4.1 Cronbach's alpha

The Cronbach's alpha, α (or coefficient alpha), is a measure of the internal consistency (or reliability) of the responses to multiple questions that are meant to measure a specific latent variable in a survey using a Likert scale. This indicator aims to tell whether the survey was accurately designed, and the questions were not answered randomly.

For each latent variable, it is necessary to have at least two indirect questions. The higher the number of questions, the better the latent variable would be measured. These questions, if possible, should be a mix of "+keyed" and "-keyed", depending if each one is positively correlated to the latent variable or negatively (Table 6).

When a survey intends to measure more than one latent variable, it is recommended to alternate their questions. This strategy combined to the mix of "+ keyed" and " keyed" is employed in order to encourage respondents to be more aware of each item and the response provided and, therefore, increases the probability of gathering valid responding. If not, respondents may realize which latent


#### Table 6.

Scores for a five-point Likert scale.

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

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

$$a = \frac{K}{K-1} \left( 1 - \frac{\sum\_{i=1}^{K} \sigma\_{Y\_i}^2}{\sigma\_X^2} \right) \tag{1}$$

considered independent from each other) and error terms (i.e. εi). Mathematically,

l<sup>11</sup> ⋯ l1<sup>j</sup> ⋯ l1<sup>k</sup> ⋮⋱⋮ ⋮ ⋮ li<sup>1</sup> ⋯ lij ⋯ lik ⋮⋯⋮⋱⋮ lp<sup>1</sup> ⋯ lpj ⋯ lpk

� �<sup>T</sup> is the vector of p observable random variables.

is the vector of the mean values of Y.

� �<sup>T</sup> is a vector of unobserved stochastic error terms, with zero

� �<sup>T</sup> is a vector of k unobserved random variables, called

Yi � μ<sup>i</sup> ¼ li1F<sup>1</sup> þ … þ likFk þ ε<sup>i</sup> (2)

Y � μ ¼ L F þ ε (3)

F1 ⋮ Fj ⋮ Fk

ε1 ⋮ εi ⋮ εp

i .

(5)

(4)

is a matrix of unknown constants, called

that can be expressed as

In matrix terms

where

L ¼

Y1 ⋮ Yi ⋮ Yp

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

μ1 ⋮ μi ⋮ μp

"common factors" as they influence all the observed Yi.

mean and finite variance, that can assume different values for each i.

� � <sup>¼</sup> 0 and Var Fj

• Cov Fð Þ¼ I so the factors are uncorrelated.

F is defined as factors, and L is the loading matrix.

2

Var Yð Þ¼<sup>i</sup> li<sup>1</sup>

by the common factors F1, …, Fj, …, Fk and shared with other variables.

Var Yð Þ¼<sup>i</sup> li<sup>1</sup>

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> lik

• <sup>ε</sup><sup>i</sup> are independent from one another, and <sup>E</sup>ð Þ¼ <sup>ε</sup><sup>i</sup> 0 and Varð Þ¼ <sup>ε</sup><sup>i</sup> <sup>σ</sup><sup>2</sup>

• F are independent from one another, as there is no relationship between factors, and are also independent from the error terms. They are also

� � <sup>¼</sup> 1.

• k≤ p: the number of observed variables Yi is larger or equal to the number of

Any solution for the unknown values lij of Eq. (2) or (3) with the constraints for

<sup>2</sup> <sup>þ</sup> … <sup>þ</sup> lik

2

Var Fð Þþ <sup>k</sup> 12

<sup>2</sup> <sup>þ</sup> <sup>σ</sup><sup>2</sup> i

<sup>2</sup> is the communality of the variance: the part that is explained

Varð Þ ε<sup>i</sup>

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

Var Fð Þþ <sup>1</sup> … þ lik

Approaches for Modelling User's Acceptance of Innovative Transportation…

Y ¼ Y1; …; Yi; …; Yp

μ ¼ μ1; …; μi; …; μ<sup>p</sup> h i<sup>T</sup>

F ¼ F1; …; Fj; …; Fk

ε ¼ ε1; …; εi; …; ε<sup>p</sup>

standardized to E Fj

common factors Fj.

where li<sup>1</sup>

27

Assuming that:

l<sup>11</sup> ⋯ l1<sup>j</sup> ⋯ l1<sup>k</sup> ⋮⋱⋮ ⋮ ⋮ li<sup>1</sup> ⋯ lij ⋯ lik ⋮⋯⋮⋱⋮ lp<sup>1</sup> ⋯ lpj ⋯ lpk

"loadings" that have to be calculated.

where:

K is the number of items (questions).

σ2 <sup>X</sup> is the variance of the observed total scores.

σ2 Yi is the variance of the scores of each item i.

A rule commonly accepted to interpret the values of Cronbach's alpha when used with a Likert scale is (Table 7):
