Editing Cronbach's alpha (section)

==Which reliability coefficient to use==
<math>\rho_T</math> is used in an overwhelming proportion. A study estimates that approximately 97% of studies use <math>\rho_T</math> as a reliability coefficient.<ref name = Cho/>

However, simulation studies comparing the accuracy of several reliability coefficients have led to the common result that <math>\rho_T</math> is an inaccurate reliability coefficient.<ref name=KTD>Kamata, A., Turhan, A., & Darandari, E. (2003). Estimating reliability for multidimensional composite scale scores. Annual Meeting of American Educational Research Association, Chicago, April 2003, April, 1–27.</ref><ref name="Osburn">{{cite journal|first=H. G.|last=Osburn|title=Coefficient alpha and related internal consistency reliability coefficients|journal=Psychological Methods|volume=5|issue=3|pages=343–355|date=2000|doi=10.1037/1082-989X.5.3.343|pmid=11004872}}</ref><ref name=RZ/><ref name=TC>Tang, W., & Cui, Y. (2012). A simulation study for comparing three lower bounds to reliability. Paper Presented on April 17, 2012 at the AERA Division D: Measurement and Research Methodology, Section 1: Educational Measurement, Psychometrics, and Assessment, 1–25.</ref><ref name=VVS>{{cite journal|last1=van der Ark|first1=L. A.|last2=van der Palm|first2=D. W.|last3=Sijtsma|first3=K.|title=A latent class approach to estimating test-score reliability|journal=Applied Psychological Measurement|volume=35|issue=5|pages=380–392|date=2011|doi=10.1177/0146621610392911|s2cid=41739445|url=https://research.tilburguniversity.edu/en/publications/becd6ed2-4796-4959-9f72-e0ae24faa8a6|access-date=2023-06-04|archive-date=2023-08-13|archive-url=https://web.archive.org/web/20230813121350/https://research.tilburguniversity.edu/en/publications/a-latent-class-approach-to-estimating-test-score-reliability|url-status=live}}</ref>

Methodological studies are critical of the use of <math>\rho_T</math>. Simplifying and classifying the conclusions of existing studies are as follows.
# Conditional use: Use <math>\rho_T</math> only when certain conditions are met.<ref name=Cho/><ref name=ChoKim/><ref name=RM/>
# Opposition to use: <math>\rho_T</math> is inferior and should not be used.<ref name=DBB>{{cite journal|last1=Dunn|first1=T. J.|last2=Baguley|first2=T.|last3=Brunsden|first3=V.|title=From alpha to omega: A practical solution to the pervasive problem of internal consistency estimation|journal=British Journal of Psychology|volume=105|issue=3|pages=399–412|date=2014|doi=10.1111/bjop.12046|pmid=24844115|url=http://irep.ntu.ac.uk/id/eprint/4853/1/215051_Dunn.pdf|access-date=2023-06-04|archive-date=2023-03-24|archive-url=https://web.archive.org/web/20230324013930/http://irep.ntu.ac.uk/id/eprint/4853/1/215051_Dunn.pdf|url-status=live}}</ref><ref name=GY/><ref name=Peters>{{cite journal|first=G. Y.|last=Peters|title=The alpha and the omega of scale reliability and validity comprehensive assessment of scale quality|journal=The European Health Psychologist|volume=1|issue=2|pages=56–69|date=2014|doi=}}</ref><ref name=RZ/><ref name=Sijtsma/><ref name=YG>Yang, Y., & Green, S. B.{{cite journal|title=Coefficient alpha: A reliability coefficient for the 21st century?|journal=Journal of Psychoeducational Assessment|volume=29|issue=4|pages=377–392|date=2011|doi=10.1177/0734282911406668|last1=Yanyun Yang|last2=Green|first2=Samuel B.|s2cid=119926199}}</ref>

===Alternatives to Cronbach's alpha===
Existing studies are practically unanimous in that they oppose the widespread practice of using <math>\rho_T</math> unconditionally for all data. However, different opinions are given on which reliability coefficient should be used instead of <math>\rho_T</math>.

