5. Mathematical formulation

### 5.1 Choice function and structural and measurement equations

The utility choice function in the hybrid choice model is based on the assumption that each individual is faced with a set of alternatives, i, and each alternative expressed as a function of a vector of observed instrumental attributes, Xi; the users' attributes, Xi,SE; a vector of latent variables, LVi; and the error term εi:

$$
\Delta U^{i} = \beta\_{\text{x}} X^{i} + \beta\_{\text{SE}} X\_{\text{SE}}^{i} + \beta\_{LV} LV^{i} + \varepsilon^{j} \tag{10}
$$

With reference to the LV<sup>i</sup> vector, two equations have to be specified: the structural and the measurement equations. The structural equations are introduced in order to specify the latent variables, while the measurement equations are introduced in order to specify the perception indicators.

In particular, if p is the generic latent variable, the structural equation for each latent variable may be expressed as follows:

$$LV\_p^i = \gamma\_p + \sum\_{j} \beta\_{\rm SE,j} X\_{\rm SE,j}^i + o\_p^i \tag{11}$$

where γ<sup>p</sup> is the intersect, X<sup>i</sup> SE,j is the vector of the users' characteristics attributes, βSE,j is the vector of the coefficients associated with the users' characteristics (to be estimated), ω<sup>i</sup> <sup>p</sup> is the error term which is usually normally distributed with zero mean and σω,p is the standard deviation.

Furthermore, let I i <sup>n</sup> be a vector of perceptions indicators associated to each latent variable. Each perception indicator (i.e. vector component) may be specified by a measurement equation as follows:

$$I\_{p,k}^i = \alpha\_{p,k} + \lambda\_{p,k} L V\_p^i + \nu\_{p,k}^i \tag{12}$$

5.2.1 Parameters of the choice utility function

DOI: http://dx.doi.org/10.5772/intechopen.87088

following equation:

Age

ZonRes

CarAge

Δcost

LV1

LV2

LV3

1

33

Table 10.

Statistics

\*in parenthesis the t-test values.

[def: age of the respondent]

[def: 1 for users achieved this educational attainment]

[def: mode choice "Car" and trip purpose "Shopping"]

the year of the vehicle owned by the respondent]

[def: latent variable representing design attitude]

[def: latent variable representing consumption attitude]

[def: latent variable representing environment attitude]

[def: mode choice "Car" and trip purpose "Personal Services"]

[def: 0 for users living to the historical centre, 1 if in the outskirts]

[def: age of the owned car on which the respondent would install the kit]

[def: In order to compare the scenarios with and without the kit, the financial gain was expressed in terms of weekly costs, rather than yearly costs or life costs based on

Number of respondents 1364 Number of observations 1364 Init-log-likelihood<sup>1</sup> �944.760 Final log-likelihood �779.81 Rho-square 0.212

Only the log-likelihood associated with the discrete choice component is considered.

Attribute coefficients of the choice model. HySolarKit case study.

Master's degree

by Car-Shopping

by Car-Personal Services

The utility choice functions were analytically specified in accordance with the

The results are shown in Table 10. In particular, the estimation results underline the following latent variables as statistically significant: attitudes towards fuel consumption (LV1), towards the vehicle design (LV2) and towards the environment (LV3). The coefficients related to the parameters in the measurement equation for an ordinal specification are estimated in the considered model. As the measurements

SE <sup>þ</sup> <sup>β</sup>LVLV<sup>i</sup> <sup>þ</sup> <sup>ε</sup><sup>i</sup> (15)

coefficients (betas) Install Not-install

+0.160 (+0.960)

+ 0.156 (+1.16)

+0.0761 (+0.960)

+0.0272 (+1.55)

+0.669 (+1.490)

+0.192 (+0.53)

+0.548 (+2.55)

+0.104 (+0.98) +0.0638 (+8.16)

+0.0682 (+0.46)

<sup>U</sup><sup>i</sup> <sup>¼</sup> <sup>β</sup>xX<sup>i</sup> <sup>þ</sup> <sup>β</sup>SEX<sup>i</sup>

Approaches for Modelling User's Acceptance of Innovative Transportation…

are using a Likert scale with seven levels, six parameters τ<sup>i</sup> are needed

Attributes Attributes

where αp,k is the intersect, λp,k is the coefficient associated with the latent variable (to be estimated), ν<sup>i</sup> p,k is the error terms usually assumed normally distributed with zero mean and σνpk is the standard deviation of the error term.

