The third stage of the research methodology is data analysis. During this stage four steps were conducted, including developing the base model, investigating the best distribution, analysing covariate effects and interpreting covariate effects. This section explains these steps in more detail.
Data analysis commenced by developing a base model for each interval time of the total accident duration, including reporting time, response time, and clearance time. Separate models were used because this research aims to determine the factors that influenced each interval time of accident duration. The analytical approach proposed here is important to assist accident responders to obtain more insight into what affects each interval time of the total accident duration, which may assist in improving various elements of the TIM process. Although it is possible to look at the whole duration as one, this approach would lose information in both estimation and interpretation of results (Nam and Mannering, 2000a).
65 A fully parametric approach was especially selected in this study to investigate accident duration. The merit of using a fully parametric approach, as opposed to semi- parametric, is that the latter does not produce a parameter that tells the shape of the baseline hazard, making it difficult to determine duration effects (Nam and Mannering, 2000a).
Before developing HBDMs, it is necessary to select explanatory variables for each model (reporting time model, response time model and clearance time model). To identify the most relevant variables, three steps, as shown below, were conducted (Collett, 2003):
1. The first step was to analyse the models considering Weibull, Log-normal and Log-logistic distribution without any explanatory variables in order to check the value of the log-likelihood before convergence. This is referred to as a null or base model.
2. The second step aims to identify which explanatory variables significantly reduce −2 𝑙𝑜𝑔𝐿� statistic. This is done by considering one variable at a time. In this step, all variables that are significant at the level of 85% were selected.
3. The third step aims to check whether any of the excluded variables in step 2 are significant in the model. Since any of the excluded explanatory variables from the initial model could be significant when put back into the model, the models of the significant variables were fitted with one of the excluded variables at a time. In this step, all variables significant at the level of 90% were selected.
As a result of this, the structure of model development was as follows:
Model 1: Reporting Time
The dependent variable in this model is the time until an accident is reported to the collision investigator. In other words, the model relates to the time until the reporting of an accident, given that the accident has not been reported up to time t. The hazard function is defined as the rate at which the accident is being reported at time t, given that no accident has been reported until time t.
Model 2: Response Time
The dependent variable in this model is the time until an accident is responded to by a collision investigator. In other words, the model relates to the time until the response to
66 the accident, given that the accident has not been responded to up to time t. The hazard function is defined as the rate at which the accident is responded to at time t, given that no accident has been responded to until time t.
Model 3: Clearance Time
The dependent variable in this model is the time until an accident is cleared. So, the model relates to the time until the clearance of the accident, given that the accident has not been cleared up to time t. The hazard function is defined as the rate at which the accident is being cleared at time t, given that no accident has been cleared until time t.
Once the models have been estimated, the plots that compare observed and predicted durations for the incidents in the dataset will be presented for each interval time. These comparisons will be conducted for the three distributions used in this study. Following that, a goodness-of-fit test should be carried out to select the best fit distribution prior to the interpretation of results. In fact, several tests can be conducted, such as a Likelihood-ratio test, Wald test and Akaike Information Criterion (AIC). The first two tests are appropriate when the models are nested, such as Weibull against Exponential, whereas the last test can be selected when the models are not nested. Because the models used in this study are not nested, AIC was used. Akaike (1974) proposed ‘penalizing each model’s log-likelihood to reflect the number of parameters being estimated and then comparing them’. This test can be written as the following:
AIC=-2InL+2(k+c) (32) Where In L refers to the model’s log-likelihood at convergence, k denotes the number of covariates in the model, and c is the number of distribution parameters. The criteria is to select the distribution that has the lowest value of AIC (Akaike, 1974; Cleves et al., 2004). It should also be noted that this test should be conducted independently for each sub-model.
Following the fitting of the best-fit distribution, the outcomes of the selected model are interpreted based on the sign of the estimated coefficient and the percentage change in duration. In the AFT model, the sign of the coefficient specifies how the variable affects the interval time duration. For example, a positive coefficient in the clearance time model means that the variable increases the clearance time duration (Gloder, 2008).
67 On the other hand, the percentage change in the interval time by each of the explanatory variables can also be calculated. This could be done by taking the exponent of the estimated coefficient of the significant variable (Chung, 2009; Nam and Mannering, 2000). Generally, when the exponent of the estimated coefficient is greater than 1.0, the relevant explanatory variable adds more time to the respective accident interval time (reporting, response, and clearance) and vice-versa. Finally, the estimated results were interpreted in terms of the covariate effects on the interval time and their relation to the current practices of the TIM process in Abu Dhabi.