### Example 18.5 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 that is displayed in Output 18.5.1. 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 18.5.1: 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 following statements add the upalt variable, which describes the choice at the upper level of the nested tree to the data set.

data small;
set small;
upalt=1;
if alt=9 then upalt=2;
if alt>9 then upalt=3;
run;


The following statements show and alternative specification, which uses the CHOICE= option with two nested levels that are represented by upalt and alt:

proc mdc data=upalt maxit=200;
model decision = r15 r10 ttime ttime_cp sde sde_cp
sdl sdlx d2l /
type=nlogit
choice=(upalt,alt);
id id;
utility u(1, ) = r15 r10 ttime ttime_cp sde sde_cp
sdl sdlx d2l;
run;


The estimation summary, discrete response profile, and the FIML estimates are displayed in Output 18.5.2 through Output 18.5.4.

Output 18.5.2: Nested Logit Estimation Summary

The MDC Procedure

Nested Logit Estimates

Model Fit Summary
Dependent Variable decision
Number of Observations 527
Number of Cases 6324
Log Likelihood -990.81912
Log Likelihood Null (LogL(0)) -1310
Number of Iterations 18
Optimization Method Newton-Raphson
AIC 2006
Schwarz Criterion 2057

Output 18.5.3: Discrete Choice Characteristics

Discrete Response Profile
Index alt Frequency Percent
0 1 6 1.14
1 2 10 1.90
2 3 61 11.57
3 4 15 2.85
4 5 27 5.12
5 6 80 15.18
6 7 55 10.44
7 8 64 12.14
8 9 187 35.48
9 10 13 2.47
10 11 8 1.52
11 12 1 0.19

Output 18.5.4: Nested Logit Estimates

The MDC Procedure

Nested Logit Estimates

Parameter Estimates
Parameter DF Estimate Standard
Error
t Value Approx
Pr > |t|
r15_L1 1 1.1034 0.1221 9.04 <.0001
r10_L1 1 0.3931 0.1194 3.29 0.0010
ttime_L1 1 -0.0465 0.0235 -1.98 0.0474
ttime_cp_L1 1 -0.0498 0.0305 -1.63 0.1028
sde_L1 1 -0.6618 0.0833 -7.95 <.0001
sde_cp_L1 1 0.0519 0.1278 0.41 0.6850
sdl_L1 1 -2.1006 0.5062 -4.15 <.0001
sdlx_L1 1 -3.5240 1.5346 -2.30 0.0217
d2l_L1 1 -1.0941 0.3273 -3.34 0.0008
INC_L2G1C1 1 0.6762 0.2754 2.46 0.0141
INC_L2G1C2 1 1.0906 0.3090 3.53 0.0004
INC_L2G1C3 1 0.7622 0.1649 4.62 <.0001

Now policy makers are particularly interested in predicting shares of each alternative to be chosen by population. One application of such predictions are market shares. Going even further, it is extremely useful to predict choice probabilities out of sample; that is, under alternative policies.

Suppose that in this particular transportation example you are interested in projecting the effect of a new program that indirectly shifts individual preferences with respect to late arrival to work. This means that you manage to decrease the coefficient for the “late dummy” D2L, which is a penalty for violating some margin of arriving on time. Suppose that you alter it from an estimated to almost twice that level, .

But first, in order to have a benchmark share, you predict probabilities to choose each particular option and output them to the new data set with the following additional statement:

   /*-- Create new data set with predicted probabilities --*/
output out=predicted1 p=probs;


Having these in sample predictions, you sort the data by alternative and aggregate across each of them as shown in the following statements:

/*-- Sort the data by alternative --*/
proc sort data=predicted1;
by alt;
run;

/*-- Calculate average probabilities of each alternative --*/
proc means data=predicted1 nonobs mean;
var probs;
class alt;
run;


Output 18.5.5 shows the summary table that is produced by the preceding statements.

