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 and severity i level with the following ordering conditions: j, . 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 While many parametric expressions may be used for this probability expression (the s. 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:
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: