mokken (version 3.0.3)

coefZ: 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 delta method (Kuijpers, Van der Ark, & Croon, 2013; Koopman, Zijlstra, & Van der Ark, 2020a). The delta method can also handle nested data and can test other lowerbounds than zero. The test uses range-preserving asymptotic theory (Koopman, Zijlstra, & Van der Ark, 2020b)

Used in the function aisp

Usage

coefZ(X, lowerbound = 0, type.se = "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.se

Indicates which type of standard errors is used to compute the Z-score: "delta": uses standard errors approximated by the delta method (Kuijpers, Van der Ark, Kroon, 2013; Koopman, Zijlstra, Van der Ark, 2020a); "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., 2020a).

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

Details

The Z-score with type.se = "delta" for the estimated item-pair coefficient \(Hij\) with standard error \(SE(Hij)\) is computed as $$Z = -(log(1 - Hij) - log(1 - lowerbound)) / (SE(Hij) / (1 - Hij)) (Koopman, Zijlstra, Van der Ark, 2020b)$$ Unlike coefH, standard errors are not provided.

References

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

See Also

coefH, aisp

Examples

# 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.se = "delta")

# Z-scores for nested data
data(autonomySupport)
scores <- autonomySupport[, -1]
classes <- autonomySupport[, 1]
coefH(scores, level.two.var = classes)
coefZ(scores, type.se = "delta", level.two.var = classes)


# }