S‐distribution [Voit, E. O. (1992) Biom. J. 7 , 855–878] provides a highly flexible mathematical form in which the density is defined as a function of the cumulative. S‐distributions can accurately approximate many known continuous and unimodal distributions, preserving the well known limit relationships between them. Besides representing well‐known distributions, S‐distributions provide an infinity of new possibilities that do not correspond with known classical distributions. Although the utility and performance of this general form has been clearly proved in different applications, its definition as a differential equation is a potential drawback for some problems. In this paper we obtain an analytical solution for the quantile equation that highly simplifies the use of S‐distributions. We show the utility of this solution in different applications. After classifying the different qualitative behaviors of the S‐distribution in parameter space, we show how to obtain different S‐distributions that accomplish specific constraints. One of the most interesting cases is the possibility of obtaining distributions that acomplish P(X ≤ X_c) = 0. Then, we demonstrate that the quantile solution facilitates the use of S‐distributions in Monte‐Carlo experiments through the generation of random samples. Finally, we show how to fit an S‐distribution to actual data, so that the resulting distribution can be used as a statistical model for them.     

Frequency responses of cat rapidly adapting mechanoreceptive fibers

The frequency responses of 11 rapidly adapting (RA) fibers in cat were studied by representing the average firing rate as a function of sinusoidal stimulus amplitude and stimulus frequency. Specifically, rate-intensity functions at different stimulation frequencies were fitted by four-parameter (a0, a1, a2, a3), piece-wise linear functions using nonlinear regression (n = 59; R2 > 0.877). Rate-intensity functions at intermediate frequencies were found by linear interpolation. The result of this analysis is rate-amplitude-frequency functions plotted as two-dimensional surfaces. The surfaces consist of five regions separated and sufficiently defined by four space curves. At 14 different frequencies, the statistical distribution of each rate-intensity-function parameter could be approximated by a particular lognormal distribution (n = 56; R2 > 0.796). The Kolmogorov-Smirnov test fails to reject this hypothesis for each combination of frequency and parameter (56 tests; p > 0.39). Therefore, at a given frequency, the variation of the parameters can be represented by lognormal distributions with specific means and standard deviations. Responses of six RA fibers, which are different from the data-set used for modeling, were compared with the stochastic model at different frequencies. The parameters of those fibers were tested against the null hypotheses that they were sampled from the particular parameter distributions dictated by the model. The Kolmogorov-Smirnov test fails to reject all the hypotheses at the alpha = 0.05 level (44 tests). At the alpha = 0.10 level, only a few test parameters were found to be departing from the model (a0 and a1 at 5 Hz; a2 at 20 Hz; a2 and a3 at 50 Hz). The remaining test parameters could be accurately described by the model. Having confirmed the validity of the model, the logarithmic means and the logarithmic standard deviations of the lognormally distributed rate-intensity-function parameters were estimated in the frequency range of 4-200 Hz. The rate-amplitude-frequency surfaces sampled from the established stochastic model completely characterize the rate responses of RA fibers to sinusoidal stimuli and are superior to tuning curves which require selecting criterion responses. The current rate-response model is promising for future computational work, especially on population modeling.     

Probits of mixtures

The tolerances of individuals (insects, parasites) in a population have a frequency or probability distribution called a tolerance distribution. Many tolerance distributions in bioassay studies can be the result of a rather heterogeneous population of individuals and can often be modelled as a mixture of a number of standard unimodal distributions. A probit analysis can be generalized to the case where the tolerance distribution is a mixture of location and scale parameter distributions. In this article, the existence and determination of the maximum likelihood estimates are investigated. An expectation-maximization (EM) algorithm for probits of mixtures is developed and it is shown that by application of the EM algorithm, the problem of probits of mixtures can be separated into a series of probits of individual component tolerance distributions.     