The psychometric indicators that reveal the latent variables may be coded using a Likert scale [19]. These indicators can be considered to be a linear continuous expression of the LV's or an ordered discrete variable. The first approach has been historically chosen because simpler and more practical with lower computational cost. However, assuming these indicators as continuous variables are in contrast with the real nature of the Likert scale (the Likert scale is a discrete measure) [30], such an approach may introduce some biases in the parameters' estimation. In recent years, several studies have treated them as discrete variables, but with a higher computational cost [31]. In particular, if the measurement is represented by an ordered discrete variable J taking the values j <sup>1</sup>, j2, …, jM, we have

$$\mathbf{J} = \begin{cases} j\_1 & \text{if } \mathbf{I} < \mathbf{r}\_1 \\ j\_2 & \text{if } \mathbf{r}\_1 \le \mathbf{I} < \mathbf{r}\_2 \\ \vdots \\ j\_i & \text{if } \mathbf{r}\_{i-1} \le \mathbf{I} < \mathbf{r}\_i \\ \vdots \\ j\_M & \text{if } \mathbf{r}\_{M-1} \le \mathbf{I} \end{cases} \tag{13}$$

where I is defined by Eq. (12) and τ1, …, τ<sup>M</sup>�<sup>1</sup> are parameters to be estimated, such that

$$
\pi\_1 \le \pi\_2 \le \dots \le \pi\_i \le \dots \le \pi\_{M-1}
$$

If the measurements use a Likert scale with M = 5 levels, four parameters τ<sup>i</sup> are needed. But, in order to account for the symmetry of the indicators, two positive parameters δ<sup>1</sup> and δ<sup>2</sup> are specified instead, in order to define

$$\begin{aligned} \tau\_1 &= -\delta\_1 - \delta\_2 \\\\ \tau\_2 &= -\delta\_1 \\\\ \tau\_3 &= \delta\_1 \\\\ \tau\_4 &= \delta\_1 + \delta\_2 \end{aligned} \tag{14}$$

Then, the probability of a given response j <sup>i</sup> is given by the ordered probit model [5]. For completeness, in the following section, the estimation results related to the HySolarKit and the electric vehicle case study are shown.

In this research report, the model parameters were estimated in accordance with the maximum simulated likelihood statistical approach.

#### 5.2 Example: the HySolarKit case study

The first results shown in this section refer to the HySolarKit case study. As already anticipated in Section 3.1, the choice set was composed of two alternatives: "install" and "not-install". In the following the estimation results are presented, distinguishing the choice utility function, the structural equations and the measurement equations.

Approaches for Modelling User's Acceptance of Innovative Transportation… DOI: http://dx.doi.org/10.5772/intechopen.87088

#### 5.2.1 Parameters of the choice utility function

I i

an ordered discrete variable J taking the values j

J ¼

parameters δ<sup>1</sup> and δ<sup>2</sup> are specified instead, in order to define

Then, the probability of a given response j

5.2 Example: the HySolarKit case study

surement equations.

32

HySolarKit and the electric vehicle case study are shown.

the maximum simulated likelihood statistical approach.

variable (to be estimated), ν<sup>i</sup>

Transportation Systems Analysis and Assessment

such that

p,k <sup>¼</sup> <sup>α</sup>p,k <sup>þ</sup> <sup>λ</sup>p,kLV<sup>i</sup>

uted with zero mean and σνpk is the standard deviation of the error term.

j

8

>>>>>>>>><

>>>>>>>>>:

j

⋮ j

⋮ j

<sup>1</sup> if I<τ<sup>1</sup>

<sup>2</sup> if τ<sup>1</sup> ≤I<τ<sup>2</sup>

<sup>i</sup> if τ<sup>i</sup>�<sup>1</sup> ≤I<τ<sup>i</sup>

<sup>M</sup> if τ<sup>M</sup>�<sup>1</sup> ≤I

where I is defined by Eq. (12) and τ1, …, τ<sup>M</sup>�<sup>1</sup> are parameters to be estimated,

τ<sup>1</sup> ≤ τ<sup>2</sup> ≤ … ≤τ<sup>i</sup> ≤ … ≤τ<sup>M</sup>�<sup>1</sup>

If the measurements use a Likert scale with M = 5 levels, four parameters τ<sup>i</sup> are needed. But, in order to account for the symmetry of the indicators, two positive

