# composite_rel_matrix: Matrix formula to estimate the reliability of a weighted or... In psychmeta: Psychometric Meta-Analysis Toolkit

## Description

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

## Usage

 1 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:

\mjdeqn\rho

_XX=\frac\mathbfw^T\left(\mathbfr\circ\mathbfs\right)+\mathbfw^T\mathbfS\mathbfw-\mathbfw^T\mathbfs\mathbfw^T\mathbfS\mathbfwrel_composite = (t(wt^2) (rel_vec * var_vec) + S - var_sum) / (t(wt) S wt)

where \mjeqn\rho_XXrel_composite is a composite reliability estimate, \mjeqn\mathbfrrel_vec is a vector of reliability estimates, \mjeqn\mathbfwwt is a vector of weights, \mjeqn\mathbfSS is a covariance matrix, and \mjeqn\mathbfsvar_vec is a vector of variances (i.e., the diagonal elements of \mjeqn\mathbfSS).

## 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

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

### Example output

-----------------------------------------------------  psychmeta version 2.3.3  --

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We work hard to produce these open-source tools for the R community,
Dahlke, J. A., & Wiernik, B. M. (2018). psychmeta: An R package for
psychometric meta-analysis. Applied Psychological Measurement.