.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/hybrid_choice_models/plot_h04_mode_lv_gauss_simult.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_hybrid_choice_models_plot_h04_mode_lv_gauss_simult.py: Gaussian hybrid mode choice model: simultaneous maximum likelihood estimation ============================================================================= This example estimates a hybrid mode choice model with one latent variable and Gaussian measurement equations. The observed Likert indicators are treated as continuous responses and modeled with Gaussian measurement equations. Responses using the type's neutral or missing labels (6 and -1 in this case) contribute a neutral factor to the measurement likelihood and are therefore excluded from the corresponding Gaussian measurement equation. In contrast with the sequential specification introduced previously, the measurement component and the mode-choice component are estimated jointly. The conditional likelihood of the indicators and the conditional likelihood of the observed choice are multiplied conditionally on the latent variable, and the resulting expression is integrated over the latent-variable distribution. The latent-variable structure is imported from a separate semantic specification file. This script adds the Gaussian 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 a Gaussian 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 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 Mon Jun 15 2026, 09:54:37 .. GENERATED FROM PYTHON SOURCE LINES 39-78 .. code-block:: Python 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, BuiltBiogemeModel, EstimationMode, Fixing, IndicatorMeasurementSpec, MeasurementConfiguration, MeasurementIntercept, MeasurementLoading, MeasurementModel, 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 .. GENERATED FROM PYTHON SOURCE LINES 79-83 Gaussian measurement configuration for all Likert indicators. ------------------------------------------------------------- The neutral-label semantics are supplied by ``likert_types`` and are applied by the latent-variable builder to these Gaussian measurement equations. .. GENERATED FROM PYTHON SOURCE LINES 83-96 .. code-block:: Python measurement_configuration = MeasurementConfiguration( specifications=[ IndicatorMeasurementSpec( indicator_name=indicator.name, measurement_model=MeasurementModel.GAUSSIAN, measurement_sigma=PositiveParameterSpec( start=DEFAULT_MEASUREMENT_SIGMA_START ), ) for indicator in likert_indicators ] ) .. GENERATED FROM PYTHON SOURCE LINES 97-99 Load the Optima data. --------------------- .. GENERATED FROM PYTHON SOURCE LINES 99-101 .. code-block:: Python database = read_data() .. GENERATED FROM PYTHON SOURCE LINES 102-108 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. .. GENERATED FROM PYTHON SOURCE LINES 108-118 .. code-block:: Python 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, ) .. GENERATED FROM PYTHON SOURCE LINES 119-135 Identification constraints for the latent variable --------------------------------------------------- A latent variable is not directly observed, so its location and scale are not identified unless normalization constraints are imposed. We use a reference-indicator strategy based on ``Envir01``: - the intercept of ``Envir01`` is fixed to 0.0, which anchors the location of the latent-variable scale; - the loading of ``Envir01`` 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 and loadings are interpreted relative to the reference indicator. .. GENERATED FROM PYTHON SOURCE LINES 135-152 .. code-block:: Python normalization_plan = NormalizationPlan() 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', ) ) .. GENERATED FROM PYTHON SOURCE LINES 153-158 Resolve the semantic specification. ----------------------------------- The resolver combines the latent-variable specification, the indicator definitions, the Gaussian measurement configuration, and the normalization plan into an internal resolved model. .. GENERATED