INTERNAL.CONSISTENCY {EFA.dimensions}R Documentation

Internal consistency reliability coefficients

Description

Internal consistency reliability coefficients

Usage

INTERNAL.CONSISTENCY(data, extraction = 'ML', reverse_these = NULL, 
	                        auto_reverse = TRUE, verbose=TRUE, factormodel)

Arguments

data

An all-numeric dataframe where the rows are cases & the columns are the variables.

extraction

(optional) The factor extraction method to be used in the omega computations. The options are: 'ML' for maximum likelihood (the default); and 'PAF' for principal axis / common factor analysis.

reverse_these

(optional) A vector of the names of items that should be reverse-coded

auto_reverse

(optional) Should reverse-coding of items be conducted when warranted? TRUE (default) or FALSE

verbose

(optional) Should detailed results be displayed in console? TRUE (default) or FALSE

factormodel

(Deprecated.) Use 'extraction' instead.

Details

When 'auto_reverse = TRUE', the item loadings on the first principal component are computed and items with negative loadings are reverse-coded.

If error messages are produced, try using 'auto_reverse = FALSE'.

If item names are provided for the 'reverse_these' argument, then auto_reverse is not conducted.

The following helpful descriptions of Cronbach's alpha and of omega total are direct quotes from McNeish (2017, pp. 414-417):

Cronbach's Alpha

"One can interpret the value of Cronbach's alpha in one of many different ways:

1. Cronbach's alpha is the correlation of the scale of interest with another scale of the same length that intends to measure the same construct, with different items, taken from the same hypothetical pool of items (Kline, 1986).

2. The square root of Cronbach's alpha is an estimate of the correlation between observed scores and true scores (Nunnally & Bernstein, 1994).

3. Cronbach's alpha is the proportion of the variance of the scale that can be attributed to a common source (DeVellis, 1991).

4. Cronbach's alpha is the average of all possible split-half reliabilities from the set of items (Pedhazur & Schmelkin, 1991). (It is important to note the correlation between the two parts is not the split half reliability, but is used to find the split half reliability found by the Spearman-Brown prophecy formula.)

Under certain assumptions, Cronbach's alpha is a consistent estimate of the population internal consistency; however, these assumptions are quite rigid and are precisely why methodologists have argued against the use of Cronbach's alpha.

The assumptions of Cronbach's alpha are:

1. The scale adheres to tau equivalence, i.e., that each item on a scale contributes equally to the total scale score. Tau equivalence tends to be unlikely for most scales that are used in empirical research some items strongly relate to the construct while some are more weakly related.

2. Scale items are on a continuous scale and normally distributed. Cronbach's alpha is largely based on the observed covariances (or correlations) between items. In most software implementations of Cronbach's alpha (such as in SAS and SPSS), these item covariances are calculated using a Pearson covariance matrix. A well-known assumption of Pearson covariance matrices is that all variables are continuous in nature. Otherwise, the elements of the matrix can be substantially biased downward. However, it is particularly common for psychological scales to contain items that are discrete (e.g., Likert or binary response scales), which violates this assumption. If discrete items are treated as continuous, the covariance estimates will be attenuated, which ultimately results in underestimation of Cronbach's alpha because the relations between items will appear smaller than they actually are. To accommodate items that are not on a continuous scale, the covariances between items can instead be estimated with a polychoric covariance (or correlation) matrix rather than with a Pearson covariance matrix. Polychoric covariance matrices assume that there is an underlying normal distribution to discrete responses.

3. The errors of the items do not covary. Correlated errors occur when sources other than the construct being measured cause item responses to be related to one another.

