Cronbach's Alpha

The Cronbachsche (Alpha) is a measure named after Lee Cronbach for the internal consistency of a scale and describes the extent to which the tasks or questions on a scale are related to one another (interrelatedness). However, it is not a measure of the one-dimensionality of a scale. The Cronbach Alpha is mainly used in the social sciences and in psychology - especially in test construction and evaluation. It is used to estimate the internal consistency of a psychometric instrument. In the more recent literature, the term Cronbach's is rejected and the term tau-equivalent reliability ( ) is proposed instead . The tau-equivalent reliability is u. a. important in psychometrics . ${\displaystyle \,{\boldsymbol {\alpha }}}$ ${\displaystyle \alpha }$ ${\displaystyle \rho _{T}}$

history

A tau-equivalent measurement model is a special case of the congeneric measurement model shown here, in which all factor loads are assumed to be identical, i.e. H. . ${\displaystyle \lambda =\lambda _{1}=\lambda _{2}=\lambda _{3}=...=\lambda _{k}}$

The first designation as alpha was given by Cronbach in 1951, although the Kuder-Richardson formula is an older version for dichotomous items and Louis Guttman had already developed the same measure in 1945 under the name Lambda-3. Recently, the use of Cronbach's alpha and the term has been increasingly criticized. Cho (2016) suggests using tau-equivalent reliability instead of Cronbach's alpha ; Cho also makes it clear that it is appropriate to use congeneric reliability instead of in many cases (see there). ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{C}}$

definition

If one assumes that a sample was examined with regard to a group of k items, then Cronbach's is defined as the average correlation between these items, corrected upwards by k using the Spearman-Brown formula . This is why Cronbach's alpha is also called the measure of the internal consistency of a scale . Cronbach's is related to the result of an analysis of variance of the item data with regard to the variance between the test persons and the variance between the items. The higher the proportional variance between the test persons, the higher the Cronbach's . ${\displaystyle \alpha \,}$ ${\displaystyle \alpha \,}$ ${\displaystyle \alpha \,}$

interpretation

Rule of thumb for interpreting the alpha values

${\displaystyle \alpha \,}$	meaning
> 0.9	excellent
> 0.8	Well
> 0.7	acceptable
> 0.6	questionable
> 0.5	bad
${\displaystyle \leq }$ 0.5	unacceptable

${\displaystyle \alpha \,}$can assume values ​​between minus infinity and 1 (although only positive values ​​can be meaningfully interpreted). As a rule of thumb , any psychometric tool should only be used when it reaches a value of 0.65 or more. However, a value that is too high (e.g. 0.95) is also rated as critical, as this indicates that several items are redundant. In the case of smaller values, a factor analysis can be used to check whether the items are distributed over several factors. ${\displaystyle \alpha \,}$

Very often there is a reference in scientific papers to Nunnally (1978), who allegedly suggested that a value of 0.7 or more is considered acceptable. In fact, Nunnally discussed the use of the coefficient very carefully and by no means made a strict requirement. For Therefore, note, is that strict rules automatically reject the measurement models below a threshold value and automatically accept above a threshold, prohibit generally. The table in this section can therefore only be used as a guide. In particular, indicators should not be removed too quickly due to a low value, as this could affect the validity of the content . A framework for the elimination of indicators from measurement scales, which includes not only statistical criteria but also evaluative criteria, is described in Wieland et al. (2017). ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{T}}$

Another problem with such specifications is that the reliability of an instrument can very easily be achieved at the expense of bandwidth. This problem is also known as the bandwidth fidelity dilemma or reliability-validity dilemma . The broader and more general an instrument measures, the more chances there are usually to predict broad and distant criteria. On the other hand, reliability suffers due to the breadth. The only solution to this problem is usually to extend the test.

