MDC Procedure

A more general model can be specified using the nested logit model.

Consider, for example, the following random utility function:

Suppose the set of all alternatives indexed by j is partitioned into K nests, . The nested logit model is obtained by assuming that the error term in the utility function has the GEV cumulative distribution function

where is a measure of a degree of independence among the alternatives in nest k. When for all k, the model reduces to the standard logit model.

Since the public transportation modes, 1 and 2, tend to be correlated, these two choices can be grouped together. The decision tree displayed in Figure 8 is constructed.

The two-level decision tree is specified in the NEST statement. The NCHOICE= option is not allowed for nested logit estimation. Instead, the CHOICE= option needs to be specified, as in the following statements:

/*-- nested logit estimation --*/
proc mdc data=newdata;
   model decision = ttime /
            type=nlogit
            choice=(mode 1 2 3)
            covest=hess;
   id pid;
   utility u(1,) = ttime;
   nest level(1) = (1 2 @ 1, 3 @ 2),
        level(2) = (1 2 @ 1);
run;

In Figure 9, estimates of the inclusive value parameters, INC_L2G1C1 and INC_L2G1C2, are indicative of a nested model structure. For more information about inclusive values, see the sections Nested Logit and Decision Tree and Nested Logit.

The nested logit model is estimated with the restriction INC_L2G1C1 = INC_L2G1C2 by specifying the SAMESCALE option, as in the following statements:

/*-- nlogit with samescale option --*/
proc mdc data=newdata;
   model decision = ttime /
            type=nlogit
            choice=(mode 1 2 3)
            samescale
            covest=hess;
   id pid;
   utility u(1,) = ttime;
   nest level(1) = (1 2 @ 1, 3 @ 2),
        level(2) = (1 2 @ 1);
run;

The estimation result is displayed in Figure 10.

The nested logit model is equivalent to the conditional logit model if INC_L2G1C1 = INC_L2G1C2 = 1. You can verify this relationship by estimating a constrained nested logit model as shown in the following statements. (For more information about imposing linear restrictions on parameter estimates, see the section RESTRICT Statement.)

/*-- constrained nested logit estimation --*/
proc mdc data=newdata;
   model decision = ttime /
            type=nlogit
            choice=(mode 1 2 3)
            covest=hess;
   id pid;
   utility u(1,) = ttime;
   nest level(1) = (1 2 @ 1, 3 @ 2),
        level(2) = (1 2 @ 1);
   restrict INC_L2G1C1 = 1, INC_L2G1C2 =1;
run;

The parameter estimates and the active linear constraints for the constrained nested logit model are displayed in Figure 11.