4. The scale is unidimensional. Though Cronbach's alpha is sometimes thought to be a measure of unidimensionality because its colloquial definition is that it measures how well items stick together, unidimensionality is an assumption that needs to be verified prior to calculating Cronbach's alpha rather than being the focus of what Cronbach's alpha measures. Internal consistency is necessary for unidimensionality but that internal consistency is not sufficient for demonstrating unidimensionality. That is, items that measure different things can still have a high degree of interrelatedness, so a large Cronbach's alpha value does not necessarily guarantee that the scale measures a single construct. As a result, violations of unidimensionality do not necessarily bias estimates of Cronbach's alpha. In the presence of a multidimensional scale, Cronbach's alpha may still estimate the interrelatedness of the items accurately and the interrelatedness of multidimensional items can in fact be quite high."

Omega total

"Omega total is an internal consistency coefficient that assumes that the scale is unidimensional. Omega estimates the reliability for the composite of items on the scale (which is conceptually similar to Cronbach's alpha). Under the assumption that the construct variance is constrained to 1 and that there are no error covariances, omega total is calculated from factor analysis output (loadings and error/uniqueness values). Tau equivalence is no longer assumed and the potentially differential contribution of each item to the scale must be assessed. Omega total is a more general version of Cronbach's alpha and actually subsumes Cronbach's alpha as a special case. More simply, if tau equivalence is met, omega total will yield the same result as Cronbach's alpha but omega total has the flexibility to accommodate congeneric scales, unlike Cronbach's alpha."

Root Mean Square Residual (rmsr)

rmsr is an index of the overall badness-of-fit. It is the square root of the mean of the squared residuals (the residuals being the simple differences between original correlations and the correlations implied by the N-factor model). rmsr is 0 when there is perfect model fit. A value less than .08 is generally considered a good fit. The rmsr coefficient is included in the internal consistency output as an index of the degree of fit of a one-factor model to the item data.

Standardized Cronbach's Alpha

Standardized alpha should be used when items have different scale ranges, e.g., some items are 1-to-7, and other items are 1-to-4, or 1-to-100. Regular alpha is based on covariances, whereas standardized alpha is based on correlations, wherein the items have identical standard deviations. Items in different metrics should be standardized before computing scale scores.

Value

A list with the following elements:

int.consist_scale

A vector with the scale omega, Cronbach's alpha, standardized Cronbach's alpha, the mean of the off-diagonal correlations, the median of the off-diagonal correlations, and the rmsr fit coefficient for a 1-factor model

int.consist_dropped

A matrix of the int.consist_scale values for when each item, in turn, is int.consist_dropped from the analyses

item_stats

The item means, standard deviations, and item-total correlations

resp_opt_freqs

The response option frequencies

resp_opt_props

The response option proportions

new_data

The data that was used for the analyses, including any item reverse-codings

Author(s)

Brian P. O'Connor

References

Flora, D. B. (2020). Your coefficient alpha is probably wrong, but which coefficient omega is right? A tutorial on using R to obtain better reliability estimates. Advances in Methods and Practices in Psychological Science, 3(4), 484501.

McNeish, D. (2018). Thanks coefficient alpha, we'll take it from here. Psychological Methods, 23(3), 412433.

Revelle, W., & Condon, D. M. (2019). Reliability from alpha to omega: A tutorial. Psychological Assessment, 31(12), 13951411.

Examples

# Rosenberg Self-Esteem scale items -- without reverse-coding
INTERNAL.CONSISTENCY(data_RSE, extraction = 'PAF', 
                     reverse_these = NULL, auto_reverse = FALSE, verbose=TRUE)

# Rosenberg Self-Esteem scale items -- with auto_reverse-coding
INTERNAL.CONSISTENCY(data_RSE, extraction = 'PAF',
                     reverse_these = NULL, auto_reverse = TRUE, verbose=TRUE)

# Rosenberg Self-Esteem scale items -- another way of reverse-coding
INTERNAL.CONSISTENCY(data_RSE, extraction = 'PAF',
                     reverse_these = c('Q1','Q2','Q4','Q6','Q7'), verbose=TRUE)

[Package EFA.dimensions version 0.1.8.1 Index]