composite_rel_matrix {psychmeta}

Matrix formula to estimate the reliability of a weighted or unweighted composite variable

Description

This function computes the reliability of a variable that is a weighted or unweighted composite of other variables.

Usage

composite_rel_matrix(rel_vec, r_mat, sd_vec, wt_vec = rep(1, length(rel_vec)))

Arguments

rel_vec

Vector of reliabilities associated with variables in the composite to be formed.

r_mat

Correlation matrix from which the composite is to be computed.

sd_vec

Vector of standard deviations associated with variables in the composite to be formed.

wt_vec

Weights to be used in forming the composite (by default, all variables receive equal weight).

Details

This function treats measure-specific variance as reliable.

The Mosier composite formula is computed as:

\rho_{XX}=\frac{\mathbf{w}^{T}\left(\mathbf{r}\circ\mathbf{s}\right)+\mathbf{w}^{T}\mathbf{S}\mathbf{w}-\mathbf{w}^{T}\mathbf{s}}{\mathbf{w}^{T}\mathbf{S}\mathbf{w}}

where \rho_{XX} is a composite reliability estimate, \mathbf{r} is a vector of reliability estimates, \mathbf{w} is a vector of weights, \mathbf{S} is a covariance matrix, and \mathbf{s} is a vector of variances (i.e., the diagonal elements of \mathbf{S}).

Value

The estimated reliability of the composite variable.

References

Mosier, C. I. (1943). On the reliability of a weighted composite. Psychometrika, 8(3), 161–168. doi:10.1007/BF02288700

Schmidt, F. L., & Hunter, J. E. (2015). Methods of meta-analysis: Correcting error and bias in research findings (3rd ed.). Thousand Oaks, CA: Sage. doi:10.4135/9781483398105. pp. 441 - 447.

Examples

composite_rel_matrix(rel_vec = c(.8, .8),
r_mat = matrix(c(1, .4, .4, 1), 2, 2), sd_vec = c(1, 1))