Interpretation of PCA Composite Index Scores

Similar (and follow up questions) to my last posts. Calculating a composite index in PCA using several principal components

I have created a composite index to rank counties in Colorado based on cancer registry, hospitalization and ER visit, community health survey data, and poverty/race ethnicity data at the county level. The variables underwent descriptive stats, were tested for normality, standardized, and underwent PCA in SAS including varimax rotation. Relevant components (by way of Cattell plot "elbow" and Kaiser criteria) were retained and combined to make an index score. These new components could be thought of as "disease" and "demographics" respectively.

Outputs of one of these indices is as follows:

COUNTY  NSI
Alamosa -0.87
Arapahoe 3.18
Archuleta -0.12
Baca    -2.1
Bent    -2.22


These aren't the easiest for me or my audience to understand, so I've ranked them 1st to 64th (1st being the best) based on what I know of the underlying variables within the components retained for the index:

COUNTY  Rank
Alamosa 4
Arapahoe    1
Archuleta   3
Baca    5
Bent    6


My question is, what inferences can I make from this index? It was my understanding that I could only make inferences into relative differences (e.g. county ranked 15th is better than county ranked 30th) but that I can't quantify those differences (e.g. county ranked 15th is #% better than county ranked 16th). Would I be able to make more inferences if I standardized the scores to fall between 0-100? My gut says no but I wanted to check. Even if I could then say something like "the county with a score of 100 is 2 points better than the county with a score of 98". that still doesn't define what the magnitude of a difference of 2 means.

• Since I do not keep track of your questions, can you shortly explain what you mean by a "composite index"? – January Apr 27 '17 at 7:03
• Or at least link to a relevant question when talking about it – January Apr 27 '17 at 7:07
• Similar (and follow up questions) to my last posts OK, Ben, please add links to those questions to your question text. – ttnphns Apr 27 '17 at 7:43
• You often see comments and inferences (e.g. % change) made about indices from anything from the Happiness index happyplanetindex.org to Financial Indices! – Jeremy Voisey Apr 27 '17 at 8:36
• Sorry about that. I've added more explaination and a link to my previous question. – BenW Apr 27 '17 at 13:44

In general, I doubt that you can. Think of it as follows: you are creating a linear combination of different variables, each of which is in different units. Say, you compare cars; you take into account several variables, such as MPG (in $miles \cdot gallons^{-1}$), weight ($kg$), acceleration ($m\cdot s^{-2}$), top cruising speed ($m\cdot s{-1}$), price ($\$$), CO2 emission ($g\cdot km^{-1}\$) etc. Now you are building a composite score, a linear combination of these variables, and ask the following question: if car A is better than B, then how much better?

Well, if you expect an answer to your question, then in what units? Say, the answer is "A is better than B by forty two". Forty two what? Gallons? Kilometers? Milligrams per hour? Furlongs per fortnight?