Editing Cronbach's alpha

{{Short description|Statistical measure of reliability}} 
{{confusing|date=May 2023|talk=Talk:Cronbach's_alpha#"Common misconceptions" section is misleading}}

'''Cronbach's alpha''' (Cronbach's <math>\alpha</math>), also known as '''tau-equivalent reliability''' (<math>\rho_T</math>) or '''coefficient alpha''' (coefficient <math>\alpha</math>), is a [[reliability coefficient]] and a measure of the [[internal consistency]] of tests and measures.<ref name=c1951>{{cite journal|last=Cronbach|first=Lee J.|title=Coefficient alpha and the internal structure of tests|journal=Psychometrika|publisher=Springer Science and Business Media LLC|volume=16|issue=3|year=1951|doi=10.1007/bf02310555|pages=297–334| hdl=10983/2196|s2cid=13820448|hdl-access=free}}</ref><ref name=c1978>{{cite journal|last=Cronbach|first=L. J.|year=1978|title=Citation Classics|journal=Current Contents|volume=13|pages=263|url=http://garfield.library.upenn.edu/classics1978/A1978EQ39200002.pdf|access-date=2021-03-22|archive-date=2022-01-20|archive-url=https://web.archive.org/web/20220120235644/http://garfield.library.upenn.edu/classics1978/A1978EQ39200002.pdf|url-status=live}}</ref><ref name=Cho>{{cite journal|last=Cho|first=Eunseong|title=Making Reliability Reliable|journal=Organizational Research Methods|publisher=SAGE Publications|volume=19|issue=4|date=2016-07-08|issn=1094-4281|doi=10.1177/1094428116656239|pages=651–682| s2cid=124129255}}</ref> It was named after the American psychologist [[Lee Cronbach]]. 

Numerous studies warn against using Cronbach's alpha unconditionally. Statisticians regard reliability coefficients based on [[structural equation modeling]] (SEM) or [[generalizability theory]] as superior alternatives in many situations.<ref name="Sijtsma">{{cite journal|first=K.|last=Sijtsma|title=On the use, the misuse, and the very limited usefulness of Cronbach's alpha|journal=Psychometrika|volume=74|issue=1|pages=107–120|date=2009|doi=10.1007/s11336-008-9101-0|pmid=20037639|pmc=2792363}}</ref><ref name="GY">{{cite journal|last1=Green|first1=S. B.|last2=Yang|first2=Y.|title=Commentary on coefficient alpha: A cautionary tale|journal=Psychometrika|volume=74|issue=1|pages=121–135|date=2009|doi=10.1007/s11336-008-9098-4|s2cid=122718353}}</ref><ref name="RZ">{{cite journal|last1=Revelle|first1=W.|last2=Zinbarg|first2=R. E.|title=Coefficients alpha, beta, omega, and the glb: Comments on Sijtsma|journal=Psychometrika|volume=74|issue=1|pages=145–154|date=2009|doi=10.1007/s11336-008-9102-z|s2cid=5864489}}</ref><ref name="ChoKim">{{cite journal|last1=Cho|first1=E.|last2=Kim|first2=S.|title=Cronbach's coefficient alpha: Well known but poorly understood|journal=Organizational Research Methods|volume=|issue=2|pages=207–230|date=2015|doi=10.1177/1094428114555994|s2cid=124810308}}</ref><ref name="RM">{{cite journal|last1=Raykov|first1=T.|last2=Marcoulides|first2=G. A.|title=Thanks coefficient alpha, we still need you!|journal=Educational and Psychological Measurement|volume=79|issue=1|pages=200–210|date=2017|doi=10.1177/0013164417725127|pmid=30636788|pmc=6318747}}</ref><ref name="c2004">{{cite journal|last1=Cronbach|first1=L. J.|last2=Shavelson|first2=R. J.|title=My Current Thoughts on Coefficient Alpha and Successor Procedures|journal=Educational and Psychological Measurement|volume=64|issue=3|pages=391–418|date=2004|doi=10.1177/0013164404266386|s2cid=51846704}}</ref>

