Understanding Alpha and Beta and Sources of Common Variance: Theoretical Underpinnings and a Practical Example

Coefficient alpha estimates the degree to which scale scores reflect systematic variation due to one or more common dimensions. Coefficient beta, on the other hand, estimates the degree to which scale scores reflect a single dimension common among all the items; that is, the target construct a scale attempts to measure. As such, the magnitude of beta, relative to alpha, informs on the ability to meaningfully interpret derived scale scores as reflecting a single construct. Despite its clear interpretative usefulness, coefficient beta is rarely reported and, perhaps, not well understood. As such, we first describe how coefficient alpha and beta are analogues to model-based reliability coefficients omega total and omega hierarchical. We then demonstrate with simulated data how these indices function under a variety of data structures. Finally, we perform a hierarchical cluster analysis of the Multidimensional Personality Questionnaire's Stress Reaction Scale, estimating alpha and beta, as clusters form. This demonstrates a chief advantage of alpha and beta; they do not require a formal structural model. Moreover, we illustrate how scales that primarily are based on sets of homogeneous item clusters can "ramp up" to yield reliable scores with conceptual breadth and predominantly reflect the intended target construct.