Cronbach's Coefficient Alpha

Cronbach's alpha is a measure of internal consistency and often used for validating psychometric tests. The unstandardized form implemented here is computed from the item variances and the variance of the total score, expressing the proportion of total-score variance not attributable to item-specific variance. This reduces to Kuder-Richardson formula 20 (KR-20) when the columns of the data matrix are dichotomous.

Usage

cronbachAlpha(
  x,
  conf.level = NA,
  sides = c("two.sided", "left", "right"),
  returnConditional = FALSE,
  na.rm = FALSE
)

Arguments

x

a \(n \times m\) matrix or data frame with item responses, \(n\) subjects (in rows) and \(m\) items (in columns)

conf.level

confidence level of the interval. If set to NA (the default), only the point estimate is returned.

sides

character string specifying the sidedness of the confidence interval (one of "two.sided" (default), "left" or "right"). See ConfidenceIntervals().

returnConditional

logical; if TRUE, alpha is additionally calculated for the dataset with each item left out

na.rm

logical; if TRUE, incomplete cases are removed before the computation proceeds

Value

a named numeric vector, or a list when returnConditional = TRUE.

If na.rm = FALSE and x contains missing values, the same structure is returned with NA_real_ throughout.

If conf.level = NA, the numeric vector contains only est; otherwise it has elements:

est

point estimate.

lci

lower confidence interval bound.

uci

upper confidence interval bound.

If returnConditional = TRUE, a list with the components:

unconditional

alpha for the full set of items, as above

conditional

a data frame with one row per item, giving the alpha that would be realized if that item were excluded. NULL when x has fewer than 3 items, since dropping one would leave too few to compute alpha.

Details

The confidence interval follows Feldt (1965) and is based on the \(F\) distribution with \(n - 1\) and \((m - 1)(n - 1)\) degrees of freedom, where \(n\) is the number of subjects (rows) and \(m\) the number of items (columns). It inherits the assumptions of the underlying ANOVA derivation - in particular normally distributed scores and essentially parallel items with homogeneous variances and covariances - and should be read with more caution than the point estimate when these are doubtful, for instance with markedly skewed or heterogeneous items.

sides names the side on which the finite bound lies: "left" yields an interval bounded below and "right" one bounded above. Alpha cannot exceed 1, so the open upper side is reported at that boundary rather than as \(\infty\) (design_rules.md 4.1), while the open lower side stays \(-\infty\) because alpha is unbounded below.

Missing values are handled according to package conventions: if na.rm = FALSE and x contains missing values, the usual structure is returned with NA_real_ in place of every estimate. If na.rm = TRUE, complete cases are used. Infinite values leave the variances undefined and are rejected with an error.

Note

Based on code of Harold C. Doran, adapted to conform to package standards.

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334. doi:10.1007/BF02310555

Feldt, L. S. (1965). The approximate sampling distribution of Kuder-Richardson reliability coefficient twenty. Psychometrika, 30(3), 357-370. doi:10.1007/BF02289499

See also

Other assoc.agreement: ccc(), cohenKappa(), icc(), kappaM(), krippAlpha(), pabak(), percAgreement(), randolphKappa()

Examples

set.seed(1234)
tmp <- data.frame(
  item1 = sample(c(0, 1), 20, replace = TRUE),
  item2 = sample(c(0, 1), 20, replace = TRUE),
  item3 = sample(c(0, 1), 20, replace = TRUE),
  item4 = sample(c(0, 1), 20, replace = TRUE),
  item5 = sample(c(0, 1), 20, replace = TRUE)
)

cronbachAlpha(tmp[, 1:4])
#>       est 
#> 0.4029851 

cronbachAlpha(tmp[, 1:4], conf.level = 0.95)
#>        est        lci        uci 
#>  0.4029851 -0.1799176  0.7377321 

# the conditional table is labelled with the column names of x
cronbachAlpha(tmp[, 1:4], returnConditional = TRUE, conf.level = 0.95)
#> $unconditional
#>        est        lci        uci 
#>  0.4029851 -0.1799176  0.7377321 
#> 
#> $conditional
#>    item       est          lci       uci
#> 1 item1 0.1574074 -0.774604418 0.6403307
#> 2 item2 0.5241379 -0.002224489 0.7968734
#> 3 item3 0.2537764 -0.571639304 0.6814668
#> 4 item4 0.3455056 -0.378446277 0.7206224
#> 

# fewer than 3 items: the conditional component is NULL
cronbachAlpha(tmp[, 1:2], returnConditional = TRUE, conf.level = 0.95)
#> $unconditional
#>         est         lci         uci 
#> -0.04678363 -1.64464747  0.58567031 
#> 
#> $conditional
#> NULL
#>