Rules of thumb for creating composite scales Normally when I create an index I basically just take the or averages of the individual variables that comprise the index. But I'm wondering if one should / could not, take the product?
E.g., a scale "friendship" is comprised of the variables 1) "fun to be with" and 2) "good to talk to when I am sad". (just a hypothetical example).
Lets say that variable 1 is answered on a 1-5 scale, and and variable 2 is answered on a 1-10 scale. 
Lets say that the two variables correlate nicely with a Pearson coefficient of 0.6.
Now, with both variables scaled from eg. 0 to 1, what would be the best friendship measure? The average of the two variables or the product?
If I use the product measure, friendship has a much larger variance than if I use the standard averages measure.
Is there a simple mathematical reason why I should not take the product? Or does it depend on theoretical assumptions about "friendship"?
 A: 
what would be the best friendship measure? The average of the two
  variables or the product?

If you can provide a definition of "best", then you may have your criteria right there. If you had the luxury of conducting a leisurely experiment, you could ask your questions, compute multiple ways of summarizing the level of friendship, and see how closely it correlates with a variable of interest (such as frequency of spending time together, likelihood to be tagged in the same picture on facebook, etc.). You could also develop an idea of what the distribution of a friendship measure "should" look like: if you think that each person should on average tend to have a small number of very good friends and a large number not-so-close friends, you could look at which measure turns out the better distribution.
A: My approach is to follow theory to develop composite scales. I would make the choices listed below for the stated reasons:

*

*Sum the individual components if the individual components are part of a whole.
Consider measuring intelligence using measures for verbal and quantitative intelligence. The overall measure of intelligence should be the sum of the verbal and quantitative components as the individual components are part of a whole.


*Average the individual components if they measure the same underlying construct.
Suppose, that we measure verbal intelligence by a battery of 10 questions. Each question measures the same aspect of intelligence (namely our ability to parse language). Thus, averaging these scores will give us a better measure of verbal intelligence as random errors in the measurements tend to even out when you average the scores.


*Multiply them if the individual components have synergistic effect on the underlying construct.
If for some reason you believe that verbal intelligence and quantitative intelligence have synergistic effects then I would multiply them. By synergistic effects I mean that a person who is verbally intelligent can leverage his language abilities to go even further in quantitative fields thus enhancing their quantitative intelligence.
