The MDC Procedure |
Parameter Heterogeneity: Mixed Logit |
One way of modeling unobserved heterogeneity across individuals in their sensitivity to observed exogenous variables is to use the mixed logit model with a random parameters or random coefficients specification. The probability of choosing alternative is written as
where is a vector of coefficients that varies across individuals and is a vector of exogenous attributes.
For example, you can specify the distribution of the parameter to be the normal distribution.
The mixed logit model uses a Monte Carlo simulation method to estimate the probabilities of choice. There are two simulation methods available. If the RANDNUM=PSEUDO option is specified in the MODEL statement, pseudo-random numbers are generated; if the RANDNUM=HALTON option is specified, Halton quasi-random sequences are used. The default value is RANDNUM=HALTON.
You can estimate the model with normally distributed random coefficients of ttime with the following SAS statements:
/*-- mixed logit estimation --*/ proc mdc data=newdata type=mixedlogit; model decision = ttime / nchoice=3 mixed=(normalparm=ttime); id pid; run;
Let and be mean and scale parameters, respectively, for the random coefficient, . The relevant utility function is
where ( and are fixed mean and scale parameters, respectively). The stochastic component, , is assumed to be standard normal since the NORMALPARM= option is given. Alternatively, the UNIFORMPARM= or LOGNORMALPARM= option can be specified. The LOGNORMALPARM= option is useful when nonnegative parameters are being estimated. The NORMALPARM=, UNIFORMPARM=, and LOGNORMALPARM= variables must be included in the right-hand side of the MODEL statement. See the section Mixed Logit Model for more details. To estimate a mixed logit model by using the transportation mode choice data, the MDC procedure requires the MIXED= option for random components. Results of the mixed logit estimation are displayed in Figure 17.21.
Parameter Estimates | |||||
---|---|---|---|---|---|
Parameter | DF | Estimate | Standard Error |
t Value | Approx Pr > |t| |
ttime_M | 1 | -0.5342 | 0.2184 | -2.45 | 0.0144 |
ttime_S | 1 | 0.2843 | 0.1911 | 1.49 | 0.1368 |
Note that the parameter ttime_M corresponds to the constant mean parameter and the parameter ttime_S corresponds to the constant scale parameter of the random coefficient .
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