==History==
In his initial 1951 publication, [[Lee Cronbach]] described the coefficient as ''Coefficient'' ''alpha''<ref name=c1951/> and included an additional derivation.<ref name="Cronbach">{{cite journal|first=L.J.|last=Cronbach|title=Coefficient alpha and the internal structure of tests|journal=Psychometrika|volume=16|issue=3|pages=297–334|date=1951|doi=10.1007/BF02310555|s2cid=13820448|hdl=10983/2196|hdl-access=free}}</ref> ''Coefficient alpha'' had been used implicitly in previous studies,<ref name="Hoyt">{{cite journal|first=C.|last=Hoyt|title=Test reliability estimated by analysis of variance|journal=Psychometrika|volume=6|issue=3|pages=153–160|date=1941|doi=10.1007/BF02289270|s2cid=122361318}}</ref><ref name="Guttman">{{cite journal|first=L.|last=Guttman|title=A basis for analyzing test-retest reliability|journal=Psychometrika|volume=10|issue=4|pages=255–282|date=1945|doi=10.1007/BF02288892|pmid=21007983|s2cid=17220260}}</ref><ref name="JF">{{cite journal|last1=Jackson|first1=R. W. B.|last2=Ferguson|first2=G. A.|title=Studies on the reliability of tests|journal=University of Toronto Department of Educational Research Bulletin|volume=12|issue=|pages=132|date=1941}}</ref><ref name="Gulliksen">{{cite book|first=H.|last=Gulliksen|title=Theory of mental tests|publisher=Wiley|date=1950|doi=10.1037/13240-000}}</ref> but his interpretation was thought to be more intuitively attractive relative to previous studies and it became quite popular.<ref>{{Cite journal|last=Cronbach|first=Lee|date=1978|title=Citation Classics|url=http://www.garfield.library.upenn.edu/classics1978/A1978EQ39200002.pdf|journal=[[Current Contents]]|volume=13|issue=8|access-date=2022-10-21|archive-date=2022-10-22|archive-url=https://web.archive.org/web/20221022201253/http://www.garfield.library.upenn.edu/classics1978/A1978EQ39200002.pdf|url-status=live}}</ref> 

* In 1967, [[Melvin R. Novick|Melvin Novick]] and Charles Lewis proved that it was equal to reliability if the true scores{{efn|The true score is the difference between the score observed during the test or measurement and the error in that observation. See [[classical test theory]] for further information.|name=Footnote|group=lower-roman}} of the compared tests or measures vary by a constant, which is independent of the people measured. In this case, the tests or measurements were said to be "essentially tau-equivalent."<ref name="NL">{{cite journal|last1=Novick|first1=M. R.|last2=Lewis|first2=C.|title=Coefficient alpha and the reliability of composite measurements|journal=Psychometrika|volume=32|issue=1|pages=1–13|date=1967|doi=10.1007/BF02289400|pmid=5232569|s2cid=186226312}}</ref>
* In 1978, Cronbach asserted that the reason the initial 1951 [[publication]] was widely cited was "mostly because [he] put a brand name on a common-place coefficient."<ref name="c1978" />{{rp|263}}<ref name="Cho" /> He explained that he had originally planned to name other types of reliability coefficients, such as those used in [[inter-rater reliability]] and [[Repeatability|test-retest reliability]], after consecutive Greek letters (i.e., <math>\beta</math>, <math>\gamma</math>, etc.), but later changed his mind.
* Later, in 2004, Cronbach and [[Richard Shavelson]] encouraged readers to use [[generalizability theory]] rather than <math>\rho_{T}</math>. Cronbach opposed the use of the name "Cronbach's alpha" and explicitly denied the existence of studies that had published the general formula of [[Kuder–Richardson formulas|KR-20]] before Cronbach's 1951 publication of the same name.<ref name="c2004" />

