Example 17.5 Choice of Time for Work Trips: Nested Logit Analysis

This example uses sample data of 527 automobile commuters in the San Francisco Bay Area to demonstrate the use of nested logit model.

Brownstone and Small (1989) analyzed a two-level nested logit model displayed in Output 17.5.1. The probability of choosing at level 2 is written as

where is an inclusive value and is computed as

The probability of choosing an alternative is denoted as

The full information maximum likelihood (FIML) method maximizes the following log-likelihood function:

where if a decision maker chooses , and 0 otherwise.

Output 17.5.1 Decision Tree for Two-Level Nested Logit

Sample data of 527 automobile commuters in the San Francisco Bay Area have been analyzed by Small (1982) and Brownstone and Small (1989). The regular time of arrival is recorded as between 42.5 minutes early and 17.5 minutes late, and indexed by 12 alternatives, using five-minute interval groups. Refer to Small (1982) for more details on these data.

The following statements estimate the two-level nested logit model:

   /*-- Two-level Nested Logit --*/
   proc mdc data=small maxit=200 outest=a;
      model decision = r15 r10 ttime ttime_cp sde sde_cp
                       sdl sdlx d2l /
               type=nlogit
               choice=(alt);
      id id;
      utility u(1, ) = r15 r10 ttime ttime_cp sde sde_cp
                       sdl sdlx d2l;
      nest level(1) = (1 2 3 4 5 6 7 8 @ 1, 9 @ 2, 10 11 12 @ 3),
           level(2) = (1 2 3 @ 1);
   run;

The estimation summary, discrete response profile, and the FIML estimates are displayed in Output 17.5.2 through Output 17.5.4.