Decision curve analysis revisited: overall net benefit, relationships to ROC curve analysis, and application to case-control studies.

ABSTRACT: BACKGROUND: Decision curve analysis has been introduced as a method to evaluate prediction models in terms of their clinical consequences if used for a binary classification of subjects into a group who should and into a group who should not be treated. The key concept for this type of evaluation is the "net benefit", a concept borrowed from utility theory. METHODS: We recall the foundations of decision curve analysis and discuss some new aspects. First, we stress the formal distinction between the net benefit for the treated and for the untreated and define the concept of the "overall net benefit". Next, we revisit the important distinction between the concept of accuracy, as typically assessed using the Youden index and a receiver operating characteristic (ROC) analysis, and the concept of utility of a prediction model, as assessed using decision curve analysis. Finally, we provide an explicit implementation of decision curve analysis to be applied in the context of case-control studies. RESULTS: We show that the overall net benefit, which combines the net benefit for the treated and the untreated, is a natural alternative to the benefit achieved by a model, being invariant with respect to the coding of the outcome, and conveying a more comprehensive picture of the situation. Further, within the framework of decision curve analysis, we illustrate the important difference between the accuracy and the utility of a model, demonstrating how poor an accurate model may be in terms of its net benefit. Eventually, we expose that the application of decision curve analysis to case-control studies, where an accurate estimate of the true prevalence of a disease cannot be obtained from the data, is achieved with a few modifications to the original calculation procedure. CONCLUSIONS: We present several interrelated extensions to decision curve analysis that will both facilitate its interpretation and broaden its potential area of application.

Stimulus-response curve of human motor nerves: multicenter assessment of various indexes.

The value of various indexes to characterize the stimulus-response curve of human motor nerves was assessed in 40 healthy subjects recruited from four European centers of investigation (Créteil, Lausanne, Liège, Marseille). Stimulus-response curves were established by stimulating the right median and ulnar motor nerves at the wrist, with stimulus durations of 0.05 and 0.5 ms. The following parameters were studied: the threshold intensity of stimulation to obtain 10% (I 10), 50% (I 50), and 90% (I 90) of the maximal compound muscle action potential, the ratios I 10/I 50, I 90/I 50, (I 90 - I 10)/I 10, (I 90-I 50)/I 50, and (I 50 - I 10)/I 10, and the slopes of the stimulus-response curves with or without normalization to I 50. For each parameter, within-center variability and reproducibility (in a test-retest study) were assessed and between-center comparisons were made. For most of the parameters, the results varied significantly within and between the centers. Within the centers, only the ratios I 10/I 50 and I 90/I 50 were found constant and reproducible. Between the centers, the absolute intensity thresholds (I 10, I 50, I 90) and the ratio I 90/I 50 did not show significant differences at stimulus duration of 0.5 ms, whatever the stimulated nerve. The reduced variability and good reproducibility of the ratios I 10/I 50 and I 90/I 50 open perspectives in neurophysiological practice for the use of these indexes of the stimulus-response curve, a rapid and noninvasive test.

J2 effect and elliptic inclined periodic orbits in the collision three-body problem

The existence of a new class of inclined periodic orbits of the collision restricted three-body problem is shown. The symmetric periodic solutions found are perturbations of elliptic kepler orbits and they exist only for special values of the inclination and are related to the motion of a satellite around an oblate planet

Sierpinski curve Julia sets for quadratic rational maps

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.

The role of smoking in changes in the survival curve: an empirical study in 10 European countries.

PURPOSE: We examined the role of smoking in the two dimensions behind the time trends in adult mortality in European countries, that is, rectangularization of the survival curve (mortality compression) and longevity extension (increase in the age-at-death). METHODS: Using data on national sex-specific populations aged 50 years and older from Denmark, Finland, France, West Germany, Italy, the Netherlands, Norway, Sweden, Switzerland, and the United Kingdom, we studied trends in life expectancy, rectangularity, and longevity from 1950 to 2009 for both all-cause and nonsmoking-related mortality and correlated them with trends in lifetime smoking prevalence. RESULTS: For all-cause mortality, rectangularization accelerated around 1980 among men in all the countries studied, and more recently among women in Denmark and the United Kingdom. Trends in lifetime smoking prevalence correlated negatively with both rectangularization and longevity extension, but more negatively with rectangularization. For nonsmoking-related mortality, rectangularization among men did not accelerate around 1980. Among women, the differences between all-cause mortality and nonsmoking-related mortality were small, but larger for rectangularization than for longevity extension. Rectangularization contributed less to the increase in life expectancy than longevity extension, especially for nonsmoking-related mortality among men. CONCLUSIONS: Smoking affects rectangularization more than longevity extension, both among men and women.

