Editing Cronbach's alpha (section)

===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/>