Editing Internal consistency (section)

==Cronbach's alpha==
{{main|Cronbach's alpha}}
Internal consistency is usually measured with Cronbach's alpha, a [[statistic]] calculated from the pairwise correlations between items. Internal consistency ranges between negative infinity and one. Coefficient alpha will be negative whenever there is greater within-subject variability than between-subject variability.<ref>{{cite journal |last=Knapp |first=T. R. |year=1991 |title=Coefficient alpha: Conceptualizations and anomalies |journal=Research in Nursing & Health |volume=14 |pages=457-480 |doi=10.1002/nur.4770140610 }}</ref>

A commonly accepted rule of thumb for describing internal consistency is as follows:<ref>{{cite book |last1=George |first1=D. |last2=Mallery |first2=P. |year=2003 |title=SPSS for Windows step by step: A simple guide and reference. 11.0 update |edition=4th |location=Boston |publisher=Allyn & Bacon |isbn=978-0205375523 }}</ref>
{| class="wikitable"
|-
! [[Cronbach's alpha]] !! Internal consistency
|-
| 0.9 ≤ α || Excellent
|-
| 0.8 ≤ α < 0.9 || Good
|-
| 0.7 ≤ α < 0.8 || Acceptable
|-
| 0.6 ≤ α < 0.7 || Questionable
|-
| 0.5 ≤ α < 0.6 || Poor
|-
| α < 0.5 || Unacceptable
|}

Very high reliabilities (0.95 or higher) are not necessarily desirable, as this indicates that the items may be redundant.<ref>{{cite journal |last=Streiner |first=D. L. |year=2003 |title=Starting at the beginning: an introduction to coefficient alpha and internal consistency |journal=Journal of Personality Assessment |volume=80 |pages=99-103 |doi=10.1207/S15327752JPA8001_18 }}</ref> The goal in designing a reliable instrument is for scores on similar items to be related (internally consistent), but for each to contribute some unique information as well. Note further that Cronbach's alpha is necessarily higher for tests measuring more narrow constructs, and lower when more generic, broad constructs are measured. This phenomenon, along with a number of other reasons, argue against using objective cut-off values for internal consistency measures.<ref>{{cite journal |last=Peters |first=G.-J. Y |year=2014 |title=The alpha and the omega of scale reliability and validity: Why and how to abandon Cronbach’s alpha and the route towards more comprehensive assessment of scale quality |journal=European Health Psychologist |volume=16 |issue=2 |doi=10.31234/osf.io/h47fv |url=http://ehps.net/ehp/index.php/contents/article/download/ehp.v16.i2.p56/1 }}</ref> Alpha is also a [[function (mathematics)|function]] of the number of items, so shorter scales will often have lower reliability estimates yet still be preferable in many situations because they are lower burden. It is also worth noting that, unlike [[Item response theory|Item Response Theory (IRT)]] based perspectives on reliability, Cronbach's alpha only provides a single summary value which may not accurately represent the variability of reliability across different points of the score distribution (e.g., it may overstate the reliability of measurement at extremes of the score distribution)<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>.

