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Ordered-logit hybrid mode choice model: simultaneous maximum likelihood estimation¶
This example estimates a hybrid mode choice model with one latent variable and ordered-logit measurement equations. The observed Likert indicators are modeled as ordinal responses, and the associated threshold parameters are estimated jointly with the latent-variable and choice-model parameters.
Compared with the previous Gaussian specification, only the measurement model changes: the hybrid structure, the latent variable, and the simultaneous maximum-likelihood estimation strategy remain the same.
The latent-variable structure is imported from a separate semantic specification file. This script adds the ordered-logit 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-logit 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-logit 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
from __future__ import annotations
from choice_latent_variables import 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-logit measurement configuration for all Likert indicators.¶
measurement_configuration = MeasurementConfiguration(
specifications=[
IndicatorMeasurementSpec(
indicator_name=indicator.name,
measurement_model=MeasurementModel.ORDERED_LOGIT,
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 are relevant here because the measurement equations use ordered-logit probabilities.
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-logit specification.
Ordered-logit 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-logit 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
Envir01as the reference indicator: - its intercept is fixed to 0.0, which anchors the location of the latentvariable;
its loading on
car_centric_attitudeis 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-logit 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-logit 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¶
conditional_choice_likelihood = logit(utilities, None, Choice)
Combined conditional likelihood¶
The ordered-logit measurement component and the mode-choice component are combined conditionally on the latent variable. As in the previous step, both parts are estimated jointly in a single 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_h05_mode_lv_ordlogit_simult'
yaml_file_name = f'saved_results/{biogeme.model_name}.yaml'
results = biogeme.estimate_or_load(yaml_file_name=yaml_file_name)
Biogeme parameters read from biogeme.toml.
Estimation results are read from saved_results/plot_h05_mode_lv_ordlogit_simult.yaml. No estimation is performed.
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)
Results for model plot_h05_mode_lv_ordlogit_simult
Nbr of parameters: 41
Sample size: 889
Excluded data: 0
Final log likelihood: -10505.44
Akaike Information Criterion: 21092.88
Bayesian Information Criterion: 21289.27
Structural equation
Name ... BHHH p-value
0 struct_car_centric_attitude_intercept ... 1.476169e-01
1 struct_car_centric_attitude_top_manager ... 2.773784e-01
2 struct_car_centric_attitude_car_oriented_parents ... 9.310337e-04
3 struct_car_centric_attitude_high_education ... 9.496054e-05
4 struct_car_centric_attitude_low_education ... 1.452991e-01
5 struct_car_centric_attitude_used_to_go_to_scho... ... 8.783886e-01
6 struct_car_centric_attitude_sigma_log ... 2.220446e-16
[7 rows x 5 columns]
Measurement equation: Envir02
Name ... BHHH p-value
9 measurement_intercept_Envir02 ... 4.011680e-12
10 measurement_coefficient_car_centric_attitude_E... ... 0.000000e+00
11 measurement_Envir02_sigma_log ... 1.344989e-01
[3 rows x 5 columns]
Measurement equation: Envir06