==Prerequisites for using Cronbach's alpha==
To use Cronbach's alpha as a reliability coefficient, the following conditions must be met:<ref>{{Cite journal|last=Spiliotopoulou|first=Georgia|date=2009|title=Reliability reconsidered: Cronbach's alpha and paediatric assessment in occupational therapy|url=https://onlinelibrary.wiley.com/doi/10.1111/j.1440-1630.2009.00785.x|journal=Australian Occupational Therapy Journal|language=en|volume=56|issue=3|pages=150–155|doi=10.1111/j.1440-1630.2009.00785.x|pmid=20854508|access-date=2022-10-21|archive-date=2022-10-21|archive-url=https://web.archive.org/web/20221021235248/https://onlinelibrary.wiley.com/doi/10.1111/j.1440-1630.2009.00785.x|url-status=live}}</ref><ref>{{Cite journal|last=Cortina|first=Jose M.|date=1993|title=What is coefficient alpha? An examination of theory and applications.|url=http://doi.apa.org/getdoi.cfm?doi=10.1037/0021-9010.78.1.98|journal=Journal of Applied Psychology|language=en|volume=78|issue=1|pages=98–104|doi=10.1037/0021-9010.78.1.98|issn=1939-1854|access-date=2022-10-21|archive-date=2023-08-13|archive-url=https://web.archive.org/web/20230813121351/https://psycnet.apa.org/doiLanding?doi=10.1037/0021-9010.78.1.98|url-status=live|url-access=subscription}}</ref>
# The data is [[Normal distribution|normally distributed]] and [[Linearity|linear]]{{Efn|This implicitly requires that the data can be ordered, and thus requires that it is not [[Level_of_measurement#Nominal_level|nominal]].|group=lower-roman|name=Footnote2}};
# The compared tests or measures are essentially tau-equivalent;
# Errors in the measurements are [[Independence (statistics)|independent]].

==Formula and calculation==
Cronbach's alpha is calculated by taking a score from each scale item and correlating it with the total score for each observation. The resulting correlations are then compared with the [[variance]] for all individual item scores. Cronbach's alpha is best understood as a function of the number of questions or items in a measure, the average [[covariance]] between pairs of items, and the overall variance of the total measured score.<ref>{{Cite web|last=Goforth|first=Chelsea|date=November 16, 2015|title=Using and Interpreting Cronbach's Alpha - University of Virginia Library Research Data Services + Sciences|url=https://data.library.virginia.edu/using-and-interpreting-cronbachs-alpha/|access-date=2022-09-06|website=University of Virginia Library|archive-date=2022-08-09|archive-url=https://web.archive.org/web/20220809031644/https://data.library.virginia.edu/using-and-interpreting-cronbachs-alpha/|url-status=live}}</ref><ref name=RM/>

<math display="block">\alpha = {k \over k-1 } \left(1 - {\sum_{i=1}^k \sigma^2_{y_i} \over \sigma^2_y} \right)</math>

where:

* <math>k</math> represents the number of items in the measure
* <math>\sigma_{y_i}^2</math> the variance associated with each item ''i''
* <math>\sigma_y^2</math> the variance associated with the total scores, <math>y = \sum_{i=1}^k y_i</math>

Alternatively, it can be calculated through the following formula:<ref>{{Cite AV media|title=Cronbach's Alpha (Simply explained)|url=https://www.youtube.com/watch?v=W9uPvAmtTOk&t=248|date=October 27, 2021|access-date=2023-08-01|author=DATAtab|publisher=YouTube|time=4:08}}</ref>
:<math> \alpha = {k \bar c \over \bar v + (k - 1) \bar c} </math>

where:

* <math>\bar v</math> represents the average variance
* <math>\bar c</math> represents the average inter-item covariance.

