4.2 Mean and standard deviation

An easy quantitative/qualitative analysis to perform on the data collected consists of calculating the mean value and the standard deviation of the responses to the Likert scale for each question and then evaluating whether the responses are consistent (they have a low deviation) or if the mean value corresponds to the one expected.

#### 4.3 Factor analysis

Given Yi is an observable random variable, with mean μi, with i ¼ 1, …, p where p is the number of observed variables (the number of indirect questions), the factor analysis is a statistical method to investigate if each observed variable can be reduced to a linear combination of k unobservable factors (i.e. Fk, latent variables,


Table 7. Scales of values for the Cronbach's alpha. Approaches for Modelling User's Acceptance of Innovative Transportation… DOI: http://dx.doi.org/10.5772/intechopen.87088

considered independent from each other) and error terms (i.e. εi). Mathematically, that can be expressed as

$$Y\_i - \mu\_i = l\_{i1} F\_1 + \dots + l\_{ik} F\_k + \varepsilon\_i \tag{2}$$

In matrix terms

$$Y - \mu = L\,F + \varepsilon \tag{3}$$

$$
\begin{bmatrix} Y\_1 \\ \vdots \\ Y\_i \\ \vdots \\ Y\_p \end{bmatrix} - \begin{bmatrix} \mu\_1 \\ \vdots \\ \mu\_i \\ \vdots \\ \mu\_p \end{bmatrix} = \begin{bmatrix} l\_{11} & \cdots & l\_{1j} & \cdots & l\_{1k} \\ \vdots & \ddots & \vdots & \vdots & \vdots \\ l\_{i1} & \cdots & l\_{ij} & \cdots & l\_{ik} \\ \vdots & \cdots & \vdots & \ddots & \vdots \\ l\_{p1} & \cdots & l\_{pj} & \cdots & l\_{pk} \end{bmatrix} \begin{bmatrix} F\_1 \\ \vdots \\ F\_j \\ \vdots \\ F\_k \end{bmatrix} + \begin{bmatrix} e\_1 \\ \vdots \\ e\_i \\ \vdots \\ e\_p \end{bmatrix} \tag{4}
$$

where

variable is being measured, and they might tend to answer with the same response if the questions are all equally keyed. In this case, the total variance will be lower,

+ keyed 1 2 3 4 5 � keyed 5 4 3 2 1

The Cronbach's alpha is calculated for each group of questions that measure a specific latent variable. Given X ¼ Y<sup>1</sup> þ Y<sup>2</sup> þ … þ YK the sum of the scores of the K

<sup>1</sup> � <sup>∑</sup><sup>K</sup>

A rule commonly accepted to interpret the values of Cronbach's alpha when used

An easy quantitative/qualitative analysis to perform on the data collected consists of calculating the mean value and the standard deviation of the responses to the Likert scale for each question and then evaluating whether the responses are consistent (they have a low deviation) or if the mean value corresponds to the one

Given Yi is an observable random variable, with mean μi, with i ¼ 1, …, p where p is the number of observed variables (the number of indirect questions), the factor analysis is a statistical method to investigate if each observed variable can be reduced to a linear combination of k unobservable factors (i.e. Fk, latent variables,

Cronbach's alpha Internal consistency 0.9 ≤ α Excellent 0.8 ≤ α < 0.9 Good 0.7 ≤ α < 0.8 Acceptable 0.6 ≤ α < 0.7 Questionable 0.5 ≤ α < 0.6 Poor α < 0.5 Unacceptable

!

Neither inaccurate nor accurate

Moderately accurate

Very accurate

<sup>i</sup>¼<sup>1</sup>σ<sup>2</sup> Yi σ2 X

(1)

and the relation with other variables in the study will be underestimated.

questions for each respondent, Cronbach's alpha is obtained as

Moderately inaccurate

K is the number of items (questions).

with a Likert scale is (Table 7):

4.2 Mean and standard deviation

<sup>X</sup> is the variance of the observed total scores.

Yi is the variance of the scores of each item i.

where:

Scoring Very

inaccurate

Transportation Systems Analysis and Assessment

Scores for a five-point Likert scale.

σ2

Table 6.

σ2

expected.

Table 7.

26

Scales of values for the Cronbach's alpha.

4.3 Factor analysis

<sup>α</sup> <sup>¼</sup> <sup>K</sup> K � 1

Y ¼ Y1; …; Yi; …; Yp � �<sup>T</sup> is the vector of p observable random variables. μ ¼ μ1; …; μi; …; μ<sup>p</sup> h i<sup>T</sup> is the vector of the mean values of Y. F ¼ F1; …; Fj; …; Fk � �<sup>T</sup> is a vector of k unobserved random variables, called "common factors" as they influence all the observed Yi.

$$L = \begin{bmatrix} l\_{11} & \cdots & l\_{1j} & \cdots & l\_{1k} \\ \vdots & \ddots & \vdots & \vdots & \vdots \\ l\_{i1} & \cdots & l\_{ij} & \cdots & l\_{ik} \\ \vdots & \cdots & \vdots & \ddots & \vdots \\ l\_{p1} & \cdots & l\_{pj} & \cdots & l\_{pk} \end{bmatrix} \\ \text{is a matrix of unknown constants, called } l\_{i1} \text{ and } l\_{i2} \text{ in the } i \text{th row of } l\_{i1} \text{ in the } i \text{th row of } l\_{i2} \text{ in the } i \text{th row of } l\_{i2} \text{ in the } i \text{th row of } l\_{i2} \text{ in the } i \text{th row of } l\_{i1} \text{ in the } i \text{th row of } l\_{i2} \text{ in the } i \text{th row of } l\_{i1} \text{ in the } i \text{th row of } l\_{i2} \text{ in the } i \text{th}\_{i1}$$

"loadings" that have to be calculated.

ε ¼ ε1; …; εi; …; ε<sup>p</sup> � �<sup>T</sup> is a vector of unobserved stochastic error terms, with zero mean and finite variance, that can assume different values for each i.

Assuming that:


Any solution for the unknown values lij of Eq. (2) or (3) with the constraints for F is defined as factors, and L is the loading matrix.

With these assumptions, the variance of Yi in (2) can be calculated as

$$\begin{aligned} Var(Y\_i) &= l\_{i1}^{\;2} Var(F\_1) + \ldots + l\_{ik}^{\;2} Var(F\_k) + \mathbf{1}^2 Var(\varepsilon\_i) \\ Var(Y\_i) &= l\_{i1}^{\;2} + \ldots + l\_{ik}^{\;2} + \sigma\_i^2 \end{aligned} \tag{5}$$

where li<sup>1</sup> <sup>2</sup> <sup>þ</sup> … <sup>þ</sup> lik <sup>2</sup> is the communality of the variance: the part that is explained by the common factors F1, …, Fj, …, Fk and shared with other variables.

σ2 <sup>i</sup> is the specific variance: the part of the variance of Yi that is not considered in the common factors. This value would be equal to 0 if the common factors were perfect predictors of the observed variables.

Given two variables, Ym and Yn,

$$\begin{aligned} Y\_m &= \mu\_m + l\_{m1}F\_1 + \dots + l\_{mk}F\_k + \varepsilon\_m\\ Y\_n &= \mu\_n + l\_{n1}F\_1 + \dots + l\_{nk}F\_k + \varepsilon\_n \end{aligned} \tag{6}$$

The loading values may be hard to interpret at a first glance. So, in a second step of the analysis, the loadings obtained can be "rotated" in order to arrive at another set of loadings, which renders the values more understandable, while fitting the observed variances and covariances equally. The effect of rotating the factors produces that each variable loads more strongly only on one of the factors and weakly

There are several rotation methods that provide different solutions, arising to different interpretation. The interpretation of each factor and the number of factors needed are very subjective, and the researcher has the task to identify what is the meaning of each factor (i.e. which is the unknown latent variable hidden in the indicators).

From a general point of view, the rotation methods can be subdivided in orthogonal (when the factors cannot correlate) or oblique (the factors are allowed to correlate). The most common methods within each one of two groups are listed below:

◦ Varimax: it aims to minimize the complexity of each factor by relating them to few variables while discouraging the detection of factors that influence all the variables. It produces the increase of the strongest loading

◦ Quartimax: it aims to find a general factor (or a reduced amount of them), on which most variables are loaded to, while minimizing the number of factors needed to explain each variable. This is done by increasing the strongest loading values while decreasing the weaker ones in each variable. This factor structure is usually not helpful to the research

◦ Equimax: it is a method that attempts to simplify both factors and

◦ Direct oblimin is the standard method when the factors are allowed to be correlated, resulting in higher eigenvalues, but the interpretability of the

◦ Promax is an alternative to the previous one, used for large dataset as it is

An overview of the preliminary statistical analysis for the electric vehicle case study<sup>1</sup> introduced in Section 3.2 is provided. The proposed example includes analysis of Cronbach's alpha, mean and standard deviations, factor analysis with principal components as the extraction method and rotated component matrix using the

First of all, it may be observed that Cronbach's alpha is not consistent for all

<sup>1</sup> For the sake of brevity only, the results which are related to one case study, the electric vehicle case

answers; therefore, the survey reliability is confirmed only in the case of

values while decreasing the weaker ones in each factor.

on the other factors, producing the eigenvalues to vary.

