Cronbach's alpha (Cronbach's <math>\alpha</math>) or coefficient alpha (coefficient <math>\alpha</math>), is a reliability coefficient and a measure of the internal consistency of tests and measures. It was devised by the American psychometrician Lee Cronbach. Today it enjoys such wide-spread usage that numerous studies warn against using Cronbach's alpha uncritically.
History
In his initial 1951 publication, Lee Cronbach described the coefficient as Coefficient alpha Coefficient alpha had been used implicitly in previous studies, but his interpretation was thought to be more intuitively attractive relative to previous studies and it became quite popular.
- In 1967, Melvin Novick and Charles Lewis proved that it was equal to reliability if the true scores 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."
- 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."
- The "parts" (i.e. items, test parts, etc.) must be essentially tau-equivalent;
- Errors in the measurements are independent.
However, under the definition of CTT, the errors are defined to be independent.
This is often a source of confusion for users who might consider some aspect of the testing process to be an "error" (rater biases, examinee collusion, self-report faking). Anything that increases the covariance among the parts will contribute to greater true score variance. Under such circumstances, alpha is likely to over-estimate the reliability intended by the user.
:<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 (or the average covariance of "parts").
Common misconceptions
Application of Cronbach's alpha is not always straightforward and can give rise to common misconceptions.
A high value of Cronbach's alpha indicates homogeneity between the items
Many textbooks refer to <math>\alpha</math> as an indicator of homogeneity between items. This misconception stems from the inaccurate explanation of Cronbach (1951) 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>\alpha=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>\alpha=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>\alpha=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>\alpha=0.4</math>, but are uni-dimensional.
Uni-dimensionality is a prerequisite for <math>\alpha</math>. One should check uni-dimensionality before calculating <math>\alpha</math> rather than calculating <math>\alpha</math> to check uni-dimensionality.
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>\alpha</math>. Cronbach (1951) It may also reduce population-level reliability. 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.