==Common misconceptions==
{{Confusing|section|date=May 2023|talk=Talk:Cronbach's_alpha#"Common misconceptions" section is misleading|reason=it is unclear whether headings are true or false}}
Application of Cronbach's alpha is not always straightforward and can give rise to common misconceptions, some of which are detailed here.<ref name="ChoKim" />
===The value of Cronbach's alpha ranges between zero and one===
By definition, reliability cannot be less than zero and cannot be greater than one. Many textbooks mistakenly equate <math>\rho_{T}</math> with reliability and give an inaccurate [[explanation]] of its range. <math>\rho_{T}</math> can be less than reliability when applied to data that are not essentially tau-equivalent. Suppose that <math>X_2</math> copied the value of <math>X_1</math> as it is, and <math>X_3</math> copied by multiplying the value of <math>X_1</math> by -1.

The covariance matrix between items is as follows, <math>\rho_{T}=-3</math>.
{| class="wikitable" style="text-align: right;"
|+ Observed covariance matrix
|-
!
! <math>X_1</math>
! <math>X_2</math>
! <math>X_3</math>
|-
! <math>X_1</math>
| <math>1</math>||<math>1</math>||<math>-1</math>
|-
! <math>X_2</math>
| <math>1</math>|| <math>1</math>|| <math>-1</math>
|-
! <math>X_3</math>
| <math>-1</math>|| <math>-1</math>|| <math>1</math>
|}
Negative <math>\rho_{T}</math> can occur for reasons such as negative discrimination or mistakes in processing reversely scored items.

Unlike <math>\rho_{T}</math>, SEM-based reliability coefficients (e.g., <math>\rho_{C}</math>) are always greater than or equal to zero.

This anomaly was first pointed out by Cronbach (1943)<ref name="c1943">{{cite journal|first=L. J.|last=Cronbach|title=On estimates of test reliability|journal=Journal of Educational Psychology|volume=34|issue=8|pages=485–494|date=1943|doi=10.1037/h0058608}}</ref> to criticize <math>\rho_{T}</math>, but Cronbach (1951)<ref name="Cronbach"/> did not comment on this problem in his article that otherwise discussed potentially problematic issues related <math>\rho_{T}</math>.<ref name="c2004"/>{{rp|396}}<ref>{{Cite journal |last1=Waller |first1=Niels |last2=Revelle |first2=William |date=2023-05-25 |title=What are the mathematical bounds for coefficient α? |url=https://doi.apa.org/doi/10.1037/met0000583 |journal=Psychological Methods |language=en |doi=10.1037/met0000583 |pmid=37227892 |issn=1939-1463|url-access=subscription }}</ref>

===If there is no measurement error, the value of Cronbach's alpha is one.===
This anomaly also originates from the fact that <math>\rho_{T}</math> underestimates reliability. 

Suppose that <math>X_2</math> copied the value of <math>X_1</math> as it is, and <math>X_3</math> copied by multiplying the value of <math>X_1</math> by two. 

The covariance matrix between items is as follows, <math>\rho_{T}=0.9375</math>.
{| class="wikitable" style="text-align: center;"
|+ Observed covariance matrix
|-
!
! <math>X_1</math>
! <math>X_2</math>
! <math>X_3</math>
|-
! <math>X_1</math>
| <math>1</math>||<math>1</math>||<math>2</math>
|-
! <math>X_2</math>
| <math>1</math>|| <math>1</math>|| <math>2</math>
|-
! <math>X_3</math>
| <math>2</math>|| <math>2</math>|| <math>4</math>
|}
For the above data, both <math>\rho_{P}</math> and <math>\rho_{C}</math> have a value of one.