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

Approaches for Modelling User's Acceptance of Innovative Transportation…

• Orthogonal methods

purpose.

variables.

factors may be reduced.

computationally more efficient.

4.4 Example 1: the electric vehicle case study

• Oblique methods

Varimax method.

study, are displayed.

29

The covariances can be calculated as

$$\begin{aligned} Cov(Y\_m, Y\_n) &= l\_{m1} l\_{n1} Var(F\_1) + \dots + l\_{mk} l\_{nk} Var(F\_k) + (1)(0)\varepsilon\_m + (0)(1)\varepsilon\_n \\ Cov(Y\_m, Y\_n) &= l\_{m1} l\_{n1} + \dots + l\_{mk} l\_{nk} \end{aligned} \tag{7}$$

This shows that the covariance of two variables is equal to the scalar product of their loadings.

With the expressions in Eqs. (5) and (7), it is possible to construct a theoretical variance–covariance matrix, implied by the model's assumptions. Then, with the data collected in the survey, an observed variance and covariance matrix can be calculated and constructed. If the model's assumptions are correct, it is possible to estimate the loadings lij in order to obtain a theoretical matrix closer to the observed one.

To extract the first set of loadings and factors from the observed variables, there are different methods. However, principal component analysis (PCA) and common factor analysis (CFA) are the most preferred and most used:

• Principal component method, or component factor analysis, determines the loadings lij that allows a close estimation of the total communality to the sum of the observed variances, while ignoring the covariances. The principal components are chosen by extracting the maximum variance and putting it in the first factor, gathering as much of the variation in the data as possible. Then, the variance explained by the first factor is removed and then extracts the maximum variance for the second factor; repeating the process until the last factor.

This model can be written as

$$C\_1 = l\_{11}Y\_1 + l\_{21}Y\_2 + l\_{31}Y\_3 \tag{8}$$

• Common factor analysis: the factors are linear combinations that maximize the common portion of the variance and put them into factors, underlying latent constructs. This method does not include the specific part of the variance to determine the factor, and it is used for structural equation modeling.

This model can be set up as

$$\begin{aligned} Y\_1 &= l\_{11}F\_1 + \varepsilon\_1\\ Y\_2 &= l\_{21}F\_1 + \varepsilon\_2\\ Y\_3 &= l\_{31}F\_1 + \varepsilon\_3 \end{aligned} \tag{9}$$

Once the factors are extracted, their eigenvalues (or characteristic roots) provides the amount of variance explained by every factor out of the total variance. Then, the number of factors is reduced by retaining only those which have an eigenvalue larger than 1, according to Kaiser's criterion [29].

The factor loadings obtained represent the amount of variance explained by the variable on every factor. In structural equation modeling, a value of 0.7 or higher represents that the factor extracts sufficient variance from that particular variable.

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

The loading values may be hard to interpret at a first glance. So, in a second step of the analysis, the loadings obtained can be "rotated" in order to arrive at another set of loadings, which renders the values more understandable, while fitting the observed variances and covariances equally. The effect of rotating the factors produces that each variable loads more strongly only on one of the factors and weakly on the other factors, producing the eigenvalues to vary.

There are several rotation methods that provide different solutions, arising to different interpretation. The interpretation of each factor and the number of factors needed are very subjective, and the researcher has the task to identify what is the meaning of each factor (i.e. which is the unknown latent variable hidden in the indicators).

From a general point of view, the rotation methods can be subdivided in orthogonal (when the factors cannot correlate) or oblique (the factors are allowed to correlate). The most common methods within each one of two groups are listed below:

• Orthogonal methods

σ2

their loadings.

