coefZ {mokken}

Computation of Z-Values

Description

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

Usage

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).

Details

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 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.

Value

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

Author(s)

L. A. van der Ark [e-mail] L. Koopman

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) 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. 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. 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. 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

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)