coefZ function

Computation of Z-Values

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 et al., 2013; Koopman et al., in press a, in press b). 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)

Arguments

• X: matrix or data frame of numeric data containing the responses of nrow(X) respondents to ncol(X) items. Missing values are not allowed

• lowerbound: 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.

• type.z: Indicates which type of z-score is computed: "WB": Wald-based z-score based on standard errors as approximated by the delta method (Kuijpers et al., 2013; Koopman et al., in press a); "RP": Range-preserving z-score, also based on the delta method (Koopman et al., in press b); "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".

• level.two.var: 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., in press a).

Returns

• Zij: matrix containing the Z-values of the item-pairs

• Zi: vector containing Z-values of the items

• Z: Z-value of the entire scale

Details

For the estimated item-pair coefficient $Hij$ with standard error $SE(Hij)$, the Z-score is computed as

if type.z = "WB", and the Z-score is computed as

if type.z = "RP"

(Koopman et al., in press b). 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.

References

Koopman, L., Zijlstra, B. J. H., & Van der Ark, L. A. (in press a). A two-step, test-guided Mokken scale analysis for nonclustered and clustered data. Quality of Life Research. (advanced online publication) tools:::Rd_expr_doi("10.1007/s11136-021-02840-2")

Koopman, L., Zijlstra, B. J. H., & Van der Ark, L. A. (in press b). Range-preserving confidence intervals and significance tests for scalability coefficients in Mokken scale analysis. In M. Wiberg, D. Molenaar, J. Gonzalez, & Kim, J.-S. (Eds.), Quantitative Psychology; The 1st Online Meeting of the Psychometric Society, 2020. Springer. tools:::Rd_expr_doi("10.1007/978-3-030-74772-5_16")

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. tools:::Rd_expr_doi("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. tools:::Rd_expr_doi("10.18637/jss.v020.i11")

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

Author(s)

L. A. van der Ark L.A.vanderArk@uva.nl

L. Koopman

See Also

coefH, aisp

Examples

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)