How to test reliability of an index when the index is composed of distinct facets? I am measuring a concept (such as 'friendship') that is multidimensional in nature.
To capture the multidimensional nature of the concept, I have measured the concept through several aspects, each of which focus on one part (or dimension) of the concept.
Each aspect has several Likert scale items, which I have added together to arrive at sub scores for the aspect. I have then added all the sub scores to arrive at the overall score for the concept.
For example, if the total of the Likert scale items for Aspect A is 6, Aspect B is 3 and Aspect C is 4, then the overall score for the concept (which is made up of A, B and C) is 13.
In this way, the overall score is a single number, which is a unidimensional index. 
My focus of interest is the overall score (not the sub scores for each aspect). 
How do I test the reliability of my index? Would Cronbach's alpha be appropriate in this case?
 A: Cronbach alpha and its versions are measures of inter-item homogeneity. By definition, a scale should be homogeneous enough to be able to measure one construct. Reliability approach poses that items in a scale are all the "same" except for some biases that are treated as "random error", unrelated to the measured construct, and that cancel each other in the end. So, alpha is not suitable for battery of constructs: it is unidimentional. Another pair of shoes, though, is that truly unidimensional construct is somewhat unrealistic concept. In factor analysis of a scale it often splits into main factor which is the construct of interest and few minor factors which interfere. Still, reliability approach treat these systematic factors as noise. See more.
A: As ttnphns said Cronbach alpha is not suitable because your scale is not unidimensional. But you can calculate McDonald's Omega instead. There you also can check whether your dimensions are correct. Take a look at the psych package that gives you everything you need http://www.personality-project.org/r/psych/help/omega.html