Output 18.5.5: Average Probabilities of Choosing Each Particular Alternative

The MEANS Procedure

Analysis Variable : probs
alt Mean
1 0.0178197
2 0.0161712
3 0.0972584
4 0.0294659
5 0.0594076
6 0.1653871
7 0.1118181
8 0.1043445
9 0.3564940
10 0.0272324
11 0.0096334
12 0.0049677

Now you change the preference parameter for variable D2L. In order to fix all the parameters, you use the MAXIT=0 option to prevent optimization and the START= option in MODEL statement to specify initial parameters.

/*-- Two-level Nested Logit --*/
proc mdc data=small maxit=0 outest=a;
model decision = r15 r10 ttime ttime_cp sde sde_cp
sdl sdlx d2l /
type=nlogit
choice=(alt)
start=( 1.1034 0.3931 -0.0465 -0.0498
-0.6618 0.0519 -2.1006 -3.5240
-2.0941 0.6762  1.0906  0.7622);
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);
output out=predicted2 p=probs;
run;


You apply the same SORT and MEANS procedures as applied earlier to obtain the following summary table in Output 18.5.6.

Output 18.5.6: Average Probabilities of Choosing Each Particular Aalternative after Changing the Preference Parameter

The MEANS Procedure

Analysis Variable : probs
alt Mean
1 0.0207766
2 0.0188966
3 0.1138816
4 0.0345654
5 0.0697830
6 0.1944572
7 0.1315588
8 0.1228049
9 0.2560674
10 0.0236178
11 0.0090781
12 0.0045128

Comparing the two tables shown in Output 18.5.5 and Output 18.5.6, you clearly see the effect of increased dislike of late arrival. People shifted their choices towards earlier times (alternatives 1–8) from the on-time option (alternative 9).

Brownstone and Small (1989) also estimate the two-level nested logit model with equal scale parameter constraints, . Replication of their model estimation is shown in the following statements:

/*-- Nested Logit with Equal Dissimilarity Parameters --*/
proc mdc data=small maxit=200 outest=a;
model decision = r15 r10 ttime ttime_cp sde sde_cp
sdl sdlx d2l /
samescale
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 parameter estimates and standard errors are almost identical to those in Brownstone and Small (1989, p. 69). Output 18.5.7 and Output 18.5.8 display the results.

Output 18.5.7: Nested Logit Estimation Summary with Equal Dissimilarity Parameters

The MDC Procedure

Nested Logit Estimates

Model Fit Summary
Dependent Variable decision
Number of Observations 527
Number of Cases 6324
Log Likelihood -994.39402
Log Likelihood Null (LogL(0)) -1310
Number of Iterations 16
Optimization Method Newton-Raphson
AIC 2009
Schwarz Criterion 2051

Output 18.5.8: Nested Logit Estimates with Equal Dissimilarity Parameters

The MDC Procedure

Nested Logit Estimates

Parameter Estimates
Parameter DF Estimate Standard
Error
t Value Approx
Pr > |t|
r15_L1 1 1.1345 0.1092 10.39 <.0001
r10_L1 1 0.4194 0.1081 3.88 0.0001
ttime_L1 1 -0.1626 0.0609 -2.67 0.0076
ttime_cp_L1 1 0.1285 0.0853 1.51 0.1319
sde_L1 1 -0.7548 0.0669 -11.28 <.0001
sde_cp_L1 1 0.2292 0.0981 2.34 0.0195
sdl_L1 1 -2.0719 0.4860 -4.26 <.0001
sdlx_L1 1 -2.8216 1.2560 -2.25 0.0247
d2l_L1 1 -1.3164 0.3474 -3.79 0.0002
INC_L2G1 1 0.8059 0.1705 4.73 <.0001

However, the test statistic for rejects the null hypothesis at the significance level since . The -value is computed in the following statements and is equal to :

data _null_;
/*-- test for H0: tau1 = tau2 = tau3 --*/
/*   ln L(max) = -990.8191             */
/*   ln L(0)   = -994.3940             */
stat = -2 * ( -994.3940 + 990.8191 );
df = 2;
p_value = 1 - probchi(stat, df);
put stat= p_value=;
run;