Cronbach's Alpha: Review of Limitations and Associated Recommendations

This paper discusses the limitations of Cronbach's alpha as a sole index of reliability, showing how Cronbach's alpha is analytically handicapped to capture important measurement errors and scale dimensionality, and how it is not invariant under variations of scale length, interitem correlation, and sample characteristics. In addition, the limitations and strengths of several recommendations on how to ameliorate these problems were critically reviewed. It was shown that the reliance on Cronbach's alpha as a sole index of reliability is no longer sufficiently warranted. This requires that other indices of internal consistency be reported along with alpha coefficient, and that when a scale is composed of large number of items, factor analysis should be performed, and appropriate internal consistency estimation method applied. This approach, if adopted, will largely minimize and guard against uncritical use of Cronbach's alpha coefficient.     

Exact distributions of intraclass correlation and Cronbach's alpha with Gaussian data and general covariance

Intraclass correlation and Cronbach's alpha are widely used to describe reliability of tests and measurements. Even with Gaussian data, exact distributions are known only for compound symmetric covariance (equal variances and equal correlations). Recently, large sample Gaussian approximations were derived for the distribution functions. New exact results allow calculating the exact distribution function and other properties of intraclass correlation and Cronbach's alpha, for Gaussian data with any covariance pattern, not just compound symmetry. Probabilities are computed in terms of the distribution function of a weighted sum of independent chi-square random variables. NewF approximations for the distribution functions of intraclass correlation and Cronbach's alpha are much simpler and faster to compute than the exact forms. Assuming the covariance matrix is known, the approximations typically provide sufficient accuracy, even with as few as ten observations. Either the exact or approximate distributions may be used to create confidence intervals around an estimate of reliability. Monte Carlo simulations led to a number of conclusions. Correctly assuming that the covariance matrix is compound symmetric leads to accurate confidence intervals, as was expected from previously known results. However, assuming and estimating a general covariance matrix produces somewhat optimistically narrow confidence intervals with 10 observations. Increasing sample size to 100 gives essentially unbiased coverage. Incorrectly assuming compound symmetry leads to pessimistically large confidence intervals, with pessimism increasing with sample size. In contrast, incorrectly assuming general covariance introduces only a modest optimistic bias in small samples. Hence the new methods seem preferable for creating confidence intervals, except when compound symmetry definitely holds. An earlier version of this paper was submitted in partial fulfillment of the requirements for the M.S. in Biostatistics, and also summarized in a presentation at the meetings of the Eastern North American Region of the International Biometric Society in March, 2001. Kistner's work was supported in part by NIEHS training grant ES07018-24 and NCI program project grant P01 CA47 982-04. She gratefully acknowledges the inspiration of A. Calandra's “Scoring formulas and probability considerations” (Psychometrika, 6, 1–9). Muller's work supported in part by NCI program project grant P01 CA47 982-04.     

Tests for equality of several alpha coefficients when their sample estimates are dependent

In a variety of measurement situations, the researcher may wish to compare the reliabilities of several instruments administered to the same sample of subjects. This paper presents eleven statistical procedures which test the equality ofm coefficient alphas when the sample alpha coefficients are dependent. Several of the procedures are derived in detail, and numerical examples are given for two. Since all of the procedures depend on approximate asymptotic results, Monte Carlo methods are used to assess the accuracy of the procedures for sample sizes of 50, 100, and 200. Both control of Type I error and power are evaluated by computer simulation. Two of the procedures are unable to control Type I errors satisfactorily. The remaining nine procedures perform properly, but three are somewhat superior in power and Type I error control.A more detailed version of this paper is also available.     

A test of the hypothesis that Cronbach's alpha reliability coefficient is the same for two tests administered to the same sample

In measurement studies the researcher may wish to test the hypothesis that Cronbach's alpha reliability coefficient is the same for two measurement procedures. A statistical test exists for independent samples of subjects. In this paper three procedures are developed for the situation in which the coefficients are determined from the same sample. All three procedures are computationally simple and give tight control of Type I error when the sample size is 50 or greater.The author is indebted to Jerry S. Gilmer for development of the computer programs used in this study.     