The above example is presented by Cho and Kim (2015).<ref name = ChoKim/>

===A high value of Cronbach's alpha indicates homogeneity between the items===
Many textbooks refer to <math>\rho_{T}</math> as an indicator of [[Homogeneity and heterogeneity (statistics)|homogeneity]]<ref>{{Cite web|title=APA Dictionary of Psychology|url=https://dictionary.apa.org/|access-date=2023-02-20|website=dictionary.apa.org|language=en|archive-date=2019-07-31|archive-url=https://web.archive.org/web/20190731124940/http://dictionary.apa.org/|url-status=live}}</ref> between items. This misconception stems from the inaccurate explanation of Cronbach (1951)<ref name = Cronbach/> that high <math>\rho_{T}</math> values show homogeneity between the items. Homogeneity is a term that is rarely used in modern literature, and related studies interpret the term as referring to uni-dimensionality. Several studies have provided proofs or counterexamples that high <math>\rho_{T}</math> values do not indicate uni-dimensionality.<ref name=Cortina>{{cite journal|first=J. M.|last=Cortina|title=What is coefficient alpha? An examination of theory and applications|journal=Journal of Applied Psychology|volume=78|issue=1|pages=98–104|date=1993|doi=10.1037/0021-9010.78.1.98}}</ref><ref name=ChoKim/><ref name=GLM>{{cite journal|last1=Green|first1=S. B.|last2=Lissitz|first2=R. W.|last3=Mulaik|first3=S. A.|title=Limitations of coefficient alpha as an Index of test unidimensionality|journal=Educational and Psychological Measurement|volume=37|issue=4|pages=827–838|date=1977|doi=10.1177/001316447703700403|s2cid=122986180}}</ref><ref>{{cite journal|first=R. P.|last=McDonald|title=The dimensionality of tests and items|journal=The British Journal of Mathematical and Statistical Psychology|volume=34|issue=1|pages=100–117|date=1981|doi=10.1111/j.2044-8317.1981.tb00621.x}}</ref><ref>{{cite journal|first=N.|last=Schmitt|title=Uses and abuses of coefficient alpha|journal=Psychological Assessment|volume=8|issue=4|pages=350–3|date=1996|doi=10.1037/1040-3590.8.4.350}}</ref><ref name=TBC>{{cite journal|last1=Ten Berge|first1=J. M. F.|last2=Sočan|first2=G.|title=The greatest lower bound to the reliability of a test and the hypothesis of unidimensionality|journal=Psychometrika|volume=69|issue=4|pages=613–625|date=2004|doi=10.1007/BF02289858|s2cid=122674001}}</ref> See counterexamples below.
{| class="wikitable" style="text-align: right;"
|+ Uni-dimensional data
|-
!
! <math>X_1</math>
! <math>X_2</math>
! <math>X_3</math>
! <math>X_4</math>
! <math>X_5</math>
! <math>X_6</math>
|-
! <math>X_1</math>
| <math>10</math>||<math>3</math>||<math>3</math>||<math>3</math>||<math>3</math>||<math>3</math>
|-
! <math>X_2</math>
| <math>3</math>|| <math>10</math>|| <math>3</math>|| <math>3</math>|| <math>3</math>|| <math>3</math>
|-
! <math>X_3</math>
| <math>3</math>|| <math>3</math>|| <math>10</math>|| <math>3</math>|| <math>3</math>|| <math>3</math>
|-
! <math>X_4</math>
| <math>3</math>|| <math>3</math>|| <math>3</math>|| <math>10</math>|| <math>3</math>|| <math>3</math>
|-
! <math>X_5</math>
| <math>3</math>|| <math>3</math>|| <math>3</math>|| <math>3</math>|| <math>10</math>|| <math>3</math>
|-
! <math>X_6</math>
| <math>3</math>|| <math>3</math>|| <math>3</math>|| <math>3</math>|| <math>3</math>|| <math>10</math>
|-
|}
<math>\rho_{T}=0.72</math> in the uni-dimensional data above.
{| class="wikitable" style="text-align: right;"
|+ Multidimensional data
|-
!
! <math>X_1</math>
! <math>X_2</math>
! <math>X_3</math>
! <math>X_4</math>
! <math>X_5</math>
! <math>X_6</math>
|-
! <math>X_1</math>
| <math>10</math>||<math>6</math>||<math>6</math>||<math>1</math>||<math>1</math>||<math>1</math>
|-
! <math>X_2</math>
| <math>6</math>|| <math>10</math>|| <math>6</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>
|-
! <math>X_3</math>
| <math>6</math>|| <math>6</math>|| <math>10</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>
|-
! <math>X_4</math>
| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>10</math>|| <math>6</math>|| <math>6</math>
|-
! <math>X_5</math>
| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>6</math>|| <math>10</math>|| <math>6</math>
|-
! <math>X_6</math>
| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>6</math>|| <math>6</math>|| <math>10</math>
|-
|}
<math>\rho_{T}=0.72</math> in the multidimensional data above.
{| class="wikitable" style="text-align: right;"
|+ Multidimensional data with extremely high reliability
|-
!
! <math>X_1</math>
! <math>X_2</math>
! <math>X_3</math>
! <math>X_4</math>
! <math>X_5</math>
! <math>X_6</math>
|-
! <math>X_1</math>
| <math>10</math>||<math>9</math>||<math>9</math>||<math>8</math>||<math>8</math>||<math>8</math>
|-
! <math>X_2</math>
| <math>9</math>|| <math>10</math>|| <math>9</math>|| <math>8</math>|| <math>8</math>|| <math>8</math>
|-
! <math>X_3</math>
| <math>9</math>|| <math>9</math>|| <math>10</math>|| <math>8</math>|| <math>8</math>|| <math>8</math>
|-
! <math>X_4</math>
| <math>8</math>|| <math>8</math>|| <math>8</math>|| <math>10</math>|| <math>9</math>|| <math>9</math>
|-
! <math>X_5</math>
| <math>8</math>|| <math>8</math>|| <math>8</math>|| <math>9</math>|| <math>10</math>|| <math>9</math>
|-
! <math>X_6</math>
| <math>8</math>|| <math>8</math>|| <math>8</math>|| <math>9</math>|| <math>9</math>|| <math>10</math>
|-
|}

