The sensitivity of estimates of regression to the mean

Estimations of the effectiveness of remedial treatments in road safety analysis are frequently bedevilled by the problem of regression to the mean (RTM). The number of accidents x observed at a site in the “before” period is a “noisy” quantity: x is Poisson distributed about an (unknown) true mean m for that site, so that x = m + e. Sites selected for treatment tend to have a positive random error component e, which will on average be zero in the “after” period, even if no treatment is applied.Methods for estimating RTM usually require some assumption about the underlying (prior) between-site distribution of the true means f₀(m): for example, in the empirical Bayes method, a gamma distribution is assumed. The paper considers the impact of different assumptions for this distribution and, indeed, whether any distributional form needs to be assumed. Using Markov Chain Monte Carlo methods, a variety of distributional forms are assumed for f₀(m) and applied to each of a number of real data sets, including that from a major study on the effectiveness of speed cameras. It is shown that, in some cases, the size of the estimated RTM effect can be quite sensitive to the choice of distribution.     

Are speed enforcement cameras more effective than other speed management measures? An evaluation of the relationship between speed and accident reductions

In this paper, models are developed which enable a prediction of how the impact of speed management schemes on accidents varies both with speed changes and with site and scheme characteristics. It was found that, the impact of schemes with vertical deflections is independent of the change in mean speed: an accident reduction of 44% is predicted by the model irrespective of the impact on speed. For cameras and other types of engineering schemes, a simple relationship between the change in mean speed and the consequent change in accidents is available. For the range of mean speeds typically found on 30 mph roads, the percentage accident reduction per 1 mph speed reduction is around 4% for cameras and 7-8% for schemes with horizontal features. While larger percentage accident reductions are achieved per 1 mph speed reduction on lower speed roads, larger speed reductions and larger overall percentage accident reductions are obtained on roads with higher before mean speeds. It is possible to predict both changes in speeds and accidents before treatment using the models derived from this study and these models confirm that schemes with vertical deflections are most effective in reducing both speeds and accidents.     

Sources of error in road safety scheme evaluation: a method to deal with outdated accident prediction models

This paper considers the errors that arise in using outdated accident prediction models in road safety scheme evaluation. Methods to correct for regression-to-mean (RTM) effects in scheme evaluation normally rely on the use of accident prediction models. However, because accident risk tends to decline over time, such models tend to become outdated and the estimated treatment effect is then exaggerated. A new correction procedure is described which can effectively eliminate such errors.     

Sources of error in road safety scheme evaluation: a quantified comparison of current methods

This paper considers the various factors that can have a confounding effect in the evaluation of road safety schemes and examines the extent to which current methods can effectively deal with these. A modification to current methods is proposed which allows the reduction in accidents attributable to risk and flow changes to be separately evaluated. Data are presented to demonstrate the relative magnitudes of the various sources of error. It is shown that a principal source of error is normally regression-to-mean (RTM) and a correction for this effect should always be applied. Changes in traffic flow can also result in substantial accident changes and it is important to establish whether flow changes have occurred and if they are attributable to the effect of the scheme.     

Are speed enforcement cameras more effective than other speed management measures? The impact of speed management schemes on 30 mph roads

This paper presents the results of an evaluation of the impact of various types of speed management schemes on both traffic speeds and accidents. The study controls for general trends in accidents, regression-to-mean effects and migration, separately estimating the accident changes attributable to the impact of the schemes on traffic speed and on traffic volume. It was found that, when judged in absolute terms, all types of speed management scheme have remarkably similar effects on accidents, with an average fall in personal injury accidents of about 1 accident/km/year. In terms of the percentage accident reduction, however, engineering schemes incorporating vertical deflections (such as speed humps or cushions) offer the largest benefits: at 44%, the average reduction in personal injury accidents attributable to such schemes, is twice that at sites where safety cameras were used to control speeds (22%) and they were the only type of scheme to have a significant impact on fatal and serious accidents. Other types of engineering scheme (with a fall of 29% in personal injury accidents) were on average less effective in reducing accidents than schemes with vertical features but more effective than cameras. All types of scheme were generally effective in reducing speeds, with the largest reductions tending to be obtained with vertical deflections and the smallest with other types of engineering schemes.