Name ... BHHH p-value
12 measurement_intercept_Envir06 ... 0.000000
13 measurement_coefficient_car_centric_attitude_E... ... 0.000000
14 measurement_Envir06_sigma_log ... 0.000028
[3 rows x 5 columns]
Measurement equation: Mobil03
Name ... BHHH p-value
15 measurement_intercept_Mobil03 ... 1.003829e-05
16 measurement_coefficient_car_centric_attitude_M... ... 6.639182e-07
17 measurement_Mobil03_sigma_log ... 8.217192e-01
[3 rows x 5 columns]
Measurement equation: Mobil05
Name ... BHHH p-value
18 measurement_intercept_Mobil05 ... 1.250703e-07
19 measurement_coefficient_car_centric_attitude_M... ... 0.000000e+00
20 measurement_Mobil05_sigma_log ... 1.785643e-01
[3 rows x 5 columns]
Measurement equation: Mobil08
Name ... BHHH p-value
21 measurement_intercept_Mobil08 ... 1.998401e-15
22 measurement_coefficient_car_centric_attitude_M... ... 8.903989e-14
23 measurement_Mobil08_sigma_log ... 2.370670e-01
[3 rows x 5 columns]
Measurement equation: Mobil09
Name ... BHHH p-value
24 measurement_intercept_Mobil09 ... 0.000000
25 measurement_coefficient_car_centric_attitude_M... ... 0.000000
26 measurement_Mobil09_sigma_log ... 0.050072
[3 rows x 5 columns]
Measurement equation: Mobil10
Name ... BHHH p-value
27 measurement_intercept_Mobil10 ... 0.573240
28 measurement_coefficient_car_centric_attitude_M... ... 0.000000
29 measurement_Mobil10_sigma_log ... 0.000885
[3 rows x 5 columns]
Measurement equation: LifSty07
Name ... BHHH p-value
30 measurement_intercept_LifSty07 ... 0.000000
31 measurement_coefficient_car_centric_attitude_L... ... 0.000146
32 measurement_LifSty07_sigma_log ... 0.194147
[3 rows x 5 columns]
Thresholds
Name Value BHHH std err. BHHH t-stat. BHHH p-value
7 likert_delta_0_log -0.685912 0.096478 -7.109483 1.164846e-12
8 likert_delta_1_log 0.513089 0.088160 5.819999 5.884793e-09
Other parameters
Name Value ... BHHH t-stat. BHHH p-value
33 choice_scale_parameter 0.069929 ... 7.697195 1.398881e-14
34 choice_asc_pt -11.524470 ... -2.664676 7.706255e-03
35 choice_beta_time_pt -13.317507 ... -4.469754 7.830946e-06
36 choice_asc_car -4.336645 ... -1.199785 2.302229e-01
37 choice_beta_time_car -28.557934 ... -5.622829 1.878550e-08
38 choice_beta_car_centric_attitude_car 5.296329 ... 4.765432 1.884499e-06
39 choice_beta_dist_work -2.911517 ... -6.672737 2.510769e-11
40 choice_beta_dist_other_purposes -4.633415 ... -6.078639 1.212071e-09
[8 rows x 5 columns]
%% General statistics
\section{General statistics}
\begin{tabular}{ll}
Number of estimated parameters & 41 \\
Sample size & 889 \\
Excluded observations & 0 \\
Init log likelihood & -20187.62 \\
Final log likelihood & -10505.44 \\
Likelihood ratio test for the init. model & 19364.36 \\
Rho-square for the init. model & 0.48 \\
Rho-square-bar for the init. model & 0.478 \\
Akaike Information Criterion & 21092.88 \\
Bayesian Information Criterion & 21289.27 \\
Final gradient norm & 1.2226E-01 \\
Number of draws & 50000 \\
Draws generation time & 0:00:13.715341 \\
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$Relative gradient = 6e-06 <= 6.1e-06$ \\
Number of function evaluations & \verb$1629$ \\
Number of gradient evaluations & \verb$939$ \\
Number of hessian evaluations & \verb$0$ \\
Number of iterations & \verb$690$ \\
Optimization time & \verb$3:02:20.675567$ \\
Proportion of Hessian calculation & \verb$0/469 = 0.0%$ \\
Relative gradient & \verb$5.992e-06$ \\
\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 & 0&352 & 0&243 & 1&45 & 0&148 \\
1 & struct\_car\_centric\_attitude\_top\_manager & 0&320 & 0&294 & 1&09 & 0&277 \\
2 & struct\_car\_centric\_attitude\_car\_oriented\_parents & 0&706 & 0&213 & 3&31 & 0&000931 \\