τ<sup>1</sup> ¼ �δ<sup>1</sup> � δ<sup>2</sup>

τ<sup>2</sup> ¼ �δ<sup>1</sup> τ<sup>3</sup> ¼ δ<sup>1</sup>

τ<sup>4</sup> ¼ δ<sup>1</sup> þ δ<sup>2</sup>

For completeness, in the following section, the estimation results related to the

In this research report, the model parameters were estimated in accordance with

The first results shown in this section refer to the HySolarKit case study. As already anticipated in Section 3.1, the choice set was composed of two alternatives: "install" and "not-install". In the following the estimation results are presented, distinguishing the choice utility function, the structural equations and the mea-

where αp,k is the intersect, λp,k is the coefficient associated with the latent

The psychometric indicators that reveal the latent variables may be coded using a Likert scale [19]. These indicators can be considered to be a linear continuous expression of the LV's or an ordered discrete variable. The first approach has been historically chosen because simpler and more practical with lower computational cost. However, assuming these indicators as continuous variables are in contrast with the real nature of the Likert scale (the Likert scale is a discrete measure) [30], such an approach may introduce some biases in the parameters' estimation. In recent years, several studies have treated them as discrete variables, but with a higher computational cost [31]. In particular, if the measurement is represented by

<sup>p</sup> <sup>þ</sup> <sup>ν</sup><sup>i</sup>

p,k is the error terms usually assumed normally distrib-

<sup>1</sup>, j2, …, jM, we have

p,k (12)

(13)

(14)

<sup>i</sup> is given by the ordered probit model [5].

The utility choice functions were analytically specified in accordance with the following equation:

$$
\Delta U^i = \beta\_{\text{x}} X^i + \beta\_{\text{SE}} \mathbf{X}\_{\text{SE}}^i + \beta\_{LV} LV^i + \varepsilon^i \tag{15}
$$

The results are shown in Table 10. In particular, the estimation results underline the following latent variables as statistically significant: attitudes towards fuel consumption (LV1), towards the vehicle design (LV2) and towards the environment (LV3).

The coefficients related to the parameters in the measurement equation for an ordinal specification are estimated in the considered model. As the measurements are using a Likert scale with seven levels, six parameters τ<sup>i</sup> are needed


#### Table 10.

Attribute coefficients of the choice model. HySolarKit case study.

#### Transportation Systems Analysis and Assessment


#### Table 11.

Delta values of the calibrated measurement equations. HySolarKit case study.

(see Section 5.1). However, in order to account for the symmetry of the indicators, three positive parameters δ1, δ<sup>2</sup> and δ<sup>3</sup> are actually required (Table 11).

#### 5.2.2 Parameters of the structural model

The coefficients in the structural model are analytically represented by the following equation:

$$LV\_p^i = \chi\_p + \sum\_j \beta\_{\rm SE,j} X\_{\rm SE,j}^i + o\_p^i \tag{16}$$

This equation shows that each latent variable is a function of an intercept value γ<sup>p</sup> of beta-coefficients βSE,j for each of the socioeconomic attributes X<sup>i</sup> SE,j of the respondents that influence the latent variable and contains an error term ω<sup>i</sup> <sup>p</sup> normally distributed with zero mean and σω<sup>p</sup> standard deviations.

The estimation results displayed in below refers to the significant latent variables of the model, representing the attitude towards the fuel consumption (LV1), the vehicle design (LV2) and the environment (LV3) (Table 12).

#### 5.2.3 Parameters of the measurement model

Finally, with regard to the measurement model depending on the perception indicators, they are analytically represented by the following equation:

$$I\_{p,k}^i = \alpha\_{p,k} + \lambda\_{p,k} L V\_p^i + \nu\_{p,k}^i \tag{17}$$

5.3 Example 2: the electric vehicle case study

Coefficients of the calibrated structural model. HySolarKit case study.

5.3.1 Parameters of the choice utility function

by the following equation:

Structural model

by car-shopping

Master's degree

Age

Age

Table 12.

35

SE\_male

SE\_male

by car-personal services

LV2: Attitude towards design issues

[def: age of the respondent]

[def: age of the respondent]

\*in parenthesis the t-test values.

