Accident Analysis and Prevention 64 (2014) 41–51 Contents lists available at ScienceDirect Accident Analysis and Prevention journal h om epage: www.elsevier.com/locate/aap Bayesian methodology to estimate and update safety performance functions under limited data conditions: A sensitivity analysis  Shahram Heydari a,∗ , Luis F. Miranda-Moreno b,1 , Dominique Lord c,2 , Liping Fu a,3 a Department of Civil and Environmental Engineering, University of Waterloo, 200 University Avenue W., Waterloo, ON N2L 3G1, Canada b Department of Civil Engineering and Applied Mechanics, McGill University, 817 Sherbrooke St. W., Montreal, QC H3A 2K6, Canada c Zachary Department of Civil Engineering, Texas A&M University, College Station, TX, USA a r t i c l e i n f o Article history: Received 21 May 2013 Received in revised form 27 October 2013 Accepted 1 November 2013 Keywords: Full Bayes road safety analysis Prior distribution SPF parameter estimation and update Index of treatment effectiveness a b s t r a c t In road safety studies, decision makers must often cope with limited data conditions. In such circum- stances, the maximum likelihood estimation (MLE), which relies on asymptotic theory, is unreliable and prone to bias. Moreover, it has been reported in the literature that (a) Bayesian estimates might be sig- nificantly biased when using non-informative prior distributions under limited data conditions, and that (b) the calibration of limited data is plausible when existing evidence in the form of proper priors is introduced into analyses. Although the Highway Safety Manual (2010) (HSM) and other research stud- ies provide calibration and updating procedures, the data requirements can be very taxing. This paper presents a practical and sound Bayesian method to estimate and/or update safety performance func- tion (SPF) parameters combining the information available from limited data with the SPF parameters reported in the HSM. The proposed Bayesian updating approach has the advantage of requiring fewer observations to get reliable estimates. This paper documents this procedure. The adopted technique is validated by conducting a sensitivity analysis through an extensive simulation study with 15 different models, which include various prior combinations. This sensitivity analysis contributes to our under- standing of the comparative aspects of a large number of prior distributions. Furthermore, the proposed method contributes to unification of the Bayesian updating process for SPFs. The results demonstrate the accuracy of the developed methodology. Therefore, the suggested approach offers considerable promise as a methodological tool to estimate and/or update baseline SPFs and to evaluate the efficacy of road safety countermeasures under limited data conditions. © 2013 Elsevier Ltd. All rights reserved. 1. Introduction Safety performance functions (SPFs) often referred to as crash frequency models are an essential component of road safety studies. In practice, roadway or transportation agencies often need to estimate crash frequency models for limited data, that is, data with only a small number of observations and limited number of contributing factors (independent variables). In fact, limited data conditions frequently occur in road safety analyses, mainly due to the lack of funds required to involve a large sample of sites  A preliminary version of this paper has been presented at the International Road Safety and Simulation Conference in Rome, Italy (2013). ∗ Corresponding author. Tel.: +1 519 888 4567. E-mail addresses: shahram.heydari@uwaterloo.ca (S. Heydari), luis.miranda-moreno@mcgill.ca (L.F. Miranda-Moreno), d-lord@tamu.edu (D. Lord), lfu@uwaterloo.ca (L. Fu). 1 Tel.: +1 514 398 6589. 2 Tel.: +1 979 458 3949. 3 Tel.: +1 519 888 4567. in developing SPFs and/or conducting before–after observational studies (Lord and Bonneson, 2005). Hence, practitioners often need to calibrate statistical models under these restrictions to obtain baseline SPFs. The MLE that relies on asymptotic theory has been shown to be unreliable for limited data conditions (Lord, 2006; Daziano et al., 2013). However, the full Bayes (FB) paradigm can be employed as a viable alternative to the MLE. Some advantages of the FB context compared to its Frequentist counterpart are, first, that the available information (based on expert criteria, previous studies, etc.) related to the parameters of interest can be incorporated into the analysis by assigning prior distributions to these parameters. This is a vital advantage resulting in unbiased estimates for limited data (Lord and Miranda-Moreno, 2008; Miranda-Moreno et al., 2013; Heydari et al., 2013). By using suitable priors, thus, the sample size required to conduct a reliable road safety analysis may decrease. Second, Bayesian statistics have a natural characteristic of accommodating hierarchical models. Note that hierarchical models are capable of dealing with complex data structures and their use in the Bayesian methods is common and straightforward (Gelman et al., 2003). Third, solving complex 0001-4575/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.aap.2013.11.001