FROM PYTHON SOURCE LINES 158-167 .. code-block:: Python 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, ) .. GENERATED FROM PYTHON SOURCE LINES 168-174 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. .. GENERATED FROM PYTHON SOURCE LINES 174-176 .. code-block:: Python built_model: BuiltBiogemeModel = build_biogeme_model(resolved_model) .. GENERATED FROM PYTHON SOURCE LINES 177-179 Choice utilities including the latent-variable term. ---------------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 179-181 .. code-block:: Python utilities = generate_utility_functions(built_model.latent_expressions) .. GENERATED FROM PYTHON SOURCE LINES 182-184 Conditional likelihood of the mode choice model ----------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 184-187 .. code-block:: Python availability = generate_availability() conditional_choice_likelihood = logit(utilities, availability, Choice) .. GENERATED FROM PYTHON SOURCE LINES 188-193 Combined conditional likelihood ------------------------------- The measurement component and the mode-choice component are combined conditionally on the latent variable. This is the key difference with the sequential approach: both parts are estimated jointly in a single likelihood. .. GENERATED FROM PYTHON SOURCE LINES 193-197 .. code-block:: Python combined_conditional_likelihood = ( built_model.conditional_likelihood * conditional_choice_likelihood ) .. GENERATED FROM PYTHON SOURCE LINES 198-204 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. .. GENERATED FROM PYTHON SOURCE LINES 204-207 .. code-block:: Python integrated_likelihood = MonteCarlo(combined_conditional_likelihood) log_likelihood = log(integrated_likelihood) .. GENERATED FROM PYTHON SOURCE LINES 208-211 Estimate the model with Biogeme. -------------------------------- Existing results are reloaded from the YAML file when available. .. GENERATED FROM PYTHON SOURCE LINES 211-224 .. code-block:: Python 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_h04_mode_lv_gauss_simult' yaml_file_name = f'saved_results/{biogeme.model_name}.yaml' results = biogeme.estimate_or_load(yaml_file_name=yaml_file_name) .. rst-class:: sphx-glr-script-out .. code-block:: none Biogeme parameters read from biogeme.toml. Estimation results are read from saved_results/plot_h04_mode_lv_gauss_simult.yaml. No estimation is performed. .. GENERATED FROM PYTHON SOURCE LINES 225-226 Display a compact summary and the estimated parameters. .. GENERATED FROM PYTHON SOURCE LINES 226-245 .. code-block:: Python 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) .. rst-class:: sphx-glr-script-out .. code-block:: none Results for model plot_h04_mode_lv_gauss_simult Nbr of parameters: 40 Sample size: 889 Excluded data: 0 Final log likelihood: -11152.32 Akaike Information Criterion: 22384.65 Bayesian Information Criterion: 22576.25 Structural equation Name ... BHHH p-value 0 struct_car_centric_attitude_intercept ... 0.000000 1 struct_car_centric_attitude_top_manager ... 0.381211 2 struct_car_centric_attitude_car_oriented_parents ... 0.000788 3 struct_car_centric_attitude_high_education ... 0.000088 4 struct_car_centric_attitude_low_education ... 0.103819 5 struct_car_centric_attitude_used_to_go_to_scho... ... 0.777651 6 struct_car_centric_attitude_sigma_log ... 0.688529 [7 rows x 5 columns] Measurement equation: Envir01 Name Value ... BHHH t-stat. BHHH p-value 7 measurement_Envir01_sigma_log -0.073942 ... -1.547934 0.121638 [1 rows x 5 columns] Measurement equation: Envir02 Name ... BHHH p-value 8 measurement_intercept_Envir02 ... 0.000000 9 measurement_coefficient_car_centric_attitude_E... ... 0.000000 10 measurement_Envir02_sigma_log ... 0.070634 [3 rows x 5 columns] Measurement equation: Envir06 Name ... BHHH p-value 11 measurement_intercept_Envir06 ... 0.0 12 measurement_coefficient_car_centric_attitude_E... ... 0.0 13 measurement_Envir06_sigma_log ... 0.0 [3 rows x 5 columns] Measurement equation: Mobil03 Name ... BHHH p-value 14 measurement_intercept_Mobil03 ... 