item.alpha: Coefficient Alpha, Hierarchical Alpha, and Ordinal Alpha

In misty: Miscellaneous Functions 'T. Yanagida'

Coefficient Alpha, Hierarchical Alpha, and Ordinal Alpha

Description

This function computes point estimate and confidence interval for the coefficient alpha (aka Cronbach's alpha), hierarchical alpha, and ordinal alpha (aka categorical alpha) along with standardized factor loadings and alpha if item deleted. By default, the function computes coefficient alpha based on unweighted least squares (ULS) parameter estimates using pairwise deletion in the presence of missing data that provides equivalent results compared to the formula-based coefficient alpha computed by using e.g. the alpha function in the psych package by William Revelle (2025).

Usage

item.alpha(data, ..., rescov = NULL, type = c("alpha", "hierarch", "categ"),
           exclude = NULL, std = FALSE,
           estimator = c("ML", "GLS", "WLS", "DWLS", "ULS", "PML"),
           missing = c("listwise", "pairwise", "fiml"),
           print = c("all", "alpha", "item"), digits = 2, conf.level = 0.95,
           as.na = NULL, write = NULL, append = TRUE, check = TRUE,
           output = TRUE)

Arguments

data	a data frame. Note that at least two items are needed for computing coefficient alpha
...	an expression indicating the variable names in data e.g., item.alpha(dat, x1, x2, x3). Note that the operators ., +, -, ~, :, ::, and ! can also be used to select variables, see 'Details' in the df.subset function.
rescov	a character vector or a list of character vectors for specifying residual covariances when computing coefficient alpha, e.g. rescov = c("x1", "x2") for specifying a residual covariance between items x1 and x2 or rescov = list(c("x1", "x2"), c("x3", "x4")) for specifying residual covariances between items x1 and x2, and items x3 and x4.
type	a character string indicating the type of alpha to be computed, i.e., alpha (default) for coefficient alpha, hierarch for hierarchical coefficient alpha, and categ for ordinal coefficient alpha.
exclude	a character vector indicating items to be excluded from the analysis.
std	logical: if TRUE, the standardized coefficient alpha is computed.
estimator	a character string indicating the estimator to be used (see 'Details' in the item.cfa function). By default, "ULS" is used for computing (hierarchical) coefficient alpha and "DWLS" is used for computing ordinal coefficient alpha.
missing	a character string indicating how to deal with missing data. (see 'Details' in the item.cfa function). By default, pairwise deletion (missing = "pairwise") is used for computing (hierarchical) coefficient alpha and ordinal coefficient alpha. Full information maximum likelihood method is available for estimating (hierarchical) coefficient alpha and is requested by specifying missing = "fiml" along with estimator = "ML".
print	a character vector indicating which results to show, i.e. "all" for all results "alpha" (default) for the coefficient alpha, and "item" for item statistics.
digits	an integer value indicating the number of decimal places to be used for displaying alpha and standardized factor loadings.
conf.level	a numeric value between 0 and 1 indicating the confidence level of the interval.
as.na	a numeric vector indicating user-defined missing values, i.e. these values are converted to NA before conducting the analysis.
write	a character string naming a file for writing the output into either a text file with file extension ".txt" (e.g., "Output.txt") or Excel file with file extension ".xlsx" (e.g., "Output.xlsx"). If the file name does not contain any file extension, an Excel file will be written.
append	logical: if TRUE (default), output will be appended to an existing text file with extension .txt specified in write, if FALSE existing text file will be overwritten.
check	logical: if TRUE (default), argument specification is checked.
output	logical: if TRUE (default), output is shown.

Details

Coefficient alpha is computed by conducting a confirmatory factor analysis based on the essentially tau-equivalent measurement model (Graham, 2006) using the cfa() function in the lavaan package by Yves Rosseel (2019). Approximate confidence intervals are computed using the procedure by Feldt, Woodruff and Salih (1987). Note that there are at least 10 other procedures for computing the confidence interval (see Kelley and Pornprasertmanit, 2016), which are implemented in the ci.reliability() function in the MBESSS package by Ken Kelley (2019)

Ordinal coefficient alpha was introduced by Zumbo, Gadermann and Zeisser (2007). Note that Chalmers (2018) highlighted that the categorical coefficient alpha should be interpreted only as a hypothetical estimate of an alternative reliability, whereby a test's ordinal categorical response options have be modified to include an infinite number of ordinal response options and concludes that coefficient alpha should not be reported as a measure of a test's reliability. However, Zumbo and Kroc (2019) argued that Chalmers' critique of categorical coefficient alpha is unfounded and that categorical coefficient alpha may be the most appropriate quantifier of reliability when using Likert-type measurement to study a latent continuous random variable.

