# Standard Errors for Reliability Coefficients

## TL;DR

This paper introduces analytic standard errors for reliability coefficients used in psychometrics, addressing a gap in measurement precision assessment.

## Contribution

The paper provides nonparametric standard errors for reliability coefficients under a multinomial sampling scheme.

## Key findings

- Standard errors are derived for coefficients like Cronbach’s alpha and item-total correlations.
- R functions for computing these standard errors are available on the Open Science Framework.
- Simulation studies show satisfactory performance for larger sample sizes and typical parameter values.

## Abstract

Reliability analysis is one of the most conducted analyses in applied psychometrics. It entails the assessment of reliability of both item scores and scale scores using coefficients that estimate the reliability (e.g., Cronbach’s alpha), measurement precision (e.g., estimated standard error of measurement), or the contribution of individual items to the reliability (e.g., corrected item-total correlations). Most statistical software packages used in social and behavioral sciences offer these reliability coefficients, whereas standard errors are generally unavailable, which is a bit ironic for coefficients about measurement precision. This article provides analytic nonparametric standard errors for coefficients used in reliability analysis. As most scores used in behavioral sciences are discrete, standard errors are derived under the relatively unrestrictive multinomial sampling scheme. Tedious derivations are presented in appendices, and R functions for computing standard errors are available from the Open Science Framework. Bias and variance of standard errors, and coverage of the corresponding Wald-based confidence intervals are studied using simulated item scores. Bias and variance, and coverage are generally satisfactory for larger sample sizes, and parameter values are not close to the boundary of the parameter space.