An alternative way of thinking about internal consistency is that it is the extent to which all of the items of a test measure the same [[latent variable]].  The advantage of this perspective over the notion of a high average correlation among the items of a test – the perspective underlying Cronbach's alpha – is that the average item correlation is affected by [[skewness]] (in the distribution of item correlations) just as any other [[average]] is.  Thus, whereas the [[mode (statistics)|modal]] item correlation is zero when the items of a test measure several unrelated latent variables, the average item correlation in such cases will be greater than zero.  Thus, whereas the ideal of measurement is for all items of a test to measure the same latent variable, alpha has been demonstrated many times to attain quite high values even when the set of items measures several unrelated latent variables.<ref>{{cite journal |last=Cortina |first=J. M. |year=1993 |title=What is coefficient alpha? An examination of theory and applications |journal=Journal of Applied Psychology |volume=78 |pages=98&ndash;104 |doi=10.1037/0021-9010.78.1.98 |url=https://www.psycholosphere.com/what%20is%20coefficient%20alpha%20by%20Cortina.pdf }}</ref><ref>{{cite journal |last=Cronbach |first=L. J. |year=1951 |title=Coefficient alpha and the internal structure of tests |journal=Psychometrika |volume=16 |issue=3 |pages=297&ndash;334 |doi=10.1007/BF02310555 |url=http://cda.psych.uiuc.edu/psychometrika_highly_cited_articles/cronbach_1951.pdf }}</ref><ref>{{cite journal |last1=Green |first1=S. B. |last2=Lissitz |first2=R.W. |last3=Mulaik |first3=S. A. |year=1977 |title=Limitations of coefficient alpha as an index of test unidimensionality |journal=Educational and Psychological Measurement |volume=37 |pages=827&ndash;838 |doi=10.1177/001316447703700403 }}</ref><ref>{{cite journal |last=Revelle |first=W. |year=1979 |title=Hierarchical cluster analysis and the internal structure of tests |journal=Multivariate Behavioral Research |volume=14 |pages=57&ndash;74 |doi=10.1207/s15327906mbr1401_4 |url=https://personality-project.org/revelle/publications/iclust.pdf }}</ref><ref>{{cite journal |last=Schmitt |first=N. |year=1996 |title=Uses and abuses of coefficient alpha |journal=Psychological Assessment |volume=8 |pages=350&ndash;353 |doi=10.1037/1040-3590.8.4.350 |url=https://dosen.perbanas.id/wp-content/uploads/2017/05/Schmitt-1996-Uses-and-abuses-of-coefficient-alpha.pdf }}</ref><ref>{{cite journal |last1=Zinbarg |first1=R. |last2=Yovel |first2=I. |last3=Revelle |first3=W. |last4=McDonald |first4=R. |year=2006 |title=Estimating generalizability to a universe of indicators that all have an attribute in common: A comparison of estimators for ''ω<sub>h</sub>'' |journal=Applied Psychological Measurement |volume=30 |pages=121&ndash;144 |doi=10.1177/0146621605278814 |url=https://psychology.northwestern.edu/documents/faculty-publications/zinbarg-estimator.pdf }}</ref><ref>{{cite journal |last1=Trippi |first1=R. |last2=Settle |first2=R. |year=1976 |title=A Nonparametric Coefficient of Internal Consistency |journal=Multivariate Behavioral Research |volume=4 |pages=419-424 |doi=10.1207/s15327906mbr1104_3 |url=http://www.sigma-research.com/misc/Nonparametric%20Coefficient%20of%20Internal%20Consistency.htm |url-access=subscription }}</ref>
The hierarchical "coefficient omega" may be a more appropriate index of the extent to which all of the items in a test measure the same latent variable.<ref>{{cite book |last=McDonald |first=R. P. |year=1999 |title=Test theory: A unified treatment |publisher=Psychology Press |doi=10.4324/9781410601087 |isbn=0-8058-3075-8 }}</ref><ref>{{cite journal |last1=Zinbarg |first1=R. |last2=Revelle |first2=W. |last3=Yovel |first3=I. |last4=Li |first4=W. |year=2005 |title=Cronbach’s α, Revelle’s β, and McDonald’s ωH: Their relations with each other and two alternative conceptualizations of reliability |journal=Psychometrika |volume=70 |pages=123&ndash;133 |doi=10.1007/s11336-003-0974-7 |url=https://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf }}</ref> Several different measures of internal consistency are reviewed by Revelle & Zinbarg (2009).<ref>{{cite journal |last1=Revelle |first1=W. |last2=Zinbarg |first2=R. |year=2009 |title=Coefficients Alpha, Beta, Omega, and the glb: Comments on Sijtsma |journal=Psychometrika |volume=74 |issue=1 |pages=145&ndash;154 |doi=10.1007/s11336-008-9102-z |url=https://personality-project.org/revelle/publications/revelle.zinbarg.08.pdf }}</ref><ref>{{cite journal |last1=Dunn |first1=T. J. |last2=Baguley |first2=T. |last3=Brunsden |first3=V. |year=2013 |title=From alpha to omega: A practical solution to the pervasive problem of internal consistency estimation |journal=British Journal of Psychology |doi=10.1111/bjop.12046 |url=https://irep.ntu.ac.uk/id/eprint/4853/1/215051_Dunn.pdf }}</ref>