Value

Returns an object of class misty.object, which is a list with following entries:

call	function call
type	type of analysis
data	data frame used for the current analysis
args	specification of function arguments
model.fit	fitted lavaan object (mod.fit)
result	list with result tables, i.e., alpha for a table with coefficient alpha and itemstat for a table with item statistics

Note

Computation of the hierarchical and ordinal alpha is based on the ci.reliability() function in the MBESS package by Ken Kelley (2019).

Author(s)

Takuya Yanagida [e-mail]

References

Chalmers, R. P. (2018). On misconceptions and the limited usefulness of ordinal alpha. Educational and Psychological Measurement, 78, 1056-1071. https://doi.org/10.1177/0013164417727036

Cronbach, L.J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334. https://doi.org/10.1007/BF02310555

Cronbach, L.J. (2004). My current thoughts on coefficient alpha and successor procedures. Educational and Psychological Measurement, 64, 391-418. https://doi.org/10.1177/0013164404266386

Feldt, L. S., Woodruff, D. J., & Salih, F. A. (1987). Statistical inference for coefficient alpha. Applied Psychological Measurement, 11 93-103. https://doi.org/10.1177/014662168701100107

Graham, J. M. (2006). Congeneric and (essentially) tau-equivalent estimates of score reliability: What they are and how to use them. Educational and Psychological Measurement, 66(6), 930–944. https://doi.org/10.1177/0013164406288165

Kelley, K., & Pornprasertmanit, S. (2016). Confidence intervals for population reliability coefficients: Evaluation of methods, recommendations, and software for composite measures. Psychological Methods, 21, 69-92. https://doi.org/10.1037/a0040086.

Ken Kelley (2019). MBESS: The MBESS R Package. R package version 4.6.0. https://CRAN.R-project.org/package=MBESS

Revelle, W. (2025). psych: Procedures for psychological, psychometric, and personality research. Northwestern University, Evanston, Illinois. R package version 2.5.3, https://CRAN.R-project.org/package=psych.

Zumbo, B. D., & Kroc, E. (2019). A measurement is a choice and Stevens' scales of measurement do not help make it: A response to Chalmers. Educational and Psychological Measurement, 79, 1184-1197. https://doi.org/10.1177/0013164419844305

Zumbo, B. D., Gadermann, A. M., & Zeisser, C. (2007). Ordinal versions of coefficients alpha and theta for Likert rating scales. Journal of Modern Applied Statistical Methods, 6, 21-29. https://doi.org/10.22237/jmasm/1177992180

See Also

item.omega, item.cfa, item.invar, item.reverse, item.scores, write.result

Examples

## Not run: 
dat <- data.frame(item1 = c(3, NA, 3, 4, 1, 2, 4, 2), item2 = c(5, 3, 3, 2, 2, 1, 3, 1),
                  item3 = c(4, 2, 4, 2, 1, 3, 4, 1), item4 = c(4, 1, 2, 2, 1, 3, 4, 3))

# Example 1a: Coefficient alpha and item statistics, pairwise deletion
item.alpha(dat)

# Example 1b: Coefficient alpha and item statistics, listwise deletion
item.alpha(dat, missing = "listwise")

# Example 1c: Coefficient alpha and item statistics, FIML
item.alpha(dat, estimator = "ML", missing = "fiml")

# Example 2: Coefficient alpha and item statistics after excluding item3
item.alpha(dat, exclude = "item3")

# Example 3a: Coefficient alpha with a residual covariance
# and item statistics
item.alpha(dat, rescov = c("item1", "item2"))

# Example 3b: Coefficient alpha with residual covariances
# and item statistics
item.alpha(dat, rescov = list(c("item1", "item2"), c("item1", "item3")))

# Example 4: Ordinal coefficient alpha and item statistics
item.alpha(dat, type = "categ")

# Example 6: Summary of the CFA model used to compute coefficient alpha
lavaan::summary(item.alpha(dat, output = FALSE)$model.fit,
                fit.measures = TRUE, standardized = TRUE)

# Example 7a: Write Results into a text file
item.alpha(dat, write = "Alpha.txt")

# Example 7b: Write Results into a Excel file
item.alpha(dat, write = "Alpha.xlsx")

## End(Not run)

misty documentation built on June 8, 2025, 1:35 p.m.