The Cronbach Alpha is often wrongly interpreted as evidence of one-dimensionality of a scale. A scale can be multidimensional and at the same time have a high internal consistency, consequently a high Cronbach alpha. An example would be a scale that combines items on depression and anxiety, i.e. is two-dimensional, and yet has a high consistency.

formula

The formula for calculating a standardized Cronbachian is: ${\displaystyle \alpha }$

${\displaystyle \alpha _{st}={\frac {N\cdot {\bar {r}}}{1+(N-1)\cdot {\bar {r}}}}}$,

where denotes the number of components (items or subscales) and the average correlation between the items. Alternatively, the Cronbachsche results from ${\displaystyle N}$ ${\displaystyle {\bar {r}}}$ ${\displaystyle \alpha }$

${\displaystyle \alpha ={\frac {N}{N-1}}\left(1-{\frac {\sum _{i=1}^{N}\sigma _{Y_{i}}^{2}}{\sigma _{X}^{2}}}\right)\qquad {\text{mit}}\qquad X=\sum _{i=1}^{N}Y_{i}}$,

where is the number of components (items or subscales), the variance of the observed total test scores and the variance in components (item, subscale) . The following usually applies to Likert scales . ${\displaystyle N}$ ${\displaystyle \sigma _{X}^{2}}$ ${\displaystyle \sigma _{Y_{i}}^{2}}$ ${\displaystyle i}$ ${\displaystyle \alpha _{st}\leq \alpha }$

example

correlation		Classic	jazz	Opera	rap	Heavy metal	Blues / R&B
	Classic	1	0.29	0.51	0.03	0.01	0.21
	jazz		1	0.21	0.22	0.09	0.54
	Opera			1	0.08	−0.04	0.19
	rap				1	0.30	0.17
	Heavy metal					1	0.09
	Blues / R & B						1

In the General Social Survey 1993 I asked about different styles of music with the answer categories (1 = likes music style, 2 = undecided, 3 = likes music style not). If a scale Mag Musik is now formed as the sum of the individual scales for each musical genre, this results ${\displaystyle N=6}$

${\displaystyle {\bar {r}}={\frac {0{,}29+0{,}51+\dots +0{,}17+0{,}09}{15}}=0{,}193}$

and

${\displaystyle \alpha _{st}={\frac {6\cdot 0{,}193}{1+5\cdot 0{,}193}}=0{,}590\qquad }$

In this case, most of the time, the new scale is not considered reliable; is there . The reason is that the correlation matrix shows at least two subscales: classical / opera and jazz / blues / R&B, i.e. H. when using Cronbach's one should be sure that the items really only form a scale (check with the factor analysis ). ${\displaystyle \alpha <0{,}7\,}$ ${\displaystyle \alpha }$

Alternative formula

Cho (2016) suggests an alternative formula for calculating the tau-equivalent reliability . This formula is equivalent to the previous one, so it leads to the same result: ${\displaystyle \rho _{T}}$

${\displaystyle \rho _{T}={\frac {k^{2}{\bar {\sigma }}_{ij}}{\sigma _{X}^{2}}}}$

Here, the number of indicators ( English items ) of the measurement model , the average covariance between the indicators and the variance of the test result. The advantage of this formula is that it is embedded in the system of formulas presented by Cho (2016) and facilitates a comparison with other coefficients, e.g. for congeneric reliability . The previously missing systematics in the naming is also the reason why Cho dispenses with the term “Cronbach's ” and instead speaks of “tau-equivalent reliability ”. However, both terms are synonyms. ${\displaystyle k(=N)}$ ${\displaystyle {\bar {\sigma }}_{ij}}$ ${\displaystyle \sigma _{X}^{2}}$ ${\displaystyle \alpha }$ ${\displaystyle \rho _{T}}$

A calculation example for both the traditional and the alternative formula can be found in Table 9 in Cho (2016).

Calculation of Cronbach's α with common statistical software

There are several packages for the free statistics software R that contain functions for calculating Cronbach's , e.g. B . ,, and . The R package cocron is also available as a free web interface and allows the statistical comparison of two or more dependent and independent Cronbach alphas. ${\displaystyle \alpha }$multilevel::cronbachpsy::cronbachpsych::alphapsychometric::alpha

In SAS the command line is . proc corr data=variable1 variable2 … variablen alpha plots;

In SPSS you select "Analyze", then "Scaling" or "Scale", then "Reliability analysis" and select the desired variables. The Cronbach alpha is then calculated for this. The syntax command since program version 17.0 is RELIABILITY VARIABLES=[VARIABLES] /MODEL=ALPHA..