Quantitative Imaging of Human Red Blood Cells Infected with Plasmodium falciparum

During its 48 h asexual reproduction cycle, the malaria parasite Plasmodium falciparum ingests and digests hemoglobin in excess of its metabolic requirements and causes major changes in the homeostasis of the host red blood cell (RBC). A numerical model suggested that this puzzling excess consumption of hemoglobin is necessary for the parasite to reduce the colloidosmotic pressure within the host RBC, thus preventing lysis before completion of its reproduction cycle. However, the validity of the colloidosmotic hypothesis appeared to be compromised by initial conflicts between model volume predictions and experimental observations. Here, we investigated volume and membrane area changes in infected RBCs (IRBCs) using fluorescence confocal microscopy on calcein-loaded RBCs. Substantial effort was devoted to developing and testing a new threshold-independent algorithm for the precise estimation of cell volumes and surface areas to overcome the shortfalls of traditional methods. We confirm that the volume of IRBCs remains almost constant during parasite maturation, suggesting that the reported increase in IRBCs' osmotic fragility results from a reduction in surface area and increased lytic propensity on volume expansion. These results support the general validity of the colloidosmotic hypothesis, settle the IRBC volume debate, and help to constrain the range of parameter values in the numerical model.     

Model-based workflow for scale-up of process strategies developed in miniaturized bioreactor systems

Miniaturized bioreactor (MBR) systems are routinely used in the development of mammalian cell culture processes. However, scale-up of process strategies obtained in MBR- to larger scale is challenging due to mainly non-holistic scale-up approaches. In this study, a model-based workflow is introduced to quantify differences in the process dynamics between bioreactor scales and thus enable a more knowledge-driven scale-up. The workflow is applied to two case studies with antibody-producing Chinese hamster ovary cell lines. With the workflow, model parameter distributions are estimated first under consideration of experimental variability for different scales. Second, the obtained individual model parameter distributions are tested for statistical differences. In case of significant differences, model parametric distributions are transferred between the scales. In case study I, a fed-batch process in a microtiter plate (4 ml working volume) and lab-scale bioreactor (3750 ml working volume) was mathematically modeled and evaluated. No significant differences were identified for model parameter distributions reflecting process dynamics. Therefore, the microtiter plate can be applied as scale-down tool for the lab-scale bioreactor. In case study II, a fed-batch process in a 24-Deep-Well-Plate (2 ml working volume) and shake flask (40 ml working volume) with two feed media was investigated. Model parameter distributions showed significant differences. Thus, process strategies were mathematically transferred, and model predictions were simulated for a new shake flask culture setup and confirmed in validation experiments. Overall, the workflow enables a knowledge-driven evaluation of scale-up for a more efficient bioprocess design and optimization.     

Size distribution of the quasi-circular vegetation patches in the Yellow River Delta,China

Recent empirical studies have suggested that the patch-size distribution of vegetation can be fitted by a power law, truncated power law, or lognormal model to provide explanatory mechanisms for vegetation pattern formation in arid and semiarid regions. However, contradictory results have been reported. Therefore, additional empirical studies are necessary to test the patch-size distribution of vegetation over several regions before it can be considered as an indicator for assessing the discontinuous transition of ecosystems and understanding the mechanisms of vegetation pattern formation. Analogous to arid and semiarid regions of the world, vegetation patterns are characterized by a two-phase mosaic composed of dense vegetation patches interspersed with areas of bare soil, referred to as quasi-circular vegetation patches (QVPs), in the Yellow River Delta (YRD), China. However, research on the patch-size distribution of the QVPs reflecting vegetation patterns and ecosystem functioning is lacking. To fill this gap, for the first time, we examined the patch-size distribution of the QVPs using the fused IKONOS high-spatial-resolution image and evaluated the statistical distributions that better fit the patch size data of the QVPs in the YRD. We found that a power law, truncated power law, or lognormal distribution was not supported in the study area, whereas gamma distribution reasonably fits the size data of QVPs, implying that micro-depressions, combined with the water-limited and salinization environments had considerable effects on vegetation pattern formation. Our results provide helpful insights and suggest that further studies are needed to classify different types of QVPs. Additionally, more efficient approaches need to be used to fit the statistical distributions for elucidating the spatial vegetation patterns in the YRD.