where, is a row vector of exogenous variables is a corresponding column vector of unknown parameters specific to segment is a segment-specific idiosyncratic random disturbance term assumed to be identically and independently standard logistic ( represents the segment-specific upper threshold associated with driver i and severity level j, with the following ordering conditions: . To maintain the ordering conditions and allow the thresholds to vary across drivers within each segment, Eluru et al., (2008) propose the following non-linear parameterization of the thresholds as a function of exogenous variables: where is a segment-specific and injury level-specific row vector of parameters to be estimated and is a corresponding column vector of segment-specific exogenous variables ( includes a constant as its first element, with the corresponding coefficient being ; for identification, we need to be a row vector of zero values, where is a sub-vector of the vector minus the first element). The traditional ordered logit (OL) model assumes that the thresholds remain fixed across drivers ( for each segment; that is, it assumes that has all zero elements for all values and all values. Given the above set-up, the probability that driver suffers an injury severity outcome , conditional on driver belonging to segment , may be written as: where represents the standard logistic cumulative distribution function. Of course, the analyst does not observe the segment to which driver belongs. So, the analyst specifies this segment assignment to be a function of a column vector of observed crash factors . To also acknowledge the presence of unobserved factors that may influence this assignment, the analyst develops an expression for the probability of driver belonging to segment s. While many parametric expressions may be used for this probability expression (the only requirement is that the probabilities sum to one across the segments for each driver ), the most commonly used form corresponds to the multinomial logit structure (see Bhat, 1997; Greene and Hensher, 2003; Eluru et al., 2012): where is a row vector of parameters to be estimated. Then, the unconditional probability of driver leading up to injury severity level can be written as:
FIGURE 1 Schematic Diagram of Initial Point of Impact Relative to the Driver’s Seat Position TABLE 1 Crash Database Sample Statistics Variables Sample Share Frequency Percentage Crash characteristics Collision object Small Object 120 2.338 Large object 710 13.835 Collision with animals 207 4.034 Collision with Moving vehicle 4095 79.793 Trajectory of vehicle’s motions Going Straight 2689 52.397 Other movement 2443 47.603 Manners of collision
Trajectory of vehicle’s motions (Base: Other movement) Going Straight − − -0.480 -3.423 Manner of collision (Base: Rear-ender and short side-angular) Rear-ended − − -1.574 -7.172 Near-angular − − -0.826 -4.427 Head-on − − -1.055 -4.402 Far-side − − 0.530 2.197 Injury Severity Components Variables Latent Propensity Threshold Latent Propensity Threshold Estimates t-stat Estimate t-stat Estimates t-stat Estimate t-stat Constant 3.859 7.962 1.468 13.571 -1.746 -6.237 0.307 2.650 Driver characteristics Driver age (Base: Age 25 to 64) Age less than 25 -0.734 -2.478 -0.137 -1.643 − − − − Age above 65+ 0.885 3.779 − − 0.432 2.629 -0.498 -2.533 Driver gender (Base: Male) Female − − − − 1.189 8.003 0.316 2.740 Restraint system use (Base: seat belt used) Seat belt not used − − -0.197 -2.161 0.717 2.472 − − Vehicle characteristics Vehicle Type (Base: Sedan)
TABLE 5 Measures of Fit in Validation Sample Disaggregate Level (100 Validation Samples) Summary statistic LSOL LSGOL Predictions C.I. Predictions C.I. Number of observations 2547.540 − 2547.540 − Number of parameters 37.000 − 41.000 − Log-likelihood at zero -2798.759 -2807.853 | -2789.665 -2798.759 -2807.853 | -2789.665 Log-likelihood at sample shares -2696.838 -2706.066 | -2687.609 -2696.838 -2706.066 | -2687.609 Predictive Log-likelihood -2438.553 -2447.559 | -2429.546 -2433.249 -2442.324 | -2424.174 Predictive adjusted likelihood ratio index 0.082 0.081 | 0.083 0.083 0.082 | 0.084 Average probability of correct prediction 0.531 0.529 | 0.533 0.531 0.529 | 0.532 Aggregate Level (100 Validation Samples) Injury categories/ Measures of fit Actual shares LSOL LSGOL Predictions C.I. Predictions C.I. No injury 42.522799 42.309 42.247 | 42.371 42.364 42.300 | 42.428 Non-incapacitating injury 36.518347 36.595 36.559 | 36.629 36.492 36.452 | 36.531 Incapacitating/Fatal injury 20.958854 21.096 21.049 | 21.142 21.144 21.099 | 21.189 RMSE − 0.821 0.753 | 0.888 0.817 0.753 | 0.882 MAPE − 2.352 2.351 | 2.354 2.343 2.341 | 2.345