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My question is very similar to this one, which was not solved unfortunately.

I am working on a project for which I want to rank countries by means of their HIV/AIDS burden. So I collected a lot of data for all countries in the world. For simplicity let's assume that I have following variables for each country:

  • DEA: Deaths due to HIV
  • LIV: People living with HIV
  • PRV: HIV prevalence rate
  • DALY: number of healthy years lost due to HIV
  • DALY ratio: proportion of healthy years lost due to HIV in total number of healthy years lost due to disease in general.

So all these variables somehow measure the same thing: the HIV burden. Now I want to combine all these variables into one 'score', such that I can rank countries by means of their HIV burden.

The first thing that came into my mind was to to perform a principal component analysis and retain one PC. However, if we look at the loadings of this first PC we see the following:

  • DEA: 0.366
  • LIV: -0.392
  • PRV: -0.442
  • DALY: 0.466
  • DALY ratio: 0.481

Because of the high pairwise correlations between the variables I would have expected each of the loadings to have the same sign. Now countries with a high HIV burden (so scoring high on each of the variables) now get a lower score for the first PC on one side (due to the negative loadings of 'LIV' and 'PRV') and a higher score for the first PC on the other side (due to the positive effects of 'DEA', 'DALY' and 'DALY ratio').

My questions:

  • Is it correct that looking at the scores for the first PC is not a proper way to give a score for HIV burden to each of the countries because of the contrary loadings as explained above?

  • Can you suggest another (better way) to combine all the information into one single score?

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    $\begingroup$ So all these variables somehow measure the same thing: the HIV burden. Now I want to combine all these variables into one 'score'. Your variables sound to me too mixed, heterogeneous by their measurement units, quantum meaning, and probably distributional qualities. If so they should be avoided to combine directly in one score. Rather, it's better to rank countries by each variable separately and then to derive an overall rank (such as just mean or some more sofisticated). $\endgroup$ – ttnphns Apr 5 '16 at 8:40
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    $\begingroup$ And in general, pursue for a single index out of multidimensional reality is often unwarranted. The fake simplicity may corrupt minds of the audience and, worse, of decision-makers within it. What might be recommended in your place is to block the variables in few theoretically, conceptually kin ones (e.g., prevalence; survival; economical burden; etc.) and anylize by blocks; different analyses might be called for different blocks. It's not a simple, it's a creative task. $\endgroup$ – ttnphns Apr 5 '16 at 9:09
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    $\begingroup$ Sound advice in the answer & comments here. Note also that the DALY purports to measure HIV burden by itself, already incorporating information on prevalence & mortality. $\endgroup$ – Scortchi Apr 5 '16 at 9:55
  • $\begingroup$ note also that principal components searches for linear combinations of the original variables that ''contain'' as much as possible variance of the orginal variables. Is it your goal to keep as much as possible of this variance ? Or do you want to look for an as large as possible ''common part'' (for which you might try to use factor analysis). $\endgroup$ – user83346 Apr 5 '16 at 11:16
  • $\begingroup$ If people die quickly after diagnosis then both the number of people living with HIV and the prevalence in such a country will be low but the number of deaths will be high so I am surprised that you say they are highly correlated pair-wise. I would check that carefully. $\endgroup$ – mdewey Apr 5 '16 at 12:15
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Taking your example literally, I'd say the approach is problematic from the outset.

  • If the problem is assessing the total burden, then absolute numbers of deaths and people living with AIDS are key variables, but any PCA is likely to be dominated by a small number of countries with large populations. Even if you use correlation-based PCA, as you should when variables are in very different units, you will have some large outliers in there for most conceivable mixes of countries.

  • If the problem is assessing the total burden given population sizes, then the other variables are relevant.

  • It seems unlikely that mixing together different kinds of variables will help either purpose.

  • The biggest question of all is whether it's a good idea at all to seek a single scale in this way. The best that I can do is flag that statistically-minded people have very different views on this, many highly negative. My own view is that PCA of this sort will only be of interest to those capable of understanding and criticising the PCA and doing their own alternative analysis. A fallacy known under many different names, of which one is the fallacy of misplaced concreteness, is confusing a desire for a single measure with a demonstration that such a measure can be reliably and intelligibly identified from data. It's one thing to have a single name (creativity, intelligence, in this case burden) and another thing to have a single quantifiable dimension.

Turning to your results, what's most alarming, as you clearly flag, is that the loadings on the first PC don't even have the same sign. If there is one important shared dimension that justifies trying to quantify burden as a single measure, then it minimally requires all those variables to be positively correlated with each other (or for reversals of sign to be obvious consequences of some measures being direct and some inverse, which doesn't seem the case here). Without seeing the data, I can't interpret further, but I'd expect the variation in sign to be a side-effect of mushing together quite different variables that are also skewed in distribution and with outliers.

Plotting the data will help you understand why you got the results you did.

I don't have suggestions for a different way to collapse to a single score. I've seen too many applications in which such endeavours were not helpful to be positive there.

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Suggest an exploratory factor analysis. Rather than assume the dimensionality behind your HIV measures, take one half of the sample randomly, extract 4 factors, by almost any method, and plot the eigenvalues. Use the scree test or just use the "eigenvalues greater than one thumb rule": Rotate to simple structure via Varimax or Quartimax the number of factors by either of these tests. I'm guessing that a two factor solution will explain the dimensionality of your HIV measures. By studying the loadings, and thinking while taking a jog or hot shower, you will soon understand what these two dimensions actually are. Then do a confirmatory factor analysis