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If are assumed to be extreme value distributions, we have the standard multinomial logit

model (McFadden, 1981). Let ( ) be the probability of injury severity category i for

observation n. Then

( )
( ) ∑ ( ) (4.4)

In the random parameter model, to let parameter ( ) vary across observations, a mixing

distribution is introduced in this model (Train, 2003) and the resulting injury severity

probalities are given by:

∫ [ ] ( | ) (4.5)

∑ [ ]

( | ) is the density function of β and refers to a vector of parameters of the

density function (mean and variance) and other terms are as previously defined.

Equation 4.5 showed the mixed logit model. In the mixed logit model estimation, β can now

account for observation-specific variations of the effect of on injury severity probabilities,

with the density functions ( | ) used to determine β.

The random parameter model uses a weighted average for different values of β across

observations where some elements of the parameter vector β may be fixed and some are
randomly distributed. If any parameters are found to be random, then the mixed logit weight

is determined by the density function. For the functional form of the density function,
numerous distributions have been considered, such as normal, uniform and lognormal. In this

study, the normal distributions were used as a density function in the mixed logit model.
Mixed logit models are usually estimated using the simulation of maximum likelihood with

Halton draws (Train 1999; Bhat 2003). Nevertheless, preliminary analysis in this study using

the random parameter multinomial logit model found no statistically significant estimate of
the variance for any of the coefficients of the explanatory variables considered.

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