Different reliability coefficients ranked first in each simulation study<ref name=KTD/><ref name=Osburn/><ref name=RZ/><ref name=TC/><ref name=VVS/> comparing the accuracy of several reliability coefficients.<ref name=ChoKim/>

The majority opinion is to use structural equation modeling or [[Structural equation modeling|SEM]]-based reliability coefficients as an alternative to <math>\rho_T</math>.<ref name=Cho/><ref name=ChoKim/><ref name=DBB/><ref name=GY/><ref name=Peters/><ref name=RM/><ref name=RZ/><ref name=YG/>

However, there is no consensus on which of the several SEM-based reliability coefficients (e.g., uni-dimensional or multidimensional models) is the best to use.

Some people suggest <math>\omega_H</math><ref name=RZ/> as an alternative, but <math>\omega_H</math> shows information that is completely different from reliability. <math>\omega_H</math> is a type of coefficient comparable to Reveille's <math>\beta</math>.<ref name="Revelle">{{cite journal|first=W.|last=Revelle|title=Hierarchical cluster analysis and the internal structure of tests|journal=Multivariate Behavioral Research|volume=14|issue=1|pages=57–74|date=1979|doi=10.1207/s15327906mbr1401_4|pmid=26766619}}</ref><ref name=RZ/> They do not substitute, but complement reliability.<ref name=Cho/>

Among SEM-based reliability coefficients, multidimensional reliability coefficients are rarely used, and the most commonly used is <math>\rho_C</math>,<ref name = Cho/> also known as composite or [[congeneric reliability]].

In addition to single estimates of reliability, [[Item response theory]] based approaches can provide estimates of conditional reliability across the full distribution of scores.<ref>{{Cite journal |last=McNeish |first=Daniel |last2=Dumas |first2=Denis |date=2025-02-10 |title=Reliability representativeness: How well does coefficient alpha summarize reliability across the score distribution? |url=https://link.springer.com/10.3758/s13428-025-02611-8 |journal=Behavior Research Methods |language=en |volume=57 |issue=3 |doi=10.3758/s13428-025-02611-8 |issn=1554-3528}}</ref>

====Software for SEM-based reliability coefficients====
General-purpose statistical software such as [[SPSS]] and [[SAS (software)|SAS]] include a function to calculate <math>\rho_T</math>. Users who don't know the formula <math>\rho_T</math> have no problem in obtaining the estimates with just a few mouse clicks.

SEM software such as AMOS, [[LISREL]], and MPLUS does not have a function to calculate SEM-based reliability coefficients. Users need to calculate the result by inputting it to the formula. To avoid this inconvenience and possible error, even studies reporting the use of SEM rely on <math>\rho_T</math> instead of SEM-based reliability coefficients.<ref name = Cho/> There are a few alternatives to automatically calculate SEM-based reliability coefficients.

# [[R (programming language)|R]] (free): The psych package<ref>{{cite web|title=An overview of the psych package|last=Revelle|first=William|date=7 January 2017|url=http://personality-project.org/r/overview.pdf|access-date=23 April 2020|archive-date=27 August 2020|archive-url=https://web.archive.org/web/20200827020016/http://personality-project.org/r/overview.pdf|url-status=live}}</ref> calculates various reliability coefficients.
# EQS (paid):<ref>{{cite web|url=http://www.mvsoft.com/eqs60.htm|title=Multivariate Software, Inc.|website=www.mvsoft.com|url-status=dead|archive-url=https://web.archive.org/web/20010521070751/http://www.mvsoft.com/eqs60.htm|archive-date=2001-05-21}}</ref> This SEM software has a function to calculate reliability coefficients.
# RelCalc (free):<ref name = Cho/> Available with [[Microsoft Excel]]. <math>\rho_C</math> can be obtained without the need for SEM software. Various multidimensional SEM reliability coefficients and various types of <math>\omega_H</math> can be calculated based on the results of SEM software.