Interpretation of PCA Composite Index Scores

Question

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
Adams   1.81
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
Adams   2
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.

Many thanks in advance!

Answer 1

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?

Asking about the magnitude of your score, you are really asking about the unit of your linear combination. There is no such thing; you do not have an absolute scale for your scores..

However, you still can have a relative one. Is the difference between A and B large in relation to the difference between A and C? Or, better yet: is the difference between A and B large in relation to the overall differences between samples in your data set?

A natural way (for a practical statistician, that is) to express this is to express the data in standard deviations as units. Knowing that almost 70% (remember the 68-95-99.7 rule?) of the data points are at most one standard deviation away from the mean gives you a pretty good feeling how important these differences are.

Or, alternatively, simply show the data, on a rug plot for example. Also, since you are referring to PCA: I normally show a two- or three-dimensional plot of all points, and indicate the points of interest. The rest is in the eye of the beholder.

Answer 2

To make a statement about the percent difference between two observations would require your scale to be a ratio scale in the ratio, interval, ordinal, nominal classification system. That is, zero on this scale (even if there is no zero in the data) would need to reflect zero of whatever is being measured. Stated differently, it would need to be meaningful to say that 10 is twice as much of whatever is being measured as 5. Rescaling the measure between 0 and 100 does not accomplish this. There is no meaningful zero on your scale (and it is not possible to create such a point), thus ratio statements (percent change) are not meaningful.

As an example, IQ as typically measured is an interval scale, not a ratio scale. You cannot meaningfully say that someone with an IQ of 110 is 10% smarter than someone with an IQ of 100. There is no "zero" intelligence, although we may like to think that some people have none.