cronbach: Estimate sample Cronbach's alpha coefficient

In IntervalQuestionStat: Tools to Deal with Interval-Valued Responses in Questionnaires

Estimate sample Cronbach's α coefficient

Description

This function allows to calculate the sample Cronbach's α 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)

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.

Details

For both traditional Likert-type and visual analogue rating scales responses, the sample Cronbach's α coefficient (Cronbach, 1951) computed by cronbach() function is defined as follows,

\widehat{α} = \frac{k}{k - 1}≤ft(1 - \frac{∑_{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}+… +X_{k}.

Analogously, for interval-valued scale responses the sample Cronbach's α coefficient computed by this function is defined as follows,

\widehat{α} = \frac{k}{k - 1}≤ft(1 - \frac{∑_{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}+… +\mathcal{X}_{k}.

Value

This function returns the calculated sample Cronbach's α coefficient stored as a single numeric value for measuring the reliability or internal consistency of the given questionnaire's responses.

Author(s)

José García-García [e-mail]

References

Cronbach L.J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334. 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)

IntervalQuestionStat documentation built on Nov. 1, 2022, 5:06 p.m.