Werner A. Stahel

Log-normal Distributions across the Sciences: Keys and Clues

formalization, and abstraction in biology, so too does mathematics’ relevance to the field (Fagerström et al. 1996). Mathematics is particularly important for analyzing and characterizing random variation of, for example, size and weight of individuals in populations, their sensitivity to chemicals, and time-to-event cases, such as the amount of time an individual needs to recover from illness. The frequency distribution of such data is a major factor determining the type of statistical analysis that can be validly carried out on any data set. Many widely used statistical methods, such as ANOVA (analysis of variance) and regression analysis, require that the data be normally distributed, but only rarely is the frequency distribution of data tested when these techniques are used. The Gaussian (normal) distribution is most often assumed to describe the random variation that occurs in the data from many scientific disciplines; the well-known bell-shaped curve can easily be characterized and described by two values: the arithmetic mean ̄x and the standard deviation s, so that data sets are commonly described by the expression x̄ ± s. A historical example of a normal distribution is that of chest measurements of Scottish soldiers made by Quetelet, Belgian founder of modern social statistics (Swoboda 1974). In addition, such disparate phenomena as milk production by cows and random deviations from target values in industrial processes fit a normal distribution. However, many measurements show a more or less skewed distribution. Skewed distributions are particularly common when mean values are low, variances large, and values cannot be negative, as is the case, for example, with species abundance, lengths of latent periods of infectious diseases, and distribution of mineral resources in the Earth’s crust. Such skewed distributions often closely fit the log-normal distribution (Aitchison and Brown 1957, Crow and Shimizu 1988, Lee 1992, Johnson et al. 1994, Sachs 1997). Examples fitting the normal distribution, which is symmetrical, and the lognormal distribution, which is skewed, are given in Figure 1. Note that body height fits both distributions. Often, biological mechanisms induce log-normal distributions (Koch 1966), as when, for instance, exponential growth is combined with further symmetrical variation: With a mean concentration of, say, 106 bacteria, one cell division more— or less—will lead to 2 × 106—or 5 × 105—cells.Thus, the range will be asymmetrical—to be precise, multiplied or divided by 2 around the mean. The skewed size distribution may be why “exceptionally”big fruit are reported in journals year after year in autumn. Such exceptions, however, may well be the rule: Inheritance of fruit and flower size has long been known to fit the log-normal distribution (Groth 1914, Powers 1936, Sinnot 1937). What is the difference between normal and log-normal variability? Both forms of variability are based on a variety of forces acting independently of one another. A major difference, however, is that the effects can be additive or multiplicative, thus leading to normal or log-normal distributions, respectively.

Three-dimensional, task-specific robot therapy of the arm after stroke: a multicentre, parallel-group randomised trial

BACKGROUND Arm hemiparesis secondary to stroke is common and disabling. We aimed to assess whether robotic training of an affected arm with ARMin--an exoskeleton robot that allows task-specific training in three dimensions-reduces motor impairment more effectively than does conventional therapy. METHODS In a prospective, multicentre, parallel-group randomised trial, we enrolled patients who had had motor impairment for more than 6 months and moderate-to-severe arm paresis after a cerebrovascular accident who met our eligibility criteria from four centres in Switzerland. Eligible patients were randomly assigned (1:1) to receive robotic or conventional therapy using a centre-stratified randomisation procedure. For both groups, therapy was given for at least 45 min three times a week for 8 weeks (total 24 sessions). The primary outcome was change in score on the arm (upper extremity) section of the Fugl-Meyer assessment (FMA-UE). Assessors tested patients immediately before therapy, after 4 weeks of therapy, at the end of therapy, and 16 weeks and 34 weeks after start of therapy. Assessors were masked to treatment allocation, but patients, therapists, and data analysts were unmasked. Analyses were by modified intention to treat. This study is registered with ClinicalTrials.gov, number NCT00719433. FINDINGS Between May 4, 2009, and Sept 3, 2012, 143 individuals were tested for eligibility, of whom 77 were eligible and agreed to participate. 38 patients assigned to robotic therapy and 35 assigned to conventional therapy were included in analyses. Patients assigned to robotic therapy had significantly greater improvements in motor function in the affected arm over the course of the study as measured by FMA-UE than did those assigned to conventional therapy (F=4.1, p=0.041; mean difference in score 0.78 points, 95% CI 0.03-1.53). No serious adverse events related to the study occurred. INTERPRETATION Neurorehabilitation therapy including task-oriented training with an exoskeleton robot can enhance improvement of motor function in a chronically impaired paretic arm after stroke more effectively than conventional therapy. However, the absolute difference between effects of robotic and conventional therapy in our study was small and of weak significance, which leaves the clinical relevance in question. FUNDING Swiss National Science Foundation and Bangerter-Rhyner Stiftung.

Aerosol emission in a road tunnel

Abstract Continuous measurements of aerosol emissions were performed within the scope of emission measurements in the Gubrist tunnel, a 3250 m long freeway tunnel near Zurich, Switzerland, from 20 September to 26 September 1993. The particles in the respirable size range (d

A Procedure for Robust Estimation and Inference in Linear Regression

Even if robust regression estimators have been around for nearly 20 years, they have not found widespread application. One obstacle is the diversity of estimator types and the necessary choices of tuning constants, combined with a lack of guidance for these decisions. While some participants of the IMA summer program have argued that these choices should always be made in view of the specific problem at hand, we propose a procedure which should fit many purposes reasonably well. A second obstacle is the lack of simple procedures for inference, or the reluctance to use the straightforward inference based on asymptotics.

