Ordered-probit hybrid mode choice model: simultaneous maximum likelihood estimation¶

This example estimates a hybrid mode choice model with one latent variable and ordered-probit measurement equations. The observed Likert indicators are treated as ordinal responses. Their thresholds, the latent-variable model, and the choice-model parameters are estimated jointly.

The model is the ordered-probit counterpart of plot_h05_mode_lv_ordlogit_simult. The hybrid structure, the latent variable, the choice utilities, and the simultaneous maximum-likelihood estimation strategy are the same. Only the ordinal measurement distribution changes: the indicator probabilities are based on normal cumulative distributions instead of logistic cumulative distributions.

The latent-variable structure is imported from a separate semantic specification file. This script adds the ordered-probit measurement configuration, the normalization constraints required for identification, the mode-choice utilities, and the simultaneous maximum-likelihood estimation setup.

The script performs the following steps:

load the latent-variable and indicator specifications,
define an ordered-probit measurement configuration for all indicators,
define the normalization constraints used for identification,
resolve the semantic specification into an estimable model,
build the Biogeme expressions from the resolved model,
build the choice utilities, including the latent-variable term,
combine the ordered-probit measurement and choice conditional likelihoods,
integrate the combined conditional likelihood over the latent variable,
estimate the hybrid model, or reload previously saved estimation results,
display the estimated parameters as grouped pandas and LaTeX tables.

Michel Bierlaire Sat Jun 13 2026, 15:13:40

from __future__ import annotations

from choice_latent_variables import generate_availability, generate_utility_functions
from likert_spec import likert_indicators, likert_types
from number_of_draws import NUMBER_OF_DRAWS
from one_latent_variable_spec import latent_variables
from optima import Choice, read_data

import biogeme.biogeme_logging as blog
from biogeme.biogeme import BIOGEME
from biogeme.expressions import MonteCarlo, log
from biogeme.latent_variables import (
    BuildContext,
    EstimationMode,
    Fixing,
    IndicatorMeasurementSpec,
    MeasurementConfiguration,
    MeasurementIntercept,
    MeasurementLoading,
    MeasurementModel,
    MeasurementSigma,
    NormalizationPlan,
    PositiveParameterSpec,
    PositivityMode,
    build_biogeme_model,
    resolve_model,
)
from biogeme.models import logit
from biogeme.results_processing import (
    get_latex_estimated_parameters,
    get_latex_general_statistics,
    get_pandas_estimated_parameters,
)

logger = blog.get_screen_logger(level=blog.INFO)

DEFAULT_MEASUREMENT_SIGMA_START = 10.0

Ordered-probit measurement configuration for all Likert indicators.¶

The semantic specification, the latent variable, and the mode-choice component are the same as in the ordered-logit example. Only the ordinal measurement distribution changes: ordered logit uses the logistic CDF, whereas ordered probit uses the normal CDF.

measurement_configuration = MeasurementConfiguration(
    specifications=[
        IndicatorMeasurementSpec(
            indicator_name=indicator.name,
            measurement_model=MeasurementModel.ORDERED_PROBIT,
            measurement_sigma=PositiveParameterSpec(
                start=DEFAULT_MEASUREMENT_SIGMA_START
            ),
        )
        for indicator in likert_indicators
    ]
)

Load the Optima data.¶

database = read_data()

Define the build context for maximum likelihood estimation.¶

The build context specifies how the semantic latent-variable specification is translated into estimable Biogeme expressions. We start from the default maximum-likelihood context and explicitly select the log-exp parameterization for positive parameters. The ordinal options control the construction and numerical treatment of ordered-probit thresholds.

default_context = BuildContext.default(EstimationMode.MAXIMUM_LIKELIHOOD)
context = BuildContext(
    estimation_mode=default_context.estimation_mode,
    draw_type=default_context.draw_type,
    positivity_mode=PositivityMode.LOG_EXP,
    naming=default_context.naming,
    ordinal_eps=default_context.ordinal_eps,
    ordinal_enforce_order=default_context.ordinal_enforce_order,
)

Identification constraints for the latent variable¶

Two layers of normalization are needed in this ordered-probit specification. They are the same as in plot_h05_mode_lv_ordlogit_simult.

