Computes Zij-values of item pairs, Zi-values of items, and Z-value of the entire scale,
which are used to test whether Hij, Hi, and H, respectively, are significantly
greater than zero using the original method Z
(Molenaar and Sijtsma, 2000, pp. 59-62; Sijtsma and Molenaar, p. 40; Van der Ark, 2007; 2010)
or the Wald-based method (WB) or range-preserving method (RP)
(Kuijpers, Van der Ark, & Croon, 2013; Koopman, Zijlstra, & Van der Ark, 2020a, 2020b).
The Wald-based method and range-preserving method can also handle nested data and can test other lowerbounds than zero.
Used in the function aisp
coefZ(X, lowerbound = 0, type.z = "Z", level.two.var = NULL)matrix or data frame of numeric data
containing the responses of nrow(X) respondents to ncol(X) items.
Missing values are not allowed
Value of the null hypothesis to which the scalability are compared to compute the Z-score (see details),
0 <= lowerbound < 1. The default is 0.
Indicates which type of z-score is computed: "WB": Wald-based z-score based on standard errors as approximated by the delta method (Kuijpers, Van der Ark, Kroon, 2013; Koopman, Zijlstra, Van der Ark, 2020a); "RP": Range-preserving z-score, also based on the delta method (Koopman, Zijlstra, Van der Ark, 2020b); "Z": uses original Z-test and is only appropriate to test lowerbound = 0 (Mokken, 1971; Molenaar and Sijtsma, 2000; Sijtsma and Molenaar, 2002). The default is "Z".
vector of length nrow(X) or matrix with number of rows equal to nrow(X)
that indicates the level two variable for nested data (Koopman et al., 2020a).
matrix containing the Z-values of the item-pairs
vector containing Z-values of the items
Z-value of the entire scale
For the estimated item-pair coefficient \(Hij\) with standard error \(SE(Hij)\), the Z-score is computed as
$$Zij = (Hij - lowerbound) / SE(Hij)$$ if type.z = "WB", and the Z-score is computed as
$$Zij = -(log(1 - Hij) - log(1 - lowerbound)) / (SE(Hij) / (1 - Hij))$$ if type.z = "RP"
(Koopman, Zijlstra, Van der Ark, 2020b).
For the estimate item-scalability coefficients \(Hi\) and total-scalbility coefficients \(H\) a similar procedure is used.
Standard errors of the Z-scores are not provided.
Koopman, L. Zijlstra, B. J. H, & Van der Ark, L. A. (2020a). A two-step procedure for scaling multilevel data using Mokken's scalability coefficients. Manuscript submitted for publication.
Koopman, L. Zijlstra, B. J. H, & Van der Ark, L. A. (2020b). Range-preserving confidence intervals for scalability coefficients in Mokken scale analysis. Manuscript submitted for publication.
Kuijpers, R. E., Van der Ark, L. A., & Croon, M. A. (2013). Standard errors and confidence intervals for scalability coefficients in Mokken scale analysis using marginal models. Sociological Methodology, 43, 42-69. https://doi.org/10.1177/0081175013481958
Molenaar, I.W., & Sijtsma, K. (2000) User's Manual MSP5 for Windows [Software manual]. IEC ProGAMMA.
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. https://www.jstatsoft.org/article/view/v020i11
Van der Ark, L. A. (2010). Getting started with Mokken scale analysis in R. Unpublished manuscript. https://sites.google.com/a/tilburguniversity.edu/avdrark/mokken
# NOT RUN {
data(acl)
Communality <- acl[,1:10]
# Compute the Z-score of each coefficient
coefH(Communality)
coefZ(Communality)
# Using lowerbound .3
coefZ(Communality, lowerbound = .3, type.z = "WB")
# Z-scores for nested data
data(autonomySupport)
scores <- autonomySupport[, -1]
classes <- autonomySupport[, 1]
coefH(scores, level.two.var = classes)
coefZ(scores, type.z = "WB", level.two.var = classes)
# }
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