Binary probit model

Example of a binary probit model. Two alternatives: Train and Car.

Michel Bierlaire, EPFL Sat Jun 28 2025, 12:43:40

from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, Elem, NormalCdf, log
from biogeme.results_processing import get_pandas_estimated_parameters

See the data processing script: Data preparation for Swissmetro (binary choice).

from swissmetro_binary import (
    CAR_CO_SCALED,
    CAR_TT_SCALED,
    CHOICE,
    TRAIN_COST_SCALED,
    TRAIN_TT_SCALED,
    database,
)

Parameters to be estimated.

asc_car = Beta('asc_car', 0, None, None, 0)
b_time_car = Beta('b_time_car', 0, None, None, 0)
b_time_train = Beta('b_time_train', 0, None, None, 0)
b_cost_car = Beta('b_cost_car', 0, None, None, 0)
b_cost_train = Beta('b_cost_train', 0, None, None, 0)

Definition of the utility functions. We estimate a binary probit model. There are only two alternatives.

v_train = b_time_train * TRAIN_TT_SCALED + b_cost_train * TRAIN_COST_SCALED
v_car = asc_car + b_time_car * CAR_TT_SCALED + b_cost_car * CAR_CO_SCALED

Associate choice probability with the numbering of alternatives.

log_probability_dict = {
    1: log(NormalCdf(v_train - v_car)),
    3: log(NormalCdf(v_car - v_train)),
}

Definition of the model. This is the contribution of each observation to the log likelihood function.

log_probability = Elem(log_probability_dict, CHOICE)

Create the Biogeme object.

the_biogeme = BIOGEME(database, log_probability, save_iterations=False)
the_biogeme.model_name = 'b23probit'

Estimate the parameters

results = the_biogeme.estimate()

print(results.short_summary())

Results for model b23probit
Nbr of parameters:              5
Sample size:                    2232
Excluded data:                  8496
Final log likelihood:           -906.9459
Akaike Information Criterion:   1823.892
Bayesian Information Criterion: 1852.445

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

           Name     Value  Robust std err.  Robust t-stat.  Robust p-value
0  b_time_train -0.649749         0.095329       -6.815839    9.371393e-12
1  b_cost_train -0.980475         0.147040       -6.668105    2.591283e-11
2       asc_car -0.353276         0.107955       -3.272442    1.066228e-03
3    b_time_car -0.184152         0.075674       -2.433501    1.495360e-02
4    b_cost_car -0.530672         0.136053       -3.900477    9.600346e-05