This model can be written as

This model can be set up as

28

perfect predictors of the observed variables. Given two variables, Ym and Yn,

Transportation Systems Analysis and Assessment

The covariances can be calculated as

Cov Yð Þ¼ <sup>m</sup>; Yn lm1ln<sup>1</sup> þ … þ lmklnk

<sup>i</sup> is the specific variance: the part of the variance of Yi that is not considered in the common factors. This value would be equal to 0 if the common factors were

> Ym ¼ μ<sup>m</sup> þ lm1F<sup>1</sup> þ … þ lmkFk þ ε<sup>m</sup> Yn ¼ μ<sup>n</sup> þ ln1F<sup>1</sup> þ … þ lnkFk þ ε<sup>n</sup>

Cov Yð Þ¼ <sup>m</sup>; Yn lm1ln1Var Fð Þþ <sup>1</sup> … þ lmklnkVar Fð Þþ <sup>k</sup> ð Þ1 ð Þ 0 ε<sup>m</sup> þ ð Þ 0 ð Þ1 ε<sup>n</sup>

This shows that the covariance of two variables is equal to the scalar product of

To extract the first set of loadings and factors from the observed variables, there are different methods. However, principal component analysis (PCA) and common

loadings lij that allows a close estimation of the total communality to the sum of

components are chosen by extracting the maximum variance and putting it in the first factor, gathering as much of the variation in the data as possible. Then, the variance explained by the first factor is removed and then extracts the maximum

C<sup>1</sup> ¼ l11Y<sup>1</sup> þ l21Y<sup>2</sup> þ l31Y<sup>3</sup> (8)

With the expressions in Eqs. (5) and (7), it is possible to construct a theoretical variance–covariance matrix, implied by the model's assumptions. Then, with the data collected in the survey, an observed variance and covariance matrix can be calculated and constructed. If the model's assumptions are correct, it is possible to estimate the

loadings lij in order to obtain a theoretical matrix closer to the observed one.

• Principal component method, or component factor analysis, determines the

variance for the second factor; repeating the process until the last factor.

• Common factor analysis: the factors are linear combinations that maximize the common portion of the variance and put them into factors, underlying latent constructs. This method does not include the specific part of the variance to determine the factor, and it is used for structural equation modeling.

> Y<sup>1</sup> ¼ l11F<sup>1</sup> þ ε<sup>1</sup> Y<sup>2</sup> ¼ l21F<sup>1</sup> þ ε<sup>2</sup> Y<sup>3</sup> ¼ l31F<sup>1</sup> þ ε<sup>3</sup>

Once the factors are extracted, their eigenvalues (or characteristic roots) provides the amount of variance explained by every factor out of the total variance. Then, the number of factors is reduced by retaining only those which have an

The factor loadings obtained represent the amount of variance explained by the variable on every factor. In structural equation modeling, a value of 0.7 or higher represents that the factor extracts sufficient variance from that particular variable.

eigenvalue larger than 1, according to Kaiser's criterion [29].

the observed variances, while ignoring the covariances. The principal

factor analysis (CFA) are the most preferred and most used:

(6)

(7)

(9)

	- Direct oblimin is the standard method when the factors are allowed to be correlated, resulting in higher eigenvalues, but the interpretability of the factors may be reduced.
	- Promax is an alternative to the previous one, used for large dataset as it is computationally more efficient.

#### 4.4 Example 1: the electric vehicle case study

An overview of the preliminary statistical analysis for the electric vehicle case study<sup>1</sup> introduced in Section 3.2 is provided. The proposed example includes analysis of Cronbach's alpha, mean and standard deviations, factor analysis with principal components as the extraction method and rotated component matrix using the Varimax method.

First of all, it may be observed that Cronbach's alpha is not consistent for all answers; therefore, the survey reliability is confirmed only in the case of

<sup>1</sup> For the sake of brevity only, the results which are related to one case study, the electric vehicle case study, are displayed.

psychological statements referred to the environment, the technical features and the EV's advantages (i.e. Cronbach's alpha is higher than 0.5). Among them, a further analysis is provided in terms of mean and standard deviations: with respect to the attitude about the environment, for both statements the mean is higher than 3, and the standard deviation is lower than 1; regarding the attitude about technical features, higher values of mean (higher than 3) and lower values (lower than 1) of standard deviations are observed for the following statements (refer to Section 3.2 for the meaning of the following variables): <F\_tech\_fea>, <TF\_power > and < TF\_range>. Finally, the perceptions of EV's advantages show mean values higher than 3 and standard deviations lower than 1 only for the following statements: <ADRed\_CO2>, <ADEfficiency > and < ADRed\_poll>. However, even though the Cronbach's alpha is not satisfying for the perceptions of disadvantages of EVs, the mean values and the standard deviations for all statements (<DISinfr>, <DISred\_fea>, <DISbatt\_range>) highlight their relevance on users' behaviour. All significant results are in bold in Table 8.

