Synonyms

Alpha Reliability; Kuder-Richardson KR20 Formula; Reliability Coefficient

Definition

Cronbach’s alpha is an estimator of test reliability measured as the internal consistency or inter-item homogeneity of the test score.

Description

Cronbach’s alpha is an estimator of test reliability that is suitable for use in single applications of a test, typically in cross-sectional designs. Given a test composed of p items, Cronbach’s alpha assumes that all items are equivalent test units and corresponds to the reliability of the full test computed by extending the properties of one unit p times. This procedure is conceptually analogous to split-halves reliability, where two parts of the same test are correlated to estimate the reliability of the full test.

Alpha measures the degree to which the items yield similar scores and must be interpreted as the proportion of measurement variance attributable to changes in individual’s true-score ranges. As a reliability index, coefficient alpha ranges between 0 and 1, although negative values can be obtained if the average inter-item covariance is negative (usually due to reverse coding of the items). All other things held equal, alpha increases as a function of the size of item correlations and the number of items. Recent developments for the computation of the standard error of alpha allow for inferences about this reliability index.

Estimation of Cronbach’s Alpha

Given a test composed p items, the alpha coefficient (Cronbach, 1951; Guttman, 1945) estimates the population reliability of a test score as

$$ \alpha = \frac{p}{p-1}\left( {1-\frac{{\sum\limits_i {{\sigma_{ii }}} }}{{\sum\limits_i {{\sigma_{ii }}+2\sum\limits_{i< j } {{\sigma_{ij }}} } }}} \right) $$
(1)

where Σ i σ ii is the sum of item variances and Σ i<j σ ij is the sum of item covariances.

Formula (1) is closely related to the Spearman-Brown prophecy formula when taking each of p tau-equivalent items (i.e., items with equal covariances) as unit tests. Notice that as the Kuder-Richardson formula (KR20) is a particular case of alpha, both formulas would obtain identical estimates with dichotomous items.

In addition to classical test theory assumptions, Cronbach’s Alpha requires to assume tau-equivalent items (Lord & Novick, 1968), which implies that the items follow a unidimensional factor model with equal factor loadings (McDonald, 1999).

When the items are standardized and summed to obtain a test score (and therefore assuming items to have equal population variances) Alpha is computed as

$$ {\alpha_{{s\tan d}}} = \frac{{p{{\bar{\sigma}}_{ij }}}}{{1+(p-1){{\bar{\sigma}}_{ij }}}} = \frac{{p{{\bar{\sigma}}_{ij }}}}{{1+(p-1){{\bar{\sigma}}_{ij }}}} $$
(2)

where \( {{\bar{\sigma}}_{ij }},\bar{\rho} \) are the average covariances and correlations, respectively. In most cases this standardized version will be less than or equal to the unstandardized version to the extent that differences between item variances are present. However, standardized alpha is prone to bias and values well over the unstandardized version can be obtained in individual samples.

Alpha converges rapidly to its asymptotic properties, and its estimates are known to behave well even when sample size is restrictive. Coefficient alpha is known to be sufficiently accurate with n = 100 for true alpha values about 0.6, regardless of test length. Longer tests may yield quite stable estimates for alpha values over 0.7 with sample sizes of just n = 50 (Duhachek, Cughan, & Iacobucci, 2004; Maydeu-Olivares, Coffman, & Hartmann, 2007).

Alpha Standard Error and Confidence Intervals

Early derivations of a sampling distribution for alpha made restrictive assumptions on item parameters, making the corresponding standard errors highly sensitive to departures from these assumptions. More recently, van Zyl, Neudecker and Nel (2000) proved that just with the assumption of item multivariate normality, formula (1) follows an asymptotically normal distribution N(0,Q), with Q being

$$ \begin{array}{l} Q = \frac{{2{p^2}}}{{{{{\left( {p-1} \right)}}^2}{{{\left( \mathbf{{1}^\prime\boldsymbol{\Sigma} \mathbf{1}} \right)}}^8}}} \\ \qquad \times \left[ {\left( \mathbf{{1}^\prime\boldsymbol{\Sigma} \mathbf{1}} \right)\left( {tr{\boldsymbol{{\Sigma}}^2}+t{r^2}\boldsymbol{\Sigma}} \right)-2tr\boldsymbol{\Sigma} \left( \mathbf{{1}^\prime{\boldsymbol{{\Sigma}}^2}\mathbf{1}} \right)} \right] \end{array} $$
(3)

where 1 is a p × 1 column vector of ones and \( \boldsymbol{\Sigma} \) is the inter-item population covariance matrix. Alpha normal-theory asymptotic standard error can be computed as \( \sqrt{Q/n } \), where n is sample size (Duhachek & Iacobucci, 2004). Given the standard error of alpha, the lower and upper limits of a 1–c % confidence interval level can be computed as

$$ \begin{array}{l} \rm{Lower}\ \rm{limit} = \alpha - {\mathrm{{Z}}_{{\it c}}}\sqrt{Q/n };\ \mathrm{Upper}\ \rm{limit} \\ \qquad\qquad\quad\; = \alpha + {\mathrm{{Z}}_{{1-{\it c}}}}\sqrt{Q/n } \end{array} $$
(4)

where Z is the standardized normal value for 1-c and c. Robust standard errors have been derived (Maydeu-Olivares et al., 2007) and are known to be accurate with skewed items and small item intercorrelations.

Interpretation

As a reliability index, Cronbach’s alpha must be mainly interpreted in terms of the proportion of score variance attributable to changes in true score. It is agreed that values above 0.7 are generally acceptable (Nunnally & Bernstein 1994). A 0.90 cutoff has been recommended for individual assessments (Hays & Revicki, 2005), where measurement error is of greater concern.

Cronbach’s alpha is arguably the most widely used estimator of reliability. However, there is increasing concern in the psychometric literature about the use of alpha in applications (Sitjsma, 2009). Criticisms about the estimator advise to take some precautions when interpreting alpha:

  • Alpha will approach true reliability to the extent that items conform to tau-equivalence. In applications, coefficient alpha is negatively biased so that it implies a lower bound for test reliability.

  • Alpha values can never coincide with true reliability in a single administration (Sijtsma, 2009). As the point-estimate is negatively biased with respect to actual reliability, confidence intervals must be provided to indicate the precision of alpha estimates.

  • Alpha provides no information about test dimensionality. On the contrary, it requires unidimensionality to yield accurate estimates of true reliability (McDonald, 1999). As Alpha is a function of item interrelatedness, which is not necessarily a consequence of unidimensionality, the one-factor structure must be checked prior to alpha estimation.

  • Alpha is always less than or equal to the closest lower bound to the true reliability. Moreover, alpha only coincides with the closest lower bound under restrictive conditions. It is recommended that better estimators of the closest lower bound of true reliability such as Guttman’s lambda-2 (Guttman, 1945), which are available in widespread statistical packages, are provided in addition to alpha.

Cross-References

Internal Consistency

Reliability