Ordinal measurement-scale normalization The latent response underlying the ordinal indicators has an arbitrary scale. We therefore fix one measurement sigma to 1.0 in order to anchor the overall scale of the ordered-probit measurement model.
Latent-variable normalization by reference indicator The latent variable itself is not directly observed, so its location and orientation must also be fixed. We use indicator Envir01 as the reference indicator:
- its intercept is fixed to 0.0, which anchors the location of the latent variable;
- its loading on car_centric_attitude is fixed to -1.0, which anchors the scale and fixes the orientation of the latent variable.

The negative sign means that higher values of car_centric_attitude imply lower expected values for Envir01. With these constraints in place, the remaining intercepts, loadings, and thresholds are interpreted relative to the chosen reference and measurement scale.

normalization_plan = NormalizationPlan()

normalization_plan.add(
    Fixing(
        MeasurementSigma('Envir01'),
        1.0,
        note='ordered-probit measurement model: scale normalization',
    )
)

normalization_plan.add(
    Fixing(
        MeasurementIntercept('Envir01'),
        0.0,
        note='car_centric_attitude reference indicator: location',
    )
)
normalization_plan.add(
    Fixing(
        MeasurementLoading('car_centric_attitude', 'Envir01'),
        -1.0,
        note='car_centric_attitude reference indicator: scale and orientation',
    )
)

Resolve the semantic specification.¶

The resolver combines the latent-variable specification, the indicator definitions, the ordered-probit measurement configuration, and the normalization plan into an internal resolved model.

resolved_model = resolve_model(
    latent_variables=latent_variables,
    likert_indicators=likert_indicators,
    likert_types=likert_types,
    measurement_configuration=measurement_configuration,
    context=context,
    normalization_plan=normalization_plan,
)

Build the Biogeme expressions.¶

The builder translates the resolved model into expressions that can be used by Biogeme for simultaneous maximum-likelihood estimation. It also provides report-ready parameter groups based on the parameters that are actually estimated.

built_model = build_biogeme_model(resolved_model)

Choice utilities including the latent-variable term.¶

utilities = generate_utility_functions(built_model.latent_expressions)

Conditional likelihood of the mode choice model¶

availability = generate_availability()
conditional_choice_likelihood = logit(utilities, availability, Choice)

Combined conditional likelihood.¶

The ordered-probit measurement component and the mode-choice component are multiplied conditionally on the latent variable. The product is later integrated over the latent-variable distribution, so all components are estimated jointly in one likelihood.

combined_conditional_likelihood = (
    built_model.conditional_likelihood * conditional_choice_likelihood
)

Log-likelihood.¶

The combined conditional likelihood is integrated over the latent variable by Monte Carlo. Taking the logarithm of the resulting integrated likelihood yields the log-likelihood used for simultaneous maximum-likelihood estimation.

integrated_likelihood = MonteCarlo(combined_conditional_likelihood)
log_likelihood = log(integrated_likelihood)

Estimate the model with Biogeme.¶

Existing results are reloaded from the YAML file when available.

biogeme = BIOGEME(
    database,
    log_likelihood,
    number_of_draws=NUMBER_OF_DRAWS,
    calculating_second_derivatives='never',
    max_iterations=5_000,
    group_of_parameters=built_model.parameter_groups,
)
biogeme.model_name = 'plot_h06_mode_lv_ordprobit_simult'

yaml_file_name = f'saved_results/{biogeme.model_name}.yaml'
results = biogeme.estimate_or_load(yaml_file_name=yaml_file_name)

Display a compact summary and the estimated parameters.

print(results.short_summary())
pandas_results = get_pandas_estimated_parameters(
    estimation_results=results,
    group_of_parameters=built_model.parameter_groups,
)
for group_name, pandas_table in pandas_results.items():
    print(group_name if group_name else 'Estimated parameters')
    print(pandas_table)

general_statistics = get_latex_general_statistics(estimation_results=results)
print(general_statistics)

estimated_parameters = get_latex_estimated_parameters(
    estimation_results=results,
    group_of_parameters=built_model.parameter_groups,
)
for group_name, latex_table in estimated_parameters.items():
    print(group_name if group_name else 'Estimated parameters')
    print(latex_table)

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