Revisiting Cronbach 1951 via Simulation with Shiny

At the time of the creation of this blog, Cronbach’s 1951 piece on coefficient alpha has 18,132 citations according to google scholar.  The main use of coefficient alpha is to assess internal consistency reliability of a test or survey.  

Although it may have been forgotten, the proof Cronbach demonstrated established that coefficient alpha is the mean of all split half reliabilities that have an equal number of items on both splits.  The proof is often criticized and it has been said that the proof is only valid when the items exhibit tau equivalence (all of the factor loadings are equal in the population) and unidimensionality (all items load onto only one factor).  I argue that the proof is still valid if items do not fit with the two assumptions described above but the estimate of reliability will be off.  

To demonstrate this I put together a web application that shows the value of alpha, the mean of a number (can be increased or decreased) of split-half reliabilities, and the population parameter of reliability for the particular data structure that is selected by you.  Notice that in the 1 factor models the both estimates (alpha and the mean of the split-half reliablities) are close to the population value but when you go to the 3 factor and 5 factor models there is a lower bound bias present.