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

model (McFadden, 1981). Let ( ) be the probability of outcome category i for observation

n. then

( )
( ) ∑ ( ) (3.5)

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

distribution is introduced (Train, 2003) and the resulting outcome probabilities are given by:

∫ [ ] ( | ) (3.6)

∑ [ ]

( | ) 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 3.6

shows 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 probability, 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. Mixed
logit models are usually estimated using the simulation of maximum likelihood with Halton

draws (Train 1999; Bhat 2003).

However, preliminary analyses in the present study using the random parameters binary
logistic model found no statistically-significant estimate of the variance for any of the

coefficients, indicating that the fixed coefficient binary logistic model is appropriate.

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