14. Nested logit with corrections for endogeneous sampling

The sample is said to be endogenous if the probability for an individual to be in the sample depends on the choice that has been made. In that case, the ESML estimator is not appropriate anymore, and corrections need to be made. See Bierlaire, Bolduc, McFadden (2008).

This is illustrated in this example.

Michel Bierlaire, EPFL Sat Jun 21 2025, 17:13:33

import biogeme.biogeme_logging as blog
import numpy as np
from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta
from biogeme.models import get_mev_for_nested, logmev_endogenous_sampling
from biogeme.nests import NestsForNestedLogit, OneNestForNestedLogit
from biogeme.results_processing import (
    EstimationResults,
    get_pandas_estimated_parameters,
)

See the data processing script: Data preparation for Swissmetro.

from swissmetro_data import (
    CAR_AV_SP,
    CAR_CO_SCALED,
    CAR_TT_SCALED,
    CHOICE,
    SM_AV,
    SM_COST_SCALED,
    SM_TT_SCALED,
    TRAIN_AV_SP,
    TRAIN_COST_SCALED,
    TRAIN_TT_SCALED,
    database,
)

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b14_nested_endogenous_sampling.py')

Example b14_nested_endogenous_sampling.py

Parameters to be estimated.

asc_car = Beta('asc_car', 0, None, None, 0)
asc_train = Beta('asc_train', 0, None, None, 0)
asc_sm = Beta('asc_sm', 0, None, None, 1)
b_time = Beta('b_time', 0, None, None, 0)
b_cost = Beta('b_cost', 0, None, None, 0)
nest_parameter = Beta('nest_parameter', 1, 1, 10, 0)

In this example, we assume that the three modes exist, and that the sampling protocol is choice-based. The probability that a respondent belongs to the sample is R_i.

R_TRAIN = 4.42e-2
R_SM = 3.36e-3
R_CAR = 7.5e-3

The correction terms are the log of these quantities

correction = {1: np.log(R_TRAIN), 2: np.log(R_SM), 3: np.log(R_CAR)}

Definition of the utility functions.

v_train = asc_train + b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED
v_swissmetro = asc_sm + b_time * SM_TT_SCALED + b_cost * SM_COST_SCALED
v_car = asc_car + b_time * CAR_TT_SCALED + b_cost * CAR_CO_SCALED

Associate utility functions with the numbering of alternatives.

v = {1: v_train, 2: v_swissmetro, 3: v_car}

Associate the availability conditions with the alternatives.

av = {1: TRAIN_AV_SP, 2: SM_AV, 3: CAR_AV_SP}

Definition of nests. Only the non-trivial nests must be defined. A trivial nest is a nest containing exactly one alternative. In this example, we create a nest for the existing modes, that is train (1) and car (3).

existing = OneNestForNestedLogit(
    nest_param=nest_parameter, list_of_alternatives=[1, 3], name='existing'
)

nests = NestsForNestedLogit(choice_set=list(v), tuple_of_nests=(existing,))

The following elements do not appear in any nest and are assumed each to be alone in a separate nest: {2}. If it is not the intention, check the assignment of alternatives to nests.

The choice model is a nested logit, with corrections for endogenous sampling We first obtain the expression of the Gi function for nested logit.

probability_generating_function = get_mev_for_nested(v, av, nests)

Then we calculate the MEV log probability, accounting for the correction.

log_probability = logmev_endogenous_sampling(
    v, probability_generating_function, av, correction, CHOICE
)

Create the Biogeme object.

the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b14_nested_endogenous_sampling'

## %%
# Estimate the parameters.
try:
    results = EstimationResults.from_yaml_file(
        filename=f'saved_results/{the_biogeme.model_name}.yaml'
    )
except FileNotFoundError:
    results = the_biogeme.estimate()

Biogeme parameters read from biogeme.toml.

print(results.short_summary())

Results for model b14_nested_endogenous_sampling
Nbr of parameters:              5
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5202.916
Akaike Information Criterion:   10415.83
Bayesian Information Criterion: 10449.93

pandas_results = get_pandas_estimated_parameters(estimation_results=results)
display(pandas_results)

{'Estimated parameters':              Name     Value  Robust std err.  Robust t-stat.  Robust p-value
0       asc_train -2.768565         0.080533      -34.377818             0.0
1          b_time -0.974527         0.110772       -8.797606             0.0
2          b_cost -0.999261         0.064137      -15.580154             0.0
3  nest_parameter  1.631020         0.062662       26.028924             0.0
4         asc_car -1.127431         0.060629      -18.595475             0.0}