The above data have <math>\rho_{T}=0.9692</math>, but are multidimensional.
{| class="wikitable" style="text-align: right;"
|+ Uni-dimensional data with unacceptably low reliability
|-
!
! <math>X_1</math>
! <math>X_2</math>
! <math>X_3</math>
! <math>X_4</math>
! <math>X_5</math>
! <math>X_6</math>
|-
! <math>X_1</math>
| <math>10</math>||<math>1</math>||<math>1</math>||<math>1</math>||<math>1</math>||<math>1</math>
|-
! <math>X_2</math>
| <math>1</math>|| <math>10</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>
|-
! <math>X_3</math>
| <math>1</math>|| <math>1</math>|| <math>10</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>
|-
! <math>X_4</math>
| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>10</math>|| <math>1</math>|| <math>1</math>
|-
! <math>X_5</math>
| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>10</math>|| <math>1</math>
|-
! <math>X_6</math>
| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>1</math>|| <math>10</math>
|-
|}
The above data have <math>\rho_{T}=0.4</math>, but are uni-dimensional.

Uni-dimensionality is a prerequisite for <math>\rho_{T}</math>. One should check uni-dimensionality before calculating <math>\rho_{T}</math> rather than calculating <math>\rho_{T}</math> to check uni-dimensionality.<ref name = Cho/>

===A high value of Cronbach's alpha indicates internal consistency===
The term "internal consistency" is commonly used in the reliability literature, but its meaning is not clearly defined. The term is sometimes used to refer to a certain kind of reliability (e.g., internal consistency reliability), but it is unclear exactly which reliability coefficients are included here, in addition to <math>\rho_{T}</math>. Cronbach (1951)<ref name = Cronbach/> used the term in several senses without an explicit definition. Cho and Kim (2015)<ref name = ChoKim/> showed that <math>\rho_{T}</math> is not an indicator of any of these.

