error term in the count model, the probability of the treatment and of the outcome can be written as the product of the individual probabilities of the treatment and of the outcome. The unconditional probability of the treatment and outcome may be obtained by integrating out the error term of the count model (see Greene, 2009 for a discussion). Similarly, in the case of an injury severity outcomes, and assuming a binary treatment variable, one needs to have a propensity equation for the treatment (this propensity translates to the observed treatment indicator, in the usual binary model fashion) and an appropriate specification for injury severity with the treatment as an indicator variable (in the form of either a single injury-severity propensity equation that is related to the observed injury severity levels through thresholds in the ordered-response or generalized ordered-response formulation, or in the form of multiple propensity equations, one for each injury severity category, in the unordered-response formulation). The error terms in the treatment and outcome propensities are then specified to have a dependency structure. After accommodating this dependency structure, the structural parameter on the treatment in the outcome model may be viewed as the and “cleansed” “true” causal effect of the treatment. In this formulation, the joint probability of the treatment and the outcome takes a bivariate truncated distribution (if an ordered-response or generalized model is used for injury severity) or a multivariate truncated distribution (if an unordered model is used for injury severity). A methodological frontier issue in safety analysis is then first to accommodate endogeneity considerations appropriately. For the count model outcome, the recasting as a generalized ordered-response model may be particularly effective in capturing endogeneity issues, and should open up a suite of possibilities for specifying and testing endogeneity effects. Further, there is substantial room for exploring a variety of copula structures for the error
error term in the count model, the probability of the treatment and of the outcome can be written as the product of the individual probabilities of the treatment and of the outcome. The unconditional probability of the treatment and outcome may be obtained by integrating out the error term of the count model (see Greene, 2009 for a discussion). Similarly, in the case of an injury severity outcomes, and assuming a binary treatment variable, one needs to have a propensity equation for the treatment (this propensity translates to the observed treatment indicator, in the usual binary model fashion) and an appropriate specification for injury severity with the treatment as an indicator variable (in the form of either a single injury-severity propensity equation that is related to the observed injury severity levels through thresholds in the ordered-response or generalized ordered-response formulation, or in the form of multiple propensity equations, one for each injury severity category, in the unordered-response formulation). The error terms in the treatment and outcome propensities are then specified to have a dependency structure. After accommodating this dependency structure, the structural parameter on the treatment in the outcome model may be viewed as the “cleansed” and “true” causal effect of the treatment. In this formulation, the joint probability of the treatment and the outcome takes a bivariate truncated distribution (if an ordered-response or generalized model is used for injury severity) or a multivariate truncated distribution (if an unordered model is used for injury severity). A methodological frontier issue in safety analysis is then first to accommodate endogeneity considerations appropriately. For the count model outcome, the recasting as a generalized ordered-response model may be particularly effective in capturing endogeneity issues, and should open up a suite of possibilities for specifying and testing endogeneity effects. Further, there is substantial room for exploring a variety of copula structures for the error
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