0.000000 15 measurement_coefficient_car_centric_attitude_M... ... 0.000002 16 measurement_Mobil03_sigma_log ... 0.048258 [3 rows x 5 columns] Measurement equation: Mobil05 Name ... BHHH p-value 17 measurement_intercept_Mobil05 ... 0.000000e+00 18 measurement_coefficient_car_centric_attitude_M... ... 1.287859e-14 19 measurement_Mobil05_sigma_log ... 4.724055e-03 [3 rows x 5 columns] Measurement equation: Mobil08 Name ... BHHH p-value 20 measurement_intercept_Mobil08 ... 0.000000e+00 21 measurement_coefficient_car_centric_attitude_M... ... 9.350298e-13 22 measurement_Mobil08_sigma_log ... 4.542337e-01 [3 rows x 5 columns] Measurement equation: Mobil09 Name ... BHHH p-value 23 measurement_intercept_Mobil09 ... 0.000000 24 measurement_coefficient_car_centric_attitude_M... ... 0.000000 25 measurement_Mobil09_sigma_log ... 0.000002 [3 rows x 5 columns] Measurement equation: Mobil10 Name ... BHHH p-value 26 measurement_intercept_Mobil10 ... 0.000000e+00 27 measurement_coefficient_car_centric_attitude_M... ... 3.907985e-14 28 measurement_Mobil10_sigma_log ... 2.640176e-04 [3 rows x 5 columns] Measurement equation: LifSty07 Name ... BHHH p-value 29 measurement_intercept_LifSty07 ... 0.000000 30 measurement_coefficient_car_centric_attitude_L... ... 0.001102 31 measurement_LifSty07_sigma_log ... 0.963720 [3 rows x 5 columns] Other parameters Name Value ... BHHH t-stat. BHHH p-value 32 choice_scale_parameter 0.064311 ... 6.747197 1.507283e-11 33 choice_asc_pt -12.400497 ... -2.606044 9.159480e-03 34 choice_beta_time_pt -15.926095 ... -4.041636 5.307959e-05 35 choice_asc_car 30.428643 ... 3.861035 1.129077e-04 36 choice_beta_time_car -33.155875 ... -5.047578 4.474465e-07 37 choice_beta_car_centric_attitude_car 12.225250 ... 4.803138 1.561983e-06 38 choice_beta_dist_work -3.219281 ... -5.983448 2.184623e-09 39 choice_beta_dist_other_purposes -5.169025 ... -5.586791 2.313041e-08 [8 rows x 5 columns] %% General statistics \section{General statistics} \begin{tabular}{ll} Number of estimated parameters & 40 \\ Sample size & 889 \\ Excluded observations & 0 \\ Init log likelihood & -30421.53 \\ Final log likelihood & -11152.32 \\ Likelihood ratio test for the init. model & 38538.42 \\ Rho-square for the init. model & 0.633 \\ Rho-square-bar for the init. model & 0.632 \\ Akaike Information Criterion & 22384.65 \\ Bayesian Information Criterion & 22576.25 \\ Final gradient norm & 1.2536E-01 \\ Number of draws & 50000 \\ Draws generation time & 0:00:14.306388 \\ Types of draws & struct\_car\_centric\_attitude\_draws: NORMAL\_MLHS\_ANTI \\ Bootstrapping time & None \\ Algorithm & \verb$BFGS with trust region for simple bound constraints$ \\ Cause of termination & \verb$Relative gradient = 6.1e-06 <= 6.1e-06$ \\ Number of function evaluations & \verb$2370$ \\ Number of gradient evaluations & \verb$1399$ \\ Number of hessian evaluations & \verb$0$ \\ Number of iterations & \verb$971$ \\ Optimization time & \verb$2:03:59.687215$ \\ Proportion of Hessian calculation & \verb$0/699 = 0.0%$ \\ Relative gradient & \verb$6.052e-06$ \\ \end{tabular} Structural equation \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 0 & struct\_car\_centric\_attitude\_intercept & -2&67 & 0&107 & -24&9 & 0&00 \\ 1 & struct\_car\_centric\_attitude\_top\_manager & 0&120 & 0&137 & 0&876 & 0&381 \\ 2 & struct\_car\_centric\_attitude\_car\_oriented\_parents & 0&329 & 0&0981 & 3&36 & 0&000788 \\ 3 & struct\_car\_centric\_attitude\_high\_education & -0&471 & 0&120 & -3&92 & 8&76e-05 \\ 4 & struct\_car\_centric\_attitude\_low\_education & 0&191 & 0&118 & 1&63 & 0&104 \\ 5 & struct\_car\_centric\_attitude\_used\_to\_go\_to\_school\_by\_car & 0&146 & 0&519 & 0&282 & 0&778 \\ 6 & struct\_car\_centric\_attitude\_sigma\_log & -0&0282 & 0&0703 & -0&401 & 0&689 \\ \end{tabular} Measurement equation: Envir01 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 7 & measurement\_Envir01\_sigma\_log & -0&0739 & 0&0478 & -1&55 & 0&122 \\ \end{tabular} Measurement equation: Envir02 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 8 & measurement\_intercept\_Envir02 & 1&69 & 0&154 & 11&0 & 0&00 \\ 9 & measurement\_coefficient\_car\_centric\_attitude\_Envir02 & -0&632 & 0&0553 & -11&4 & 0&00 \\ 10 & measurement\_Envir02\_sigma\_log & -0&0580 & 0&0321 & -1&81 & 0&0706 \\ \end{tabular} Measurement equation: Envir06 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 11 & measurement\_intercept\_Envir06 & 3&35 & 0&0894 & 37&4 & 0&00 \\ 12 & measurement\_coefficient\_car\_centric\_attitude\_Envir06 & -0&369 & 0&0342 & -10&8 & 0&00 \\ 13 & measurement\_Envir06\_sigma\_log & -0&399 & 0&0220 & -18&2 & 0&00 \\ \end{tabular} Measurement equation: Mobil03 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 14 & measurement\_intercept\_Mobil03 & 2&44 & 0&127 & 19&1 & 0&00 \\ 15 & measurement\_coefficient\_car\_centric\_attitude\_Mobil03 & -0&217 & 0&0453 & -4&79 & 1&69e-06 \\ 16 & measurement\_Mobil03\_sigma\_log & 0&0643 & 0&0325 & 1&98 & 0&0483 \\ \end{tabular} Measurement equation: Mobil05 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 17 & measurement\_intercept\_Mobil05 & 1&97 & 0&150 & 13&1 & 0&00 \\ 18 & measurement\_coefficient\_car\_centric\_attitude\_Mobil05 & -0&415 & 0&0538 & -7&71 & 1&29e-14 \\ 19 & measurement\_Mobil05\_sigma\_log & 0&0985 & 0&0349 & 2&83 & 0&00472 \\ \end{tabular} Measurement equation: Mobil08 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 20 & measurement\_intercept\_Mobil08 & 3&12 & 0&124 & 25&2 & 0&00 \\ 21 & measurement\_coefficient\_car\_centric\_attitude\_Mobil08 & 0&330 & 0&0462 & 7&14 & 9&35e-13 \\ 22 & measurement\_Mobil08\_sigma\_log & 0&0224 & 0&0300 & 0&748 & 0&454 \\ \end{tabular} Measurement equation: Mobil09 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 23 & measurement\_intercept\_Mobil09 & 2&71 & 0&118 & 22&9 & 0&00 \\ 24 & measurement\_coefficient\_car\_centric\_attitude\_Mobil09 & -0&418 & 0&0445 & -9&39 & 0&00 \\ 25 & measurement\_Mobil09\_sigma\_log & -0&129 & 0&0270 & -4&80 & 1&58e-06 \\ \end{tabular} Measurement equation: Mobil10 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 26 & measurement\_intercept\_Mobil10 & 4&34 & 0&214 & 20&3 & 0&00 \\ 27 & measurement\_coefficient\_car\_centric\_attitude\_Mobil10 & 0&569 & 0&0752 & 7&56 & 3&91e-14 \\ 28 & measurement\_Mobil10\_sigma\_log & 0&167 & 0&0458 & 3&65 & 0&000264 \\ \end{tabular} Measurement equation: LifSty07 \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 29 & measurement\_intercept\_LifSty07 & 2&44 & 0&104 & 23&4 & 0&00 \\ 30 & measurement\_coefficient\_car\_centric\_attitude\_LifSty07 & 0&120 & 0&0367 & 3&26 & 0&00110 \\ 31 & measurement\_LifSty07\_sigma\_log & 0&00151 & 0&0331 & 0&0455 & 0&964 \\ \end{tabular} Other parameters \begin{tabular}{rlr@{.}lr@{.}lr@{.}lr@{.}l} & & \multicolumn{2}{l}{} & \multicolumn{2}{l}{BHHH} & \multicolumn{4}{l}{} \\ Parameter & & \multicolumn{2}{l}{Coeff.} & \multicolumn{2}{l}{Asympt.} & \multicolumn{4}{l}{} \\ number & Description & \multicolumn{2}{l}{estimate} & \multicolumn{2}{l}{std. error} & \multicolumn{2}{l}{$t$-stat} & \multicolumn{2}{l}{$p$-value} \\ \hline 32 & choice\_scale\_parameter & 0&0643 & 0&00953 & 6&75 & 1&51e-11 \\ 33 & choice\_asc\_pt & -12&4 & 4&76 & -2&61 & 0&00916 \\ 34 & choice\_beta\_time\_pt & -15&9 & 3&94 & -4&04 & 5&31e-05 \\ 35 & choice\_asc\_car & 30&4 & 7&88 & 3&86 & 0&000113 \\ 36 & choice\_beta\_time\_car & -33&2 & 6&57 & -5&05 & 4&47e-07 \\ 37 & choice\_beta\_car\_centric\_attitude\_car & 12&2 & 2&55 & 4&80 & 1&56e-06 \\ 38 & choice\_beta\_dist\_work & -3&22 & 0&538 & -5&98 & 2&18e-09 \\ 39 & choice\_beta\_dist\_other\_purposes & -5&17 & 0&925 & -5&59 & 2&31e-08 \\ \end{tabular} .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 1.375 seconds) .. _sphx_glr_download_auto_examples_hybrid_choice_models_plot_h04_mode_lv_gauss_simult.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_h04_mode_lv_gauss_simult.ipynb ` .. container:: sphx-glr-download 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