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Cronbach's α (alpha) has an important use as a measure of the reliability of a psychometric instrument. It indicates the extent to which a set of test items can be treated as measuring a single latent variable. It was first named as alpha by Cronbach (1951), as he had intended to continue with further instruments. It is the extension of an earlier version, the Kuder-Richardson Formula 20 (often shortened to KR-20), which is the equivalent for dichotomous items, and Guttman (1945) developed the same quantity under the name lambda-2. The mathematical foundations of statistical reliability are probability and statistics. ...
Psychometrics is the field of study concerned with the theory and technique of psychological measurement, which includes the measurement of knowledge, abilities, attitudes, and personality traits. ...
Latent variables, as opposed to observable variables, are those variables that cannot be directly observed but are rather inferred from other variables that can be observed and directly measured. ...
Lee J. Cronbach (1916 - 2001) was an American educational psychologist who made significant contributions to psychological testing and measurement. ...
In statistics, the Kuder-Richardson Formula 20 (KR-20) is a measure of reliability for measures with dichotomous choices. ...
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Cronbach's α is defined as
 ,
where N is the number of components (items or testlets),  is the variance of the observed total test scores, and  is the variance of component i.
Alternatively, the standardized Cronbach's α can also be defined as
where N is the number of components (items or testlets) and  is the average of all (Pearson) correlation coefficients between the components. In statistics, the Pearson product-moment correlation coefficient (sometimes known as the PMCC) (r) is a measure of the correlation of two variables X and Y measured on the same object or organism, that is, a measure of the tendency of the variables to increase or decrease together. ...
Cronbach's alpha and internal consistency
Cronbach's alpha will generally increase when the correlations between the items increase. For this reason the coefficient is also called the internal consistency or the internal consistency reliability of the test. Internal consistency, in gaming, refers to the consistency of the physical and social rules that affect online computer role-playing games. ...
Cronbach's alpha in classical test theory Alpha is an unbiased estimator of reliability if and only if the components essentially τ-equivalent (Lord & Novick, 1968). Under this condition the components can have different means and different variances, but their covariances should all be equal - which implies that they have 1 common factor in a factor analysis. One special case of essential τ-equivalence is that the components are parallel. Although the assumption of essential τ-equivalence may sometimes be met (at least approximately) by testlets, when applied to items it is probably never true. This is caused by the facts that (1) most test developers invariably include items with a range of difficulties (or stimuli that vary in their standing on the latent trait, in the case of personality, attitude or other non-cognitive instruments), and (2) the item scores are usually bounded from above and below. These circumstances make it unlikely that the items have a linear regression on a common factor. A factor analysis may then produce artificial factors that are related to the differential skewnesses of the components. When the assumption of essential τ-equivalence of the components is violated, alpha is not an unbiased estimator of reliability. Instead, it is a lower bound on reliability. In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...
In probability theory and statistics, the covariance between two real-valued random variables X and Y, with expected values and is defined as: where E is the expected value. ...
Factor analysis is a statistical technique that originated in psychometrics. ...
α can take values between negative infinity and 1 (although only positive values make sense). Some professionals, as a rule of thumb, require a reliability of 0.70 or higher (obtained on a substantial sample) before they will use an instrument. Obviously, this rule should be applied with caution when α has been computed from items that systematically violate its assumptions. Further, the appropriate degree of reliability depends upon the use of the instrument, e.g., an instrument designed to be used as part of a battery may be intentionally designed to be as short as possible (and thus somewhat less reliable). Other situations may require extremely precise measures (with very high reliabilities).
Cronbach's α is related conceptually to the Spearman-Brown prediction formula. Both arise from the basic classical test theory result that the reliability of test scores can be expressed as the ratio of the true score and total score (error and true score) variances: The Spearman-Brown prediction formula (also known as the Spearman-Brown prophecy formula) is a formula relating psychometric reliability to test length: where is the predicted reliability; N is the number of tests combined (see below); and is the reliability of the current test. The formula predicts the reliability of...
Classical test theory is a body of related psychometric theory that predict outcomes of psychological testing such as the difficulty of items or the ability of test-takers. ...
Alpha is most appropriately used when the items measure different substantive areas within a single construct. Conversely, alpha (and other internal consistency estimates of reliability) are inappropriate for estimating the reliability of an intentionally hetrogeneous instrument (such as screening device such as a biodata or the original MMPI). Also, α can be artificially inflated by making scales which consist of superficial changes to the wording within a set of items or by analyzing speeded tests. Within personnel selection, biographical data, or biodata is a method used to select candidates to fill jobs based on their previous work history, work preferences, work habits, and other background characteristics and interests. ...
Cronbach's alpha in generalizability theory Cronbach and others generalized some basic assumptions of classical test theory in their generalizability theory. If this theory is applied to test construction, then it is assumed that the items that constitute the test are a random sample from a larger universe of items. The expected score of a person in the universe is called the universum score, analogously to a true score. The generalizability is defined analogously as the variance of the universum scores divided by the variance of the observable scores, analogously to the concept of reliability in classical test theory. In this theory, Cronbach's alpha is an unbiased estimate of the generalizability. For this to be true the assumptions of essential τ-equivalence or parallelness are not needed. Consequently, Cronbach's alpha can be viewed as a measure of how well the sum score on the selected items capture the expected score in the entire domain, even if that domain is heterogeneous. Generalizability theory (G Theory) is a statistical framework for conceptualizing, investigating, and designing reliable observations. ...
Reliability concerns quality or consistency. ...
Classical test theory is a body of related psychometric theory that predict outcomes of psychological testing such as the difficulty of items or the ability of test-takers. ...
Cronbach's alpha and the intra-class correlation Cronbach's alpha is equal to the stepped-up consistency version of the intra-class correlation, which is commonly used in observational studies. This can be viewed as another application of generalizability theory, where the items are replaced by raters or observers who are randomly drawn from a population. Cronbach's alpha will then estimate how strongly the score obtained from the actual panel of raters correlates with the score that would have been obtained by another random sample of raters.
Cronbach's alpha and factor analysis As stated in the section about its relation with classical test theory, Cronbach's alpha has a theoretical relation with factor analysis. There is also a more empirical relation: Selecting items such that they optimize Cronbach's alpha will often result in a test that homogeneous in that they (very roughly) approximately satisfy a factor analysis with one common factor. The reason for this is that Cronbach's alpha increases with the average correlation between item, so optimization of it tends to select items that have high correlations with most other items. Factor analysis is a statistical technique that originated in psychometrics. ...
Cronbach's alpha and other disciplines Although this description of the use of α is given in terms of psychology, the statistic can be used in any discipline. A statistic (singular) is the result of applying a statistical algorithm to a set of data. ...
References - Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.
- Allen, M.J., & Yen, W. M. (2002). Introduction to Measurement Theory. Long Grove, IL: Waveland Press.
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