LV3: Attitude towards environment

LV1: Attitude towards fuel consumption

DOI: http://dx.doi.org/10.5772/intechopen.87088

[def: mode choice "Car" and trip purpose "Shopping"]

[def: 1 for users achieved this educational attainment]

[def: respondent's gender (0 = female; 1 = male)]

[def: respondent's gender (0 = female; 1 = male)]

[def: mode choice "Car" and trip purpose "Personal Services"]

As described in Section 3.2, the considered choice set is composed of two alternatives: the alternative A representing the respondent's willingness to buy an electric vehicle and the alternative B corresponding to the willingness of buying a conventional (diesel) vehicle. The provided results are referred to an ordered model specification. In the following the estimation results are presented, distinguishing the choice of

Attributes Attribute coefficients (betas)

(�29.80)

(+16.37)

�0.448 (�2.52)

+0.550 (+3.88)

+0.128 (+2.22)

(�42.17)

(+6.96)

�0.0955 (�2.37)

�0.278 (�6.65)

(+27.18)

�1.06 (�15.60)

�1.12 (�16.74)

γ<sup>1</sup> �2.32

Approaches for Modelling User's Acceptance of Innovative Transportation…

ω<sup>1</sup> +0.787

γ<sup>2</sup> �3.39

ω<sup>2</sup> +0.299

γ<sup>3</sup> ω<sup>3</sup> +1.34

The attribute coefficients in the utility choice function are analytically specified

SE <sup>þ</sup> <sup>β</sup>LVLV<sup>i</sup> <sup>þ</sup> <sup>ε</sup><sup>i</sup> (18)

utility function, the structural equations and the measurement equations.

<sup>U</sup><sup>i</sup> <sup>¼</sup> <sup>β</sup>xX<sup>i</sup> <sup>þ</sup> <sup>β</sup>SEX<sup>i</sup>

where each perception indicator is a function of an intercept value αp,k, a coefficient λp,k associated with the latent variable and an error term ν<sup>i</sup> p,k assumed normally distributed with zero mean and σνpk standard deviation.

Table 13 shows the coefficients for each perception indicator, which were specified in accordance with the preliminary analyses (not shown for the sake of brevity). In particular, the first latent variable about fuel consumption (LV1) is described by two indicators <Icons0 > and < Icons2>. The second latent variable, about vehicle design (LV2), is described by four perception indicators, <Idesign0>, <Idesign1>, <Idesign3 > and < Idesign4>. Finally, the last latent variable representing the attitudes towards the environment (LV3) is described by four indicators < Ienv0>, < Ienv3>, < Ienv4 > and < Ienv6 > .

In general, the estimation results underline the necessity to introduce two different kinds of questions: direct and indirect questions.

Approaches for Modelling User's Acceptance of Innovative Transportation… DOI: http://dx.doi.org/10.5772/intechopen.87088


#### Table 12.

(see Section 5.1). However, in order to account for the symmetry of the indicators,

The coefficients in the structural model are analytically represented by the

βSE,jX<sup>i</sup>

This equation shows that each latent variable is a function of an intercept value

The estimation results displayed in below refers to the significant latent variables of the model, representing the attitude towards the fuel consumption (LV1),

Finally, with regard to the measurement model depending on the perception

p,k <sup>¼</sup> <sup>α</sup>p,k <sup>þ</sup> <sup>λ</sup>p,kLV<sup>i</sup>

where each perception indicator is a function of an intercept value αp,k, a

Table 13 shows the coefficients for each perception indicator, which were specified in accordance with the preliminary analyses (not shown for the sake of brev-

In general, the estimation results underline the necessity to introduce two dif-

SE,j <sup>þ</sup> <sup>ω</sup><sup>i</sup>

<sup>p</sup> <sup>þ</sup> <sup>ν</sup><sup>i</sup>

<sup>p</sup> (16)

(+ 38.90)

(+ 43.85)

(+ 46.11)

p,k (17)

SE,j of the

p,k assumed

<sup>p</sup> nor-

three positive parameters δ1, δ<sup>2</sup> and δ<sup>3</sup> are actually required (Table 11).

DELTA1 + 1.46

DELTA2 + 1.34

DELTA3 + 1.52

<sup>p</sup> ¼ γ<sup>p</sup> þ ∑<sup>j</sup>

γ<sup>p</sup> of beta-coefficients βSE,j for each of the socioeconomic attributes X<sup>i</sup>

respondents that influence the latent variable and contains an error term ω<sup>i</sup>

LV<sup>i</sup>

Delta values of the calibrated measurement equations. HySolarKit case study.

mally distributed with zero mean and σω<sup>p</sup> standard deviations.

the vehicle design (LV2) and the environment (LV3) (Table 12).

indicators, they are analytically represented by the following equation:

coefficient λp,k associated with the latent variable and an error term ν<sup>i</sup>

ity). In particular, the first latent variable about fuel consumption (LV1) is described by two indicators <Icons0 > and < Icons2>. The second latent variable, about vehicle design (LV2), is described by four perception indicators, <Idesign0>, <Idesign1>, <Idesign3 > and < Idesign4>. Finally, the last latent variable representing the attitudes towards the environment (LV3) is described by four indicators

normally distributed with zero mean and σνpk standard deviation.

