blox (version 0.0.2)

hc.beta: Compute Revelle's Beta for all worst split-halves using HC-SVD.

Description

Performs HC-SVD to reveal the hierarchical variable structure using average linkage as described in Bauer (202X). For a data matrix comprising \(p\) items, this means that \(p-1\) splits are identified. The obtained structure aligns with the structure according to the worst split-half reliability and is thus used to compute a hierarchy of all Revelle's beta as described in Bauer (2026).

Usage

hc.beta(R, splits = NULL, n.splits = NULL, is.corr = TRUE, verbose = TRUE)

Value

A list with n.splits components. Each split is a list of four components:

split

The split number.

beta

Revelle's beta for this split.

A

One of the two sub-scales that has been split.

B

One of the two sub-scales that has been split.

beta.alpha

Computes the ratio of Revelle's beta and Cronbach's alpha.

Arguments

R

A correlation matrix of dimension \(p\)x\(p\) or a data matrix of dimension \(n\)x\(p\) can be provided. If a data matrix is supplied, it must be indicated by setting is.corr = FALSE, and the correlation matrix will then be calculated as cor(X).

splits

An object containing the splits identified by HC-SVD. This can either be the result of hcsvd (for all splits) or bd.approx (for a single split). If omitted, hc.beta will internally call hcsvd(R) and compute Revelle's beta for all \(p - 1\) splits.

n.splits

Number of splits for which Revelle's beta is computed. If splits is from hcsvd, the default is all \(p-1\) splits. If splits is from bd.approx, only a single split is available and n.splits is set to 1.

is.corr

Is the supplied object a correlation matrix. Default is TRUE and this parameter must be set to FALSE if a data matrix instead of a correlation matrix is supplied.

verbose

Print out progress as \(p-1\) iterations for divisive hierarchical clustering are performed. Default is TRUE.

Details

Supplementary details are in Bauer (2026).

References

Bauer, J.O. (2026). Revelle’s beta: The wait is over—computation becomes possible. Psychometrika.

Bauer, J.O. (202X). Divisive hierarchical clustering using block diagonal matrix approximations. Working paper.

See Also

bd.approx hcsvd

Examples

#We compute the worst split-half reliabilities on a correlation matrix.

# \donttest{
#Load the correlation matrix Bechtoldt from the psych
#package (see ?Bechtoldt for more information).
if (requireNamespace("psych", quietly = TRUE)) {
  data("Bechtoldt", package = "psych")
}
R <- Bechtoldt


### RUN HC-SVD FOR HIERARCHICAL VARIABLE CLUSTERING

#Compute HC-SVD (with average linkage).
hcsvd.obj <- hcsvd(R)

#The object of type hclust with corresponding dendrogram can be obtained
#directly from hcsvd(...):
hc.div <- hcsvd.obj$hclust
plot(hc.div, axes = FALSE, ylab = "", main = "Revelle's Beta Splits")


### COMPUTE REVELLE'S BETA FOR ALL IDENTIFIED SPLITS

#Compute Revelle's beta
betas <- hc.beta(R = R)

#Alternatively, you can submit the object obtained from hcsvd(). Thus,
#the hiearchy needs not to be computed again using hcsvd().
betas <- hc.beta(R = R, splits = hcsvd.obj)

#Visualize the splits, e.g., as
splits <- sapply(betas, `[[`, "split")
beta.values <- sapply(betas, `[[`, "beta")

plot(splits, beta.values,
  type = "b",
  xlab = "Split",
  ylab = "Revelle's Beta",
  main = "Revelle's Beta Across Splits",
  pch = 19)

#Visualize the ratio of Revelle's beta and Cronbach's alpha
beta.alpha <- sapply(betas, `[[`, "beta.alpha")
plot(splits, beta.values,
  type = "b",
  xlab = "Split",
  ylab = "Beta/Alpha",
  main = "Ratio of Beta and Alpha Across Splits",
  pch = 19)


### COMPUTE REVELLE'S BETA FOR THE FIRST IDENTIFIED SPLIT

#The first split can be identified using bd.approx()
#This is computationally faster, as only the first split
#is identified
hc.beta(R = R, splits = bd.approx(R))

# }