Ordinal logit model

Example of an ordinal logit model. This is just to illustrate the syntax, as the data are not ordered. But the example assume, for the sake of it, that the alternatives are ordered as 1->2->3

Michel Bierlaire, EPFL Thu Jun 26 2025, 15:52:21

import biogeme.biogeme_logging as blog
from IPython.core.display_functions import display
from biogeme.biogeme import BIOGEME
from biogeme.expressions import Beta, Elem, log
from biogeme.models import ordered_logit
from biogeme.results_processing import get_pandas_estimated_parameters

See the data processing script: Data preparation for Swissmetro.

from swissmetro_data import CHOICE, TRAIN_COST_SCALED, TRAIN_TT_SCALED, database

logger = blog.get_screen_logger(level=blog.INFO)
logger.info('Example b18ordinal_logit.py')

Example b18ordinal_logit.py

Parameters to be estimated

b_time = Beta('b_time', 0, None, None, 0)
b_cost = Beta('b_cost', 0, None, None, 0)

Threshold parameters for the ordered logit.

\(\tau_1 \leq 0\).

tau1 = Beta('tau1', -1, None, 0, 0)

\(\delta_2 \geq 0\).

delta2 = Beta('delta2', 2, 0, None, 0)

\(\tau_2 = \tau_1 + \delta_2\)

tau2 = tau1 + delta2

Utility.

utility = b_time * TRAIN_TT_SCALED + b_cost * TRAIN_COST_SCALED

Associate each discrete indicator with an interval.

1. \(-\infty \to \tau_1\),

2. \(\tau_1 \to \tau_2\),

3. \(\tau_2 \to +\infty\).

the_probability = ordered_logit(
    continuous_value=utility,
    list_of_discrete_values=[1, 2, 3],
    reference_threshold_parameter=tau1,
    scale_parameter=1.0,
)

Extract from the dict the formula associated with the observed choice.

the_chosen_proba = Elem(the_probability, CHOICE)

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

log_probability = log(the_chosen_proba)

Create the Biogeme object.

the_biogeme = BIOGEME(database, log_probability)
the_biogeme.model_name = 'b18ordinal_logit'

Biogeme parameters read from biogeme.toml.

Estimate the parameters

results = the_biogeme.estimate()

*** Initial values of the parameters are obtained from the file __b18ordinal_logit.iter
Parameter values restored from __b18ordinal_logit.iter
Starting values for the algorithm: {'b_time': -0.022080790172257413, 'b_cost': 1.2628932870283, 'tau1': -1.0300981224269388, 'tau1_diff_2': 3.233236512522347}
As the model is not too complex, we activate the calculation of second derivatives. To change this behavior, modify the algorithm to "simple_bounds" in the TOML file.
Optimization algorithm: hybrid Newton/BFGS with simple bounds [simple_bounds]
** Optimization: Newton with trust region for simple bounds
Optimization algorithm has converged.
Relative gradient: 2.9148755617971546e-06
Cause of termination: Relative gradient = 2.9e-06 <= 6.1e-06
Number of function evaluations: 1
Number of gradient evaluations: 1
Number of hessian evaluations: 0
Algorithm: Newton with trust region for simple bound constraints
Number of iterations: 0
Optimization time: 0:00:00.130996
Calculate second derivatives and BHHH
File b18ordinal_logit~00.html has been generated.
File b18ordinal_logit~00.yaml has been generated.

print(results.short_summary())

Results for model b18ordinal_logit
Nbr of parameters:              4
Sample size:                    6768
Excluded data:                  3960
Final log likelihood:           -5789.309
Akaike Information Criterion:   11586.62
Bayesian Information Criterion: 11613.9

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 -0.022081         0.040060       -0.551195          0.5815
1       b_cost  1.262893         0.058542       21.572527          0.0000
2         tau1 -1.030098         0.067967      -15.155781          0.0000
3  tau1_diff_2  3.233237         0.044509       72.642216          0.0000