Sierpinski curve Julia sets for quadratic rational maps

We investigate under which dynamical conditions the Julia set of a quadratic rational map is a Sierpiński curve.

Is the wage curve formal or informal? Evidence for Colombia

Using microdata from the 2002-2006 Colombian Continuous Household Survey, we find an elasticity of individual wages to local unemployment rates of -0.07. However, the elasticity for informal workers is significantly higher, a result which is consistent with efficiency wage theoretical models and relevant for regional labour markets analysis in developing countries.

Receiver operating characteristic (ROC) curve for classification of18F-NaF uptake on PET/CT

Abstract Objective: To assess the cutoff values established by ROC curves to classify18F-NaF uptake as normal or malignant. Materials and Methods: PET/CT images were acquired 1 hour after administration of 185 MBq of18F-NaF. Volumes of interest (VOIs) were drawn on three regions of the skeleton as follows: proximal right humerus diaphysis (HD), proximal right femoral diaphysis (FD) and first vertebral body (VB1), in a total of 254 patients, totalling 762 VOIs. The uptake in the VOIs was classified as normal or malignant on the basis of the radiopharmaceutical distribution pattern and of the CT images. A total of 675 volumes were classified as normal and 52 were classified as malignant. Thirty-five VOIs classified as indeterminate or nonmalignant lesions were excluded from analysis. The standardized uptake value (SUV) measured on the VOIs were plotted on an ROC curve for each one of the three regions. The area under the ROC (AUC) as well as the best cutoff SUVs to classify the VOIs were calculated. The best cutoff values were established as the ones with higher result of the sum of sensitivity and specificity. Results: The AUCs were 0.933, 0.889 and 0.975 for UD, FD and VB1, respectively. The best SUV cutoffs were 9.0 (sensitivity: 73%; specificity: 99%), 8.4 (sensitivity: 79%; specificity: 94%) and 21.0 (sensitivity: 93%; specificity: 95%) for UD, FD and VB1, respectively. Conclusion: The best cutoff value varies according to bone region of analysis and it is not possible to establish one value for the whole body.

Is the wage curve formal or informal? Evidence for Colombia

The objective of this paper is to analyse the existente or not of a wage curve in Colombia, paying special attention to the differences between formal and informal workers, an issue that has been systematically ignored in the wage curve literature. The obtained results using microdata from the Colombian Continuous Household Survey (CHS) between 2002 and 2006 show the existence of a wage curve with a negative slope for the Colombian economy. Using information on metropolitan areas, the estimates of the elasticity of individual wages to local unemployment rates was -0.07, a value that is very close to those obtained for other countries. However, the disaggregation of statistical information for formal and informal workers has shown significant differences among both groups of workers. In particular, for the less protected groups of the labour market, informal workers (both men and women), a high negatively sloped wage curve was found. This result is consistent with the conclusions from efficiency wage theoretical models and should be taken into account when analysing the functioning of regional labour markets in developing countries.

Propagation through fractal media: The Sierpinski gasket and the Koch curve

We present new analytical tools able to predict the averaged behavior of fronts spreading through self-similar spatial systems starting from reaction-diffusion equations. The averaged speed for these fronts is predicted and compared with the predictions from a more general equation (proposed in a previous work of ours) and simulations. We focus here on two fractals, the Sierpinski gasket (SG) and the Koch curve (KC), for two reasons, i.e. i) they are widely known structures and ii) they are deterministic fractals, so the analytical study of them turns out to be more intuitive. These structures, despite their simplicity, let us observe several characteristics of fractal fronts. Finally, we discuss the usefulness and limitations of our approa

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean n-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. Väisälä published in the early 1960’s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value p = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.