IntervalQuestionStat (version 0.2.0)

cronbach: Estimate sample Cronbach's \(\alpha\) coefficient

Description

This function allows to calculate the sample Cronbach's \(\alpha\) coefficient as an estimate of the reliability (understood in this case under the internal consistency point of view) of the responses collected through Likert-type, visual analogue, and interval-valued rating scales in questionnaires.

Usage

cronbach(data, ivs = TRUE, type = 1, theta = 1)

Value

This function returns the calculated sample Cronbach's \(\alpha\) coefficient stored as a single numeric

value for measuring the reliability or internal consistency of the given questionnaire's responses.

Arguments

data

A matrix or data.frame with the questionnaire's responses.

ivs

A logical value indicating if an interval-valued scale is used (default) or not.

type

A single numeric value specifying both the order and the characterization that is being used for storing interval-valued information on data argument. Only the following four options are allowed:

• 1: The inf/sup-characterization is used variable by variable (default).

• 2: The mid/spr-characterization is used variable by variable.

• 3: All the supremums follow all the infimums in the same variable order.

• 4: All the spreads follow all the mid-points in the same variable order.

theta

A single positive real number stored as a unique numeric value which is used for distance computations. By default, theta = 1.

Author

José García-García

Details

For both traditional Likert-type and visual analogue rating scales responses, the sample Cronbach's \(\alpha\) coefficient (Cronbach, 1951) computed by cronbach() function is defined as follows, $$\widehat{\alpha} = \frac{k}{k - 1}\left(1 - \frac{\sum_{j=1}^{k} s_{X_{j}}^2}{s_{X_{total}}^2}\right),$$ where \(k>1\) is the number of items; \(s_{X_{j}}^2\) is the sample variance of \(X_{j}\), which is the real-valued random variable modeling the responses to the \(j\)-th item; and \(s_{X_{total}}^2\) is the sample variance of the sum of all the involved items, that is, $$X_{total}=X_{1}+X_{2}+\ldots +X_{k}.$$

Analogously, for interval-valued scale responses the sample Cronbach's \(\alpha\) coefficient computed by this function is defined as follows, $$\widehat{\alpha} = \frac{k}{k - 1}\left(1 - \frac{\sum_{j=1}^{k} s_{\mathcal{X}_{j}}^2}{s_{\mathcal{X}_{total}}^2}\right),$$ where \(k>1\) is the number of items; \(s_{\mathcal{X}_{j}}^2\) is the sample Fréchet variance of \(\mathcal{X}_{j}\), which is the interval-valued random set modeling the responses to the \(j\)-th item; and \(s_{\mathcal{X}_{total}}^2\) is the sample Fréchet variance of the sum of all the involved items, that is, $$\mathcal{X}_{total} = \mathcal{X}_{1}+\mathcal{X}_{2}+\ldots +\mathcal{X}_{k}.$$

References

Cronbach L.J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334. tools:::Rd_expr_doi("1001007/BF02310555").

Examples

## These code lines illustrate Cronbach's alpha coefficient calculation
## for interval-valued, Likert-type, and visual analogue scales responses

## Some trivial cronbach() examples
## Cronbach's alpha index for interval-valued scale responses stored
## in a matrix with the inf/sup-characterization variable by variable
## using Bertoluzza's distance with Lebesgue measure (theta = 1/3)
data1 <- matrix(c(1, 2.6, 1.5, 3, 3.8, 6, 4, 7), 2, 4)
cronbach(data1, theta = 1/3)

## Cronbach's alpha index for interval-valued scale responses stored
## in a data.frame with the mid/spr-characterization saving all the
## mid-points and then all the spreads using rho2 distance (theta = 1)
data2 <- data.frame(mids1 = c(2, 3),
                    mids2 = c(4, 5),
                    sprs1 = c(1, 2),
                    sprs2 = c(2, 4))
cronbach(data2, type = 4)

## Cronbach's alpha coefficient for Likert-type
## scale responses stored in a matrix
data3 <- matrix(c(1, 3, 4, 7), 2, 2)
cronbach(data3, ivs = FALSE)

## Cronbach's alpha coefficient for visual analogue
## scale responses stored in a data.frame
data4 <- data.frame(item1 = c(1.5, 2.8),
                    item2 = c(3.9, 6.2))
cronbach(data4, ivs = FALSE)

## Real-life data example
## Load the interval-valued data
data(lackinfo, package = "IntervalQuestionStat")

## Calculate Cronbach's alpha coefficient for interval-valued responses
cronbach(lackinfo[, 3:12])

## Convert interval-valued responses into their corresponding equivalent
## Likert-type answers and then calculate Cronbach's alpha coefficient
likert <- ivs2likert(IntervalMatrix(lackinfo[, 3:12]))
cronbach(likert, ivs = FALSE)

## Analogously, interval-valued responses are transformed into their
## corresponding equivalent visual analogue scale answers and
## Cronbach's alpha coefficient is then computed
vas <- ivs2vas(IntervalMatrix(lackinfo[, 3:12]))
cronbach(vas, ivs = FALSE)