Editing Cronbach's alpha (section)

==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/>