Computes the scalability coefficients of Mokken scale analysis: the item-pair coefficients \(H_{ij}\), the item coefficients \(H_j\), and the test coefficient \(H\). For polytomous items these are the weighted Loevinger coefficients of Molenaar (1991).
Arguments
- U
Either an object of class "exametrika" or raw data. When raw data is given, it is converted to the exametrika class with the
dataFormatfunction.- na
Values to be treated as missing values.
- Z
Missing indicator matrix of type matrix or data.frame. Values of 1 indicate observed responses, while 0 indicates missing data.
- w
Item weight vector. Not used.
Value
An object of class c("exametrika", "Scalability") with
- ItemLabel
Item labels.
- Hij
Matrix of item-pair coefficients (diagonal is
NA).- Hj
Vector of item coefficients.
- H
Test coefficient.
Details
Each coefficient is a normed covariance. For a pair of items, $$H_{ij} = \mathrm{Cov}(X_i, X_j) / \mathrm{Cov}(X_i, X_j)_{\max},$$ where the denominator is the largest covariance attainable given the two marginal distributions. It is obtained by pairing the sorted scores of the two items (the comonotone arrangement). The item and test coefficients pool the pair-wise numerators and denominators (the item coefficient is Mokken's; Loevinger's own item-test coefficient was defined differently), $$H_j = \sum_{i \ne j} \mathrm{Cov}(X_i, X_j) / \sum_{i \ne j} \mathrm{Cov}(X_i, X_j)_{\max},$$ $$H = \sum_{i < j} \mathrm{Cov}(X_i, X_j) / \sum_{i < j} \mathrm{Cov}(X_i, X_j)_{\max}.$$
Unidimensionality, local independence and latent monotonicity imply \(0 \le H_{ij} \le 1\), \(0 \le H_j \le 1\) and \(0 \le H \le 1\) (Sijtsma & Molenaar, 2002). A negative coefficient therefore points to an item scored in the opposite direction (for example, a reverse-keyed item that has not been recoded) or to an item whose relation to the others is not monotone. Mokken (1971) suggested the rules of thumb \(0.3 \le H < 0.4\) (weak scale), \(0.4 \le H < 0.5\) (moderate) and \(H \ge 0.5\) (strong).
Missing responses are handled pair-wise: each pair of items uses the
examinees who answered both. With complete data the results agree with
coefH() of the mokken package. Standard errors are not
computed. Item weights w are ignored.
The function only reports; it never recodes or drops items. Whether a reverse-keyed item should be recoded is a decision for the person who prepares the data.
References
Loevinger, J. (1948). The technic of homogeneous tests compared with some aspects of "scale analysis" and factor analysis. Psychological Bulletin, 45, 507–529.
Mokken, R. J. (1971). A theory and procedure of scale analysis. De Gruyter.
Molenaar, I. W. (1991). A weighted Loevinger H-coefficient extending Mokken scaling to multicategory items. Kwantitatieve Methoden, 12(37), 97–117.
Sijtsma, K., & Molenaar, I. W. (2002). Introduction to nonparametric item response theory. Sage.
van der Ark, L. A. (2007). Mokken scale analysis in R. Journal of Statistical Software, 20(11), 1–19.
Examples
# \donttest{
sc <- Scalability(J15S3810)
sc$H
#> [1] 0.2225775
sc$Hj
#> V1 V2 V3 V4 V5 V6 V7
#> 0.28477623 0.23807893 0.21606542 0.30217293 0.25051347 0.34964498 0.19578510
#> V8 V9 V10 V11 V12 V13 V14
#> 0.15666992 0.16071280 0.15946396 0.25047250 0.08892962 0.27451454 0.09377406
#> V15
#> 0.24099386
# }