Estimating the number needed to treat from continuous outcomes in randomised controlled trials: methodological challenges and worked example using data from the UK Back Pain Exercise and Manipulation (BEAM) trial

Background Reporting numbers needed to treat (NNT) improves interpretability of trial results. It is unusual that continuous outcomes are converted to numbers of individual responders to treatment (i.e., those who reach a particular threshold of change); and deteriorations prevented are only rarely considered. We consider how numbers needed to treat can be derived from continuous outcomes; illustrated with a worked example showing the methods and challenges. Methods We used data from the UK BEAM trial (n = 1, 334) of physical treatments for back pain; originally reported as showing, at best, small to moderate benefits. Participants were randomised to receive 'best care' in general practice, the comparator treatment, or one of three manual and/or exercise treatments: 'best care' plus manipulation, exercise, or manipulation followed by exercise. We used established consensus thresholds for improvement in Roland-Morris disability questionnaire scores at three and twelve months to derive NNTs for improvements and for benefits (improvements gained+deteriorations prevented). Results At three months, NNT estimates ranged from 5.1 (95% CI 3.4 to 10.7) to 9.0 (5.0 to 45.5) for exercise, 5.0 (3.4 to 9.8) to 5.4 (3.8 to 9.9) for manipulation, and 3.3 (2.5 to 4.9) to 4.8 (3.5 to 7.8) for manipulation followed by exercise. Corresponding between-group mean differences in the Roland-Morris disability questionnaire were 1.6 (0.8 to 2.3), 1.4 (0.6 to 2.1), and 1.9 (1.2 to 2.6) points. Conclusion In contrast to small mean differences originally reported, NNTs were small and could be attractive to clinicians, patients, and purchasers. NNTs can aid the interpretation of results of trials using continuous outcomes. Where possible, these should be reported alongside mean differences. Challenges remain in calculating NNTs for some continuous outcomes

TIME-DEPENDENT CARRIER VELOCITIES IN III-V COMPOUNDS CALCULATED BY THE LEGENDRE-POLYNOMIAL ITERATIVE METHOD

Une méthode semi-analytique rapide est présentée pour la résolution de l'équation de Boltzmann par la décomposition en polynômes orthogonaux de Legendre. Cette méthode est basée sur l'introduction d'un terme "Self scattering". Cette méthode, si on la compare à l'intégration numérique est directe et ne souffre pas des difficultés dues aux instabilités numériques. Elle permet aussi de calculer la réponse de la répartition des vitesses sur un champ électrique variable dans le temps.A fast semi-analytical method is described for solving the time-dependent Boltzmann equation expanded in Legendre polynomials. The method is based upon the introduction of a self-scattering term. Compared with direct numerical integration this method is straightforward and has no difficulties due to numerical instabilities. It also allows calculation of the time-response of the distribution function to a varying electric field

A full alternative for the RTD quantum-inductance equivalent-circuit model

Under specific conditions, the small-signal series/parallel double-RC equivalent-network is a novel full mutual alternative for the resonant tunnelling diode quantum-inductance circuit model. Network optimisations to accurately match measured intrinsic impedances of stable, non-oscillating GaAs/AlAs devices, pointed at these conditions. The capacitance Cw of the series-RC branch, peaks needle-sharp at the negative dynamic conductance "maximum, indicating carrier discharge from the quantum well. The Rb Cw -time constant equals Lq Gd of the quantum-inductance model, so it is also an indication of the quasibound-state lifetime in the well. For CAD purposes, a very good RTD intrinsic impedance description in the entire bias/frequency space (0-2 V; 0.05-40.05 GHz) is obtained with frequency-independent intrinsic elements, scalable with device area