The factor analysis carried out through the principal component analysis extraction method, allowed to identify the latent factors correlated to the psychological statements. The components extracted were also rotated using the Varimax

Variables Components Rotated components

Approaches for Modelling User's Acceptance of Innovative Transportation…

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

AD\_red\_CO2 0.721 �0.306 0.773 0.125 AD\_red\_poll 0.601 �0.018 0.751 �0.156 AD\_efficiency 0.553 �0.532 0.519 0.305 F\_consumption 0.548 0.518 �0.068 0.768 F\_pollution 0.352 0.686 0.188 0.731

1 21 2

The results (significant values are highlighted in bold in Table 9) underline the correlations among the following statements referred to the advantage perceptions <ADRed\_CO>, <ADEfficiency > and < ADRed\_poll > and all statements regarding the

environmental attitude, <F\_consumption > and < F\_pollution > .

5.1 Choice function and structural and 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>

duced in order to specify the perception indicators.

LV<sup>i</sup>

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

latent variable may be expressed as follows:

zero mean and σω,p is the standard deviation.

i

where γ<sup>p</sup> is the intersect, X<sup>i</sup>

measurement equation as follows:

(to be estimated), ω<sup>i</sup>

31

Furthermore, let I

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

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 intro-

In particular, if p is the generic latent variable, the structural equation for each

butes, βSE,j is the vector of the coefficients associated with the users' characteristics

variable. Each perception indicator (i.e. vector component) may be specified by a

βSE,jX<sup>i</sup>

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

<sup>p</sup> is the error term which is usually normally distributed with

<sup>n</sup> be a vector of perceptions indicators associated to each latent

SE,j is the vector of the users' characteristics attri-

SE <sup>þ</sup> <sup>β</sup>LVLV<sup>i</sup> <sup>þ</sup> <sup>ε</sup><sup>i</sup> (10)

<sup>p</sup> (11)

method.

Table 9.

Factor analysis of the EV study.

Factor analysis

error term εi:

5. Mathematical formulation


Table 8.

Mean and standard deviations of the responses in the EV study.

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


Table 9.

psychological statements referred to the environment, the technical features and the EV's advantages (i.e. Cronbach's alpha is higher than 0.5). Among them, a further analysis is provided in terms of mean and standard deviations: with respect to the attitude about the environment, for both statements the mean is higher than 3, and the standard deviation is lower than 1; regarding the attitude about technical features, higher values of mean (higher than 3) and lower values (lower than 1) of standard deviations are observed for the following statements (refer to Section 3.2 for the meaning of the following variables): <F\_tech\_fea>, <TF\_power > and < TF\_range>. Finally, the perceptions of EV's advantages show mean values higher than 3 and standard deviations lower than 1 only for the following statements: <ADRed\_CO2>, <ADEfficiency > and < ADRed\_poll>. However, even though the Cronbach's alpha is not satisfying for the perceptions of disadvantages of EVs, the mean values and the standard deviations for all statements (<DISinfr>, <DISred\_fea>, <DISbatt\_range>) highlight their relevance on users' behaviour. All

F\_cons 4.32 0.71 F\_poll 3.13 0.98

F\_tech\_fea 3.68 0.85 TF\_power 3.51 0.84 TF\_top\_speed 2.98 0.85 TF\_accel 3.14 1.00 TF\_range 4.53 0.73

ADRed\_CO2 4.11 0.82 ADEfficiency 3.42 1.08 ADRed\_poll 3.62 0.97 ADLess\_parts 3.01 1.14

DISinfr 4.21 0.82 DISred\_fea 3.25 0.94 DISbatt\_range 4.05 0.87

mean SD

mean SD

Mean SD

Mean SD

significant results are in bold in Table 8.

Transportation Systems Analysis and Assessment

Attitudes about environment Cronbach's alpha = 0.551

Attitudes about technical features Cronbach's alpha = 0.656

Perceptions of advantages of EVs Cronbach's alpha = 0.549

Perceptions of disadvantages of EVs

Mean and standard deviations of the responses in the EV study.

Cronbach's alpha = 0.364

Table 8.

30

Factor analysis of the EV study.

The factor analysis carried out through the principal component analysis extraction method, allowed to identify the latent factors correlated to the psychological statements. The components extracted were also rotated using the Varimax method.

The results (significant values are highlighted in bold in Table 9) underline the correlations among the following statements referred to the advantage perceptions <ADRed\_CO>, <ADEfficiency > and < ADRed\_poll > and all statements regarding the environmental attitude, <F\_consumption > and < F\_pollution > .