===Removing items using "alpha if item deleted" always increases reliability===
Removing an item using "alpha if item deleted"{{Clarify|reason=What is "alpha if item deleted"?|date=August 2022}} may result in 'alpha inflation,' where sample-level reliability is reported to be higher than population-level reliability.<ref name=KL>{{cite journal|last1=Kopalle|first1=P. K.|last2=Lehmann|first2=D. R.|title=Alpha inflation? The impact of eliminating scale items on Cronbach's alpha|journal=Organizational Behavior and Human Decision Processes|volume=70|issue=3|pages=189–197|date=1997|doi=10.1006/obhd.1997.2702|doi-access=free}}</ref> It may also reduce population-level reliability.<ref name=r2007>{{cite journal|first=T.|last=Raykov|title=Reliability if deleted, not 'alpha if deleted': Evaluation of scale reliability following component deletion|journal=The British Journal of Mathematical and Statistical Psychology|volume=60|issue=2|pages=201–216|date=2007|doi=10.1348/000711006X115954|pmid=17971267}}</ref> The elimination of less-reliable items should be based not only on a statistical basis but also on a theoretical and logical basis. It is also recommended that the whole sample be divided into two and cross-validated.<ref name=KL/>

==Ideal reliability level and how to increase reliability==
===Nunnally's recommendations for the level of reliability===
Nunnally's book<ref name="n1">{{cite book|first=J. C.|last=Nunnally|title=Psychometric theory|publisher=McGraw-Hill|date=1967|isbn=0-07-047465-6|oclc=926852171}}</ref><ref name="n3">{{cite book|last1=Nunnally|first1=J. C.|last2=Bernstein|first2=I. H.|title=Psychometric theory|publisher=McGraw-Hill|edition=3rd|date=1994|isbn=0-07-047849-X|pages=|oclc=28221417}}</ref> is often mentioned as the primary source for determining the appropriate level of dependability coefficients. However, his proposals contradict his aims as he suggests that different criteria should be used depending on the goal or stage of the investigation. Regardless of the type of study, whether it is exploratory research, applied research, or scale development research, a criterion of 0.7 is universally employed.<ref name="LBM">{{cite journal|last1=Lance|first1=C. E.|last2=Butts|first2=M. M.|last3=Michels|first3=L. C.|title=What did they really say?|journal=Organizational Research Methods|volume=9|issue=2|pages=202–220|date=2006|doi=10.1177/1094428105284919|s2cid=144195175}}</ref> He advocated 0.7 as a criterion for the early stages of a study, most studies published in the journal do not fall under that category. Rather than 0.7, Nunnally's applied research criterion of 0.8 is more suited for most empirical studies.<ref name="LBM"/>
{| class="wikitable" style="text-align: right;
|+ Nunnally's recommendations on the level of reliability
|-
! !! 1st edition<ref name=n1/> !! 2nd & 3rd<ref name=n3/> edition
|-
| Early stage of research|| 0.5 or 0.6|| 0.7
|-
| Applied research|| 0.8|| 0.8
|-
| When making important decisions|| 0.95 (minimum 0.9)|| 0.95 (minimum 0.9)
|}

His recommendation level did not imply a cutoff point. If a criterion means a cutoff point, it is important whether or not it is met, but it is unimportant how much it is over or under. He did not mean that it should be strictly 0.8 when referring to the criteria of 0.8. If the reliability has a value near 0.8 (e.g., 0.78), it can be considered that his recommendation has been met.<ref name = c2020>{{cite journal|first=E.|last=Cho|title=A comprehensive review of so-called Cronbach's alpha|journal=Journal of Product Research|volume=38|issue=1|pages=9–20|date=2020|doi=}}</ref>

===Cost to obtain a high level of reliability===
Nunnally's idea was that there is a cost to increasing reliability, so there is no need to try to obtain maximum reliability in every situation.