Emission factors from road traffic from a tunnel study (Gubrist tunnel, Switzerland). Part III: Results of organic compounds, SO2 and speciation of organic exhaust emission

Emission factors (EF) of volatile hydrocarbons, oxygenated organics, polycyclic aromatic hydrocarbons (PAH) and sulfur dioxide measured in a road tunnel study (Gubrist tunnel, close to Zurich, Switzerland) in September 1993 are reported, extending the previously published list. The speciation of organic exhaust emission of gasoline powered vehicles agreed generally well with recent tunnel studies from U.S.A.

Sharpening Wald-type inference in robust regression for small samples

The datasets used in statistical analyses are often small in the sense that the number of observations n is less than 5 times the number of parameters p to be estimated. In contrast, methods of robust regression are usually optimized in terms of asymptotics with an emphasis on efficiency and maximal bias of estimated coefficients. Inference, i.e., determination of confidence and prediction intervals, is proposed as complementary criteria. An analysis of MM-estimators leads to the development of a new scale estimate, the Design Adaptive Scale Estimate, and to an extension of the MM-estimate, the SMDM-estimate, as well as a suitable @j-function. A simulation study shows and a real data example illustrates that the SMDM-estimate has better performance for small n/p and that the use the new scale estimate and of a slowly redescending @j-function is crucial for adequate inference.

Non-invasive glucose monitoring in patients with Type 1 diabetes: a Multisensor system combining sensors for dielectric and optical characterisation of skin.

In vivo variations of blood glucose (BG) are affecting the biophysical characteristics (e.g. dielectric and optical) of skin and underlying tissue (SAUT) at various frequencies. However, the skin impedance spectra for instance can also be affected by other factors, perturbing the glucose related information, factors such as temperature, skin moisture and sweat, blood perfusion as well as body movements affecting the sensor-skin contact. In order to be able to correct for such perturbing factors, a Multisensor system was developed including sensors to measure the identified factors. To evaluate the quality of glucose monitoring, the Multisensor was applied in 10 patients with Type 1 diabetes. Glucose was administered orally to induce hyperglycaemic excursions at two different study visits. For analysis of the sensor signals, a global multiple linear regression model was derived. The respective coefficients of the variables were determined from the sensor signals of this first study visit (R(2)=0.74, MARD=18.0%--mean absolute relative difference). The identical set of modelling coefficients of the first study visit was re-applied to the test data of the second study visit to evaluate the predictive power of the model (R(2)=0.68, MARD=27.3%). It appears as if the Multisensor together with the global linear regression model applied, allows for tracking glucose changes non-invasively in patients with diabetes without requiring new model coefficients for each visit. Confirmation of these findings in a larger study group and under less experimentally controlled conditions is required for understanding whether a global parameterisation routine is feasible.

Respiratory responses during affective picture viewing.

Previous research has demonstrated covariation of physiological responding with judgments of valence and arousal. However, until now links between these affective dimensions and respiratory measures have not been extensively investigated. In this study, eight picture series of different affective valence and arousal level were shown to 30 subjects, while respiration, skin conductance level (SCL), heart rate (HR) and affective judgments were measured. With increasing pleasantness, inspiratory time lengthened, mean inspiratory flow decreased and thoracic breathing increased. With increasing arousal, inspiratory time and total breath duration shortened and mean inspiratory flow, minute ventilation, thoracic breathing and electrodermal activity increased. These findings confirm the importance of arousal in respiratory responding, but also indicate a modulatory role of affective valence. We propose that the arousal effects reflect energy mobilization in preparation to act, and that the valence effects might be a manifestation of an attention bias toward negative stimuli.

Comparison of emission factors for road traffic from a tunnel study (Gubrist tunnel, Switzerland) and from emission modeling

Abstract The emission factors of NOx, VOC and CO of a road tunnel study performed in September 1993 in the Gubrist tunnel, close to Zurich (Switzerland) are compared with results of emission calculations based on recent results of dynamometric test measurements. The emission calculations are carried out with a traffic emission model taking into account the detailed composition of the vehicle fleet in the tunnel, the average speed and the gradient of the road and the special aerodynamics in a tunnel. With the exception of NOx emission factors for heavy duty vehicles no evidence for a discrepancy between the results of the tunnel study and the emission modeling was found. The measured emission factors of individual hydrocarbons of light duty vehicles were in good agreement with the expectations for most components.

Problems with Using the Normal Distribution – and Ways to Improve Quality and Efficiency of Data Analysis

Background The Gaussian or normal distribution is the most established model to characterize quantitative variation of original data. Accordingly, data are summarized using the arithmetic mean and the standard deviation, by ± SD, or with the standard error of the mean, ± SEM. This, together with corresponding bars in graphical displays has become the standard to characterize variation. Methodology/Principal Findings Here we question the adequacy of this characterization, and of the model. The published literature provides numerous examples for which such descriptions appear inappropriate because, based on the “95% range check”, their distributions are obviously skewed. In these cases, the symmetric characterization is a poor description and may trigger wrong conclusions. To solve the problem, it is enlightening to regard causes of variation. Multiplicative causes are by far more important than additive ones, in general, and benefit from a multiplicative (or log-) normal approach. Fortunately, quite similar to the normal, the log-normal distribution can now be handled easily and characterized at the level of the original data with the help of both, a new sign, x/, times-divide, and notation. Analogous to ± SD, it connects the multiplicative (or geometric) mean * and the multiplicative standard deviation s* in the form * x/s*, that is advantageous and recommended. Conclusions/Significance The corresponding shift from the symmetric to the asymmetric view will substantially increase both, recognition of data distributions, and interpretation quality. It will allow for savings in sample size that can be considerable. Moreover, this is in line with ethical responsibility. Adequate models will improve concepts and theories, and provide deeper insight into science and life.