A note on using alpha and stratified alpha to estimate the reliability of a test composed of item parcels

Several authors have suggested that prior to conducting a confirmatory factor analysis it may be useful to group items into a smaller number of item `parcels' or `testlets'. The present paper mathematically shows that coefficient alpha based on these parcel scores will only exceed alpha based on the entire set of items if W, the ratio of the average covariance of items between parcels to the average covariance of items within parcels, is greater than unity. If W is less than unity, however, and errors of measurement are uncorrelated, then stratified alpha will be a better lower bound to the reliability of a measure than the other two coefficients. Stratified alpha are also equal to the true reliability of a test when items within parcels are essentially tau-equivalent if one assumes that errors of measurement are not correlated.