====Trade-off with validity====
Measurements with perfect reliability lack validity.<ref name = ChoKim/> For example, a person who takes the test with a reliability of one will either receive a perfect score or a zero score, because if they answer one item correctly or incorrectly, they will answer all other items in the same manner. The phenomenon where validity is sacrificed to increase reliability is known as the attenuation paradox.<ref>{{cite journal|first=J.|last=Loevinger|title=The attenuation paradox in test theory|journal=Psychological Bulletin|volume=51|issue=5|pages=493–504|date=1954|doi=10.1002/j.2333-8504.1954.tb00485.x|pmid=13204488}}</ref><ref>{{cite journal|first=L.|last=Humphreys|title=The normal curve and the attenuation paradox in test theory|journal=Psychological Bulletin|volume=53|issue=6|pages=472–6|date=1956|doi=10.1037/h0041091|pmid=13370692}}</ref>

A high value of reliability can conflict with content validity. To achieve high content validity, each item should comprehensively represent the content to be measured. However, a strategy of repeatedly measuring essentially the same question in different ways is often used solely to increase reliability.<ref>{{cite journal|first=G. J.|last=Boyle|title=Does item homogeneity indicate internal consistency or item redundancy in psychometric scales?|journal=Personality and Individual Differences|volume=12|issue=3|pages=291–4|date=1991|doi=10.1016/0191-8869(91)90115-R}}</ref><ref>{{cite journal|first=D. L.|last=Streiner|title=Starting at the beginning: An introduction to coefficient alpha and internal consistency|journal=Journal of Personality Assessment|volume=80|issue=1|pages=99–103|date=2003|doi=10.1207/S15327752JPA8001_18|pmid=12584072|s2cid=3679277}}</ref>

====Trade-off with efficiency====
When the other conditions are equal, reliability increases as the number of items increases. However, the increase in the number of items hinders the efficiency of measurements.

===Methods to increase reliability===
Despite the costs associated with increasing reliability discussed above, a high level of reliability may be required. The following methods can be considered to increase reliability.

Before [[data collection]]:
* Eliminate the ambiguity of the measurement item.
* Do not measure what the respondents do not know.<ref>{{Cite journal|last1=Beatty|first1=P.|last2=Herrmann|first2=D.|last3=Puskar|first3=C.|last4=Kerwin|first4=J.|date=July 1998|title="Don't know" responses in surveys: is what I know what you want to know and do I want you to know it?|url=https://pubmed.ncbi.nlm.nih.gov/9829099/|journal=Memory (Hove, England)|volume=6|issue=4|pages=407–426|doi=10.1080/741942605|issn=0965-8211|pmid=9829099|access-date=2023-02-20|archive-date=2023-02-20|archive-url=https://web.archive.org/web/20230220140847/https://pubmed.ncbi.nlm.nih.gov/9829099/|url-status=live}}</ref>
* Increase the number of items. However, care should be taken not to excessively inhibit the efficiency of the measurement.
* Use a scale that is known to be highly reliable.<ref>Lee, H. (2017). Research Methodology (2nd ed.), Hakhyunsa.</ref>
* Conduct a pretest - discover in advance the problem of reliability.
* Exclude or modify items that are different in content or form from other items (e.g., reverse-scored items).

After data collection:
* Remove the problematic items using "alpha if item deleted". However, this deletion should be accompanied by a theoretical rationale.
* Use a more accurate reliability coefficient than <math>\rho_{T}</math>. For example, <math>\rho_{C}</math> is 0.02 larger than <math>\rho_{T}</math> on average.<ref name="PK">{{cite journal|last1=Peterson|first1=R. A.|last2=Kim|first2=Y.|title=On the relationship between coefficient alpha and composite reliability|journal=Journal of Applied Psychology|volume=98|issue=1|pages=194–8|date=2013|doi=10.1037/a0030767|pmid=23127213}}</ref>

==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.

==Notes==
{{notelist|group=lower-roman}}

==References==
{{Reflist|30em}}

==External links==
* [http://www.open.ac.uk/socialsciences/spsstutorial/files/tutorials/cronbachs-alpha.pdf Cronbach's alpha SPSS tutorial]
* The free web interface and R package [http://comparingcronbachalphas.org cocoon] allow us to statistically compare two or more dependent or independent Cronbach alpha coefficients.

{{DEFAULTSORT:Cronbach's Alpha}}
[[Category:Comparison of assessments]]
[[Category:Statistical reliability]]
[[Category:Psychometrics]]