A modification of Feldt's test of the equality of two dependent alpha coefficients

The available statistical tests of the equality of nonindependent alpha reliability coefficients require that the product of the number of test parts times the number of subjects be quite large—1000 or more. A modification of one of these tests is derived which avoids this limitation. Monte Carlo studies indicate that the modified test effectively controls the Type I error rate with as few as 2 or 3 test parts and 50 subjects. This means the modified test can be safely employed in comparisons between interrater reliabilities.     

A k-sample significance test for independent alpha coefficients

The earlier two-sample procedure of Feldt [1969] for comparing independent alpha reliability coefficients is extended to the case ofK 2 independent samples. Details of a normalization of the statistic under consideration are presented, leading to computational procedures for the overallK-group significance test and accompanying multiple comparisons. Results based on computer simulation methods are presented, demonstrating that the procedures control Type I error adequately. The results of a power comparison of the case ofK=2 with Feldt's [1969]F test are also presented. The differences in power were negligible. Some final observations, along with suggestions for further research, are noted.The authors gratefully acknowledge the assistance of Michael E. Masson, in the computations performed, and of Leonard S. Feldt, in suggesting the data generation procedures used in the study. In addition, the authors thank James Zidek and the Institute of Applied Mathematics and Statistics, University of British Columbia, for advice concerning some of the theoretical development.     

On robusiness of the normal-theory based asymptotic distributions of three reliability coefficient estimates

This paper studies the asymptotic distributions of three reliability coefficient estimates: Sample coefficient alpha, the reliability estimate of a composite score following a factor analysis, and the estimate of the maximal reliability of a linear combination of item scores following a factor analysis. Results indicate that the asymptotic distribution for each of the coefficient estimates, obtained based on a normal sampling distribution, is still valid within a large class of nonnormal distributions. Therefore, a formula for calculating the standard error of the sample coefficient alpha, recently obtained by van Zyl, Neudecker and Nel, applies to other reliability coefficients and can still be used even with skewed and kurtotic data such as are typical in the social and behavioral sciences.This research was supported by grants DA01070 and DA00017 from the National Institute on Drug Abuse and a University of North Texas faculty research grant. We would like to thank the Associate Editor and two reviewers for suggestions that helped to improve the paper.     

Coefficients Alpha,Beta, Omega,and the glb: Comments on Sijtsma

There are three fundamental problems in Sijtsma (Psychometrika, 2008): (1) contrary to the name, the glb is not the greatest lower bound of reliability but rather is systematically less than ω _t (McDonald, Test theory: A unified treatment, Erlbaum, Hillsdale, 1999), (2) we agree with Sijtsma that when considering how well a test measures one concept, α is not appropriate, but recommend ω _t rather than the glb, and (3) the end user needs procedures that are readily available in open source software.     

Internal Consistency and Power When Comparing Total Scores from Two Groups

Researchers now know that when theoretical reliability increases, power can increase, decrease, or stay the same. However, no analytic research has examined the relationship of power to the most commonly used type of reliability—internal consistency—and the most commonly used measures of internal consistency, coefficient alpha and ICC(A,k). We examine the relationship between the power of independent samples t tests and internal consistency. We explicate the mathematical model upon which researchers usually calculate internal consistency, one in which total scores are calculated as the sum of observed scores on K measures. Using this model, we derive a new formula for effect size to show that power and internal consistency are influenced by many of the same parameters but not always in the same direction. Changing an experiment in one way (e.g., lengthening the measure) is likely to influence multiple parameters simultaneously; thus, there are no simple relationships between such changes and internal consistency or power. If researchers revise measures to increase internal consistency, this might not increase power. To increase power, researchers should increase sample size, select measures that assess areas where group differences are largest, and use more powerful statistical procedures (e.g., ANCOVA).     

A generalization of coefficient alpha

It is well known that coefficient alpha can be used to estimate the reliability of a test even when the test is split into several parts. It is also known that alpha can severely underestimate test reliability when the several parts have an unequal number of items. A gernalization of alpha,β _k, is proposed to correct this defect. Several properties ofβ _k are also presented. The author gratefully acknowledges the assistance of Dr. Leonard Feldt for reviewing an earlier draft of this paper, and Ms. Rita Karwacki Bode and Mr. Dave Mansell for the analysis of the experimental data reported here. The comments of an unknown referee which contributed substantially to the clarity of the presentation are also gratefully acknowledged.     