I i

5.2.2 Parameters of the structural model

Transportation Systems Analysis and Assessment

5.2.3 Parameters of the measurement model

< Ienv0>, < Ienv3>, < Ienv4 > and < Ienv6 > .

34

ferent kinds of questions: direct and indirect questions.

following equation:

\*in parenthesis the t-test values.

Table 11.

Coefficients of the calibrated structural model. HySolarKit case study.

#### 5.3 Example 2: the electric vehicle case study

As described in Section 3.2, the considered choice set is composed of two alternatives: the alternative A representing the respondent's willingness to buy an electric vehicle and the alternative B corresponding to the willingness of buying a conventional (diesel) vehicle. The provided results are referred to an ordered model specification.

In the following the estimation results are presented, distinguishing the choice of utility function, the structural equations and the measurement equations.

#### 5.3.1 Parameters of the choice utility function

The attribute coefficients in the utility choice function are analytically specified by the following equation:

$$\mathbf{U}^{i} = \beta\_{\mathbf{x}} \mathbf{X}^{i} + \beta\_{\text{SE}} \mathbf{X}\_{\text{SE}}^{i} + \beta\_{LV} \mathbf{L} \mathbf{V}^{i} + \varepsilon^{i} \tag{18}$$


#### Table 13.

Coefficients of the calibrated measurement model. HySolarKit case study.

The results are shown in Table 14. In particular, the following attitudes were statistically significant: the attitudes towards the environment (LV1) and the perception of the advantages of EVs (LV2).

The coefficients related to the parameters in the measurement equation for an ordinal specification are estimated in the considered model. As the measurements are using a Likert scale with five levels, four parameters τ<sup>i</sup> are needed. However, in order to account for the symmetry of the indicators, two positive parameters δ<sup>1</sup> and

Attributes Attribute coefficients

[def: Variation in monthly cost [EUR] between an electric car and a conventional

Approaches for Modelling User's Acceptance of Innovative Transportation…

[def: The vehicle technical features significantly influence my choice in purchasing

LV\_1 +2.10

LV\_2 +0.435

Number of respondents 1462 Number of observations 1462 Init-log-likelihood<sup>1</sup> 1013.38 Final log-likelihood 385.55 Rho-square 0.620

DELTA1 0.531

DELTA2 1.27

Only the log-likelihood associated with the discrete choice component is considered

Delta values of the calibrated measurement equations. EV case study.

Attribute coefficients of the choice model. EV case study.

[def: Compared to a normal car, EV are inferior in terms of performances

[def: 1 if the respondent has at least 1 car in the household]

DOI: http://dx.doi.org/10.5772/intechopen.87088

(betas)

(+11.86)

(+1.57)

(+30.25)

(+38.94)

BUY - EV BUY - CV

+0.118 (+17.65)

+0.865 (+2.02)

+0.451 (+1.67)

+0.778 (+2.25)

The coefficients in the structural model are analytically represented by the

δ<sup>2</sup> are actually required (Table 15).

VAR\_monthly\_cost\_abs

a new car (5 = strongly agree)]

one]

SE\_AutoSI

F\_tech\_fea [5]

DIS\_red\_fea [5]

Statistics

1

Table 14.

Table 15.

(5 = strongly agree)]

\*in parenthesis the t-test values.

\*in parenthesis the t-test values.

5.3.2 Parameters of the structural model

following equation:

37

### Approaches for Modelling User's Acceptance of Innovative Transportation… DOI: http://dx.doi.org/10.5772/intechopen.87088


### Table 14.

Attribute coefficients of the choice model. EV case study.


Table 15.

Delta values of the calibrated measurement equations. EV case study.

The coefficients related to the parameters in the measurement equation for an ordinal specification are estimated in the considered model. As the measurements are using a Likert scale with five levels, four parameters τ<sup>i</sup> are needed. However, in order to account for the symmetry of the indicators, two positive parameters δ<sup>1</sup> and δ<sup>2</sup> are actually required (Table 15).
