where

*L*(*β*) = ∏n=1..N *ρ*(*kn*|**I**;*β*) is the likelihood function. *n* is the generic user observed (or interviewed). *N* is the number of users observed. *kn* is the alternative chosen by user *n*.

RP consists of the construction of a survey on behaviors revealed or demonstrated in real contexts; SP consists instead of the construction of a survey on behaviors declared by users in hypothetical contexts [30]. A pilot survey conducted on a small sample allows the definition of the size of an effective sample to conduct a full survey, and the results obtained in the study area are reported in Section 4.

#### *3.3. Validation*

Informal and formal tests can be considered for model validation.

Among the informal tests, the signs of the calibrated coefficients and their relationships can be verified, i.e., the value of time (VOT) or the ratio between the coefficient relating to time and that relating to monetary cost, which must be consistent with how much users are willing to pay [30]. In a transport alternative, a parameter characterized by a positive sign indicates that a user perceives positive utility in the choice relative to the corresponding attribute (i.e., comfort, quality of service or vehicles, and integration), while a parameter characterized by a negative sign indicates that a user perceives disutility (i.e., travel time or monetary cost). In the case of a model of MaaS preference, the sign of the parameter for distance (or a proxy variable) for a MaaS utility alternative may be positive because MaaS attractiveness can increase with distance.

Among the formal tests, there is that of Student's *t*-test of single coefficients, which verifies that the estimated parameters are equal to 0. This test establishes that a parameter estimate is significantly non-zero if the statistical test is not accepted [30]. This test is helpful in the calibration phase for evaluating if an attribute needs to be tested in different specifications; a high value of a constant indicates that there are other attributes of attractiveness or costs to be considered. The statistic for each parameter *h* is evaluated by the following:

$$t\_{\rm li} = \beta\_{\rm h} \star / \sigma(\beta\_{\rm h} \rm \bf \, ^\circ) \tag{7}$$

where

*β*h\* is the parameter *h* estimate using the maximum likelihood method and *σ*(*β*h\*) is the standard deviation relative to *β*<sup>h</sup> \* .

Another formal test is the goodness of fit of a model *ρ*2. This depends on the likelihood function evaluated with the vectors of calibrated parameters with optimal and 0 values. The values are in the range of [0; 1] in relation to the model reproduction of the choices of the sample [30]:

$$\boldsymbol{\phi}^2 = 1 - \ln(L(\boldsymbol{\mathcal{B}}^\*)) / \ln(L(\boldsymbol{\mathcal{B}}^0)) \tag{8}$$

where

*L* is the likelihood function; *β*\* is the vector of the optimal values of the calibrated parameters; and *β*<sup>0</sup> is the vector of the calibrated parameters equal to zero.

Other formal tests can be adopted, such as the Chi-squared test for the vectors of coefficients, the likelihood ratio test for the vectors of coefficients, and a test for the functional form of a model [30].
