- Cronbach's alpha (the reliability coefficient) Cronbach's alpha is a measure of the intercorrelation of items; the estimate of internal consistency of items in a scale, measuring the extent to which item responses obtained at the same time correlate highly with each other. If alpha is greater than or equal to .6, then the items are considered unidimensional and may be combined in an index or scale. Some researchers use the more stringent cutoff of .7. Cronbach's alpha is found in SPSS 13 under Analyze, Scale, Reliability Analysis. Strictly speaking alpha is not a measure of unidimensionality. A set of items can have a high alpha and still be multidimensional. This happens when there are separate clusters of items (separate dimensions) which intercorrelate highly, even though the clusters themselves do not intercorrelate highly. Also, a set of items can have a low alpha even when unidimensional if there is high random error. The more items, the more reliable a scale will be (when the number of items in a scale is higher, alpha will be higher). The widely-accepted social science cut-off is that alpha should be .70 or higher for a set of items to be considered a scale, but some use .75 or .80 while others are as lenient as .60. That .70 is as low as one may wish to go is reflected in the fact that when alpha is .70, the standard error of measurement will be over half (0.55) a standard deviation. Under some circumstances, alpha may be negative. This reflects a serious coding error in the data: data should be recoded if necessary to assure that all items are coded in the same conceptual direction
- Exploratory factor analysis: Principal components analysis EFA, indicators should have higher factor loadings on their own constructs than on other constructs. Some researchers also require that loadings be higher than some absolute cutoff value, such as .3. Some researchers also require that indicators not crossload on factors not their own (ex., that all loadings other than their own factor be below some absolute cutoff value, such as .3). Factor analysis is found in SPSS 13 under Analyze, Data Reduction, Factor.
- SEM Confirmatory factor analysis. In SEM, to test the unidimensionality of a concept, the fit (ex., AIC or other fit measures) of two models is compared: (1) a model with two factors (of the concept) whose correlation is estimated freely; and (2) a model in which the correlation is fixed, usually to 1.0 (meaning the two factors are the same). Model (2) is nested within model (1) because model (2) has fewer free parameters than model (1). Model (2) with fewer parameters is more restricted. The null hypothesis is whether the restriction (correlation fixed to 1) make sense. If model (2),the more parsimonious one, fits as well as model (1), then the researcher infers that there is no unshared variance and the two factors measure the same thing (are unidimensional).
- Guttman scaling and other forms of scaling. In Guttman, proximity, Mokken, and certain other types of scales, indicator variables are tested to see if they form a certain relationship to each other, such that the researcher is justified in putting the items in the same scale. The relationships, and hence the meanings of unidimensionality, differ by scale type. Note that the most common form of scale, the Likert scale, does not normally involve testing for unidimensionality. Guttman scaling is found in SPSS 13 under Analyze, Scale, Reliability Analysis.
- Unidimensionality can easily be assessed by investigating the standardized residual covariance matrix for values > 2.58 corresponding to the critical p < .01 threshold (Anderson & Gerbing, 1988; Segars, 1997). That is, if the a priori theory suggests that all measures are reflective only of their specified latent variable. However caution should be exercised since a residual exceeding the cutoff of 2.58 does not automatically denote lack of unidimensionality. One reason is that the shared variance may be attributable to random shared non-common variance between the two measurement items (Gefen, 2003).
Note that a set of items may be considered to be unidimensional using one of the methods above, even when another method would fail to find statistical justification in considering the items to measure a single construct. For instance, it would be quite common to find that items in a large Guttman scale would fail to load on a single factor using factor analysis as a method. Finding satisfactory model fit in SEM would not assure that the Cronbach's alpha criterion was met. The researcher must decide on theoretical grounds what definition and criterion for unidimensionality best serves his or her research purpose. However, some method must always be used before proceeding to use multiple indicators to measure a concept.
"Consistency criteria and unidimensionality: An attempt at clarification". Bagozzi, R. P., and Fornell, C. (1989). Adcences in Consumer Research, Vol. 1