Inequalities among lower bounds to reliability: With applications to test construction and factor analysis

A chain of lower-bound inequalities leading to the greatest lower bound to reliability is established for the internal consistency of a composite of unit-weighted components. The chain includes the maximum split-half coefficient, the lowest coefficient consistent with nonimaginary common factors, and the lowest coefficient consistent with nonimaginary common and unique factors. Optimization theory is utilized to determine the conditions that are requisite for the inequalities. Convergence proofs demonstrate that the coefficients can be attained. Rapid algorithms obtain estimates of the coefficients with sample data. The theory yields methods for splitting items into maximally similar sets and for exploratory factor analysis based on a theoretical solution to the communality problem.     

非正态分布测量数据对克隆巴赫信度α系数的影响

对“增大被试群体异质性,能提高测验信度”观点提出质疑。讨论非正态分布测量数据的偏度对克隆巴赫信度α系数的影响,以模拟方法验证Box—COX正态化变换对信度的高估现象,进而给出对克隆巴赫信度α系数估计的改进方法。     

On the Use,the Misuse,and the Very Limited Usefulness of Cronbach’s Alpha

This discussion paper argues that both the use of Cronbach’s alpha as a reliability estimate and as a measure of internal consistency suffer from major problems. First, alpha always has a value, which cannot be equal to the test score’s reliability given the interitem covariance matrix and the usual assumptions about measurement error. Second, in practice, alpha is used more often as a measure of the test’s internal consistency than as an estimate of reliability. However, it can be shown easily that alpha is unrelated to the internal structure of the test. It is further discussed that statistics based on a single test administration do not convey much information about the accuracy of individuals’ test performance. The paper ends with a list of conclusions about the usefulness of alpha.     

On the robustness of maximum likelihood scaling for violations of the error model

A Monte Carlo study was conducted to investigate the robustness of the assumed error distribution in maximum likelihood estimation models for multidimensional scaling. Data sets generated according to the lognormal, the normal, and the rectangular distribution were analysed with the log-normal error model in Ramsay's MULTISCALE program package. The results show that violations of the assumed error distribution have virtually no effect on the estimated distance parameters. In a comparison among several dimensionality tests, the corrected version of thex ² test, as proposed by Ramsay, yielded the best results, and turned out to be quite robust against violations of the error model.     

A method of estimating item characteristic functions using the maximum likelihood estimate of ability

A method of estimating item characteristic functions is proposed, in which a set of test items, whose operating characteristics are known and which give a constant test information function for a substantially wide range of ability, are used. The method is based on the maximum likelihood estimates of ability for a group of several hundred examinees. Throughout the present study the Monte Carlo method is used.     

Note on two generalizations of coefficient alpha

An important relationship is given for two generalizations of coefficient alpha, Rajaratnam, Cronbach and Gleser's generalizability formula for stratified-parallel tests and Raju's coefficient beta.The author gratefully acknowledges the generous assistance given by reviewers and the editor.     

Coefficients alpha and reliabilities of unrotated and rotated components

It has been shown by Kaiser that the sum of coefficients alpha of a set of principal components does not change when the components are transformed by an orthogonal rotation. In this paper, Kaiser's result is generalized. First, the invariance property is shown to hold for any set of orthogonal components. Next, a similar invariance property is derived for the reliability of any set of components. Both generalizations are established by considering simultaneously optimal weights for components with maximum alpha and with maximum reliability, respectively. A short-cut formula is offered to evaluate the coefficients alpha for orthogonally rotated principal components from rotation weights and eigenvalues of the correlation matrix. Finally, the greatest lower bound to reliability and a weighted version are discussed.Comments by Henk A.L. Kiers and by anonymous referees are gratefully acknowledged.     

Internal consistency and reliability of alternate short forms of the college women's assertion sample: Clinical and research implications

Assertive behavior is most often assessed with self-report or role-play measures. The latter modality is preferred because it provides for the sampling of the structure of behavior and for the consideration of the situational context. MacDonald (1978) has developed such an assessment device but it is limited by the length of time for administration and scoring. Two studies were conducted to reconstruct reliable alternate short forms. The first study describes the selection of items and demonstrates the internal consistency of the alternate forms. The second study demonstrates the alternate form and retest reliability and provides normative statistics. We conclude that reliable alternate short forms have been constructed to be used in research in clinical applications.This research was supported by the Marie Wilson Howells Fund.Alternate short forms of the CWAS may be obtained from the first author.