3 & struct\_car\_centric\_attitude\_high\_education & -1&05 & 0&269 & -3&90 & 9&50e-05 \\
4 & struct\_car\_centric\_attitude\_low\_education & 0&379 & 0&260 & 1&46 & 0&145 \\
5 & struct\_car\_centric\_attitude\_used\_to\_go\_to\_school\_by\_car & 0&182 & 1&19 & 0&153 & 0&878 \\
6 & struct\_car\_centric\_attitude\_sigma\_log & 0&756 & 0&0914 & 8&27 & 2&22e-16 \\
\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
9 & measurement\_intercept\_Envir02 & 1&07 & 0&154 & 6&94 & 4&01e-12 \\
10 & measurement\_coefficient\_car\_centric\_attitude\_Envir02 & -0&525 & 0&0379 & -13&8 & 0&00 \\
11 & measurement\_Envir02\_sigma\_log & -0&164 & 0&110 & -1&50 & 0&134 \\
\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
12 & measurement\_intercept\_Envir06 & 2&44 & 0&244 & 10&0 & 0&00 \\
13 & measurement\_coefficient\_car\_centric\_attitude\_Envir06 & -0&362 & 0&0339 & -10&7 & 0&00 \\
14 & measurement\_Envir06\_sigma\_log & -0&431 & 0&103 & -4&19 & 2&82e-05 \\
\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
15 & measurement\_intercept\_Mobil03 & 0&454 & 0&103 & 4&42 & 1&00e-05 \\
16 & measurement\_coefficient\_car\_centric\_attitude\_Mobil03 & -0&175 & 0&0352 & -4&97 & 6&64e-07 \\
17 & measurement\_Mobil03\_sigma\_log & 0&0237 & 0&105 & 0&225 & 0&822 \\
\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
18 & measurement\_intercept\_Mobil05 & 0&760 & 0&144 & 5&29 & 1&25e-07 \\
19 & measurement\_coefficient\_car\_centric\_attitude\_Mobil05 & -0&380 & 0&0434 & -8&74 & 0&00 \\
20 & measurement\_Mobil05\_sigma\_log & 0&150 & 0&111 & 1&35 & 0&179 \\
\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
21 & measurement\_intercept\_Mobil08 & -1&13 & 0&142 & -7&95 & 2&00e-15 \\
22 & measurement\_coefficient\_car\_centric\_attitude\_Mobil08 & 0&315 & 0&0423 & 7&46 & 8&90e-14 \\
23 & measurement\_Mobil08\_sigma\_log & 0&129 & 0&109 & 1&18 & 0&237 \\
\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
24 & measurement\_intercept\_Mobil09 & 1&69 & 0&192 & 8&80 & 0&00 \\
25 & measurement\_coefficient\_car\_centric\_attitude\_Mobil09 & -0&355 & 0&0338 & -10&5 & 0&00 \\
26 & measurement\_Mobil09\_sigma\_log & -0&207 & 0&106 & -1&96 & 0&0501 \\
\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
27 & measurement\_intercept\_Mobil10 & 0&0872 & 0&155 & 0&563 & 0&573 \\
28 & measurement\_coefficient\_car\_centric\_attitude\_Mobil10 & 0&667 & 0&0779 & 8&57 & 0&00 \\
29 & measurement\_Mobil10\_sigma\_log & 0&440 & 0&132 & 3&32 & 0&000885 \\
\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
30 & measurement\_intercept\_LifSty07 & -1&29 & 0&140 & -9&21 & 0&00 \\
31 & measurement\_coefficient\_car\_centric\_attitude\_LifSty07 & 0&139 & 0&0366 & 3&80 & 0&000146 \\
32 & measurement\_LifSty07\_sigma\_log & 0&147 & 0&113 & 1&30 & 0&194 \\
\end{tabular}
Thresholds
\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 & likert\_delta\_0\_log & -0&686 & 0&0965 & -7&11 & 1&16e-12 \\
8 & likert\_delta\_1\_log & 0&513 & 0&0882 & 5&82 & 5&88e-09 \\
\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
33 & choice\_scale\_parameter & 0&0699 & 0&00908 & 7&70 & 1&40e-14 \\
34 & choice\_asc\_pt & -11&5 & 4&32 & -2&66 & 0&00771 \\
35 & choice\_beta\_time\_pt & -13&3 & 2&98 & -4&47 & 7&83e-06 \\
36 & choice\_asc\_car & -4&34 & 3&61 & -1&20 & 0&230 \\
37 & choice\_beta\_time\_car & -28&6 & 5&08 & -5&62 & 1&88e-08 \\
38 & choice\_beta\_car\_centric\_attitude\_car & 5&30 & 1&11 & 4&77 & 1&88e-06 \\
39 & choice\_beta\_dist\_work & -2&91 & 0&436 & -6&67 & 2&51e-11 \\
40 & choice\_beta\_dist\_other\_purposes & -4&63 & 0&762 & -6&08 & 1&21e-09 \\
\end{tabular}
Total running time of the script: (0 minutes 1.116 seconds)