The Cronbach's can be calculated with the instruction using the Stata program package . The item test and item-rest correlations are specified by selecting the option . With the option , the determined scale is saved as a variable. If the items on the scale are to be standardized beforehand (to mean 0 and variance 1), the option must also be added. ${\displaystyle \alpha }$alpha varlist [if] [in] [, options]itemgenerate(newvar)std

Alternatives

Cronbach's , or better the tau-equivalent reliability ( ), assumes the same factor loadings for all indicators. In reality, however, this requirement is rarely met, which means that reliability is underestimated. An alternative to that explicitly takes different factor loadings into account is congeneric reliability ( ), which has traditionally also been referred to as "composite reliability", a term that has recently been criticized. ${\displaystyle \alpha }$ ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{T}}$ ${\displaystyle \rho _{C}}$

See also

• Average recorded variance (DEV; English average variance Extracted )
• kongenerische reliability ( English congeneric reliability , formerly composite reliability )

literature

• LJ Cronbach: Coefficient alpha and the internal structure of tests. (PDF) In: Psychometrika , 16, 1951, pp. 297–334, doi: 10.1007 / BF02310555 .
• K. Schermelleh-Engel, CS Werner: Methods of determining reliability . In: Helfried Moosbrugger , Augustin Kelava (Ed.): Test theory and questionnaire construction , 2nd, updated and revised edition. Springer, Berlin / Heidelberg 2012, ISBN 978-3-642-20071-7 , pp. 119-141, doi : 10.1007 / 978-3-642-20072-4_6 .
• Neal Schmitt: Uses and Abuses of Coefficient Alpha. (PDF; 435 kB) In: Psychological Assessment , 8 (4), 1996, pp. 350–353, doi: 10.1037 / 1040-3590.8.4.350 .

Web links

• Calculation in the SPSS syntax
• The free web interface and R package cocron allows the statistical comparison of two or more dependent and independent Cronbach alphas.
• Handbook of Management Scales (English) by Wikibooks collects business management constructs, their indicators and gives Cronbach's alpha.
• RelCalc. Tools for calculating the tau equivalent and congeneric reliability as well as other coefficients.

Individual evidence

1. ↑ ^{a b} Jose M. Cortina: What is Coefficient Alpha? Examination of Theory and Applications . (PDF; 1.2 MB) In: Journal of Applied Psychology , 78 (1), 1993, pp. 98-104, doi: 10.1037 / 0021-9010.78.1.98 .
2. ↑ ^{a b c d} Cho. 2016, doi: 10.1177 / 1094428116656239
3. ^ Louis Guttman: A basis for analyzing test-retest reliability . In: Psychometrika . 10, 1945, pp. 255-282. doi : 10.1007 / BF02288892 .
4. ↑ Darren George, Paul Mallery: SPSS for Windows Step by Step: A Simple Guide and Reference, 11.0 Update . 4th edition. Allyn & Bacon, 2002, ISBN 978-0-205-37552-3 , pp. 231 .
5. ↑ DL Streiner: Starting at the beginning: An introduction to coefficient alpha and internal consistency In: Journal of Personality Assessment Ban 80, 2003, pp. 99-103. doi : 10.1207 / S15327752JPA8001_18
6. ^ JC Nunnally: Psychometric theory (2nd ed.). McGraw-Hill, New York 1978.
7. ↑ Guide, Ketokivi. 2015, doi: 10.1016 / S0272-6963 (15) 00056-X
8. ^ A. Wieland, CF Durach, J. Kembro, H. Treiblmaier: Statistical and judgmental criteria for scale purification . In: Supply Chain Management: An International Journal , Vol. 22, No. 4, 2017, doi: 10.1108 / SCM-07-2016-0230
9. ↑ K. Schermelleh-Engel, CS Werner: Methods of determining reliability . In: H. Moosbrugger, A. Kelava (Ed.): Test theory and questionnaire construction . Springer, Berlin / Heidelberg 2012, pp. 119–141, doi : 10.1007 / 978-3-642-20072-4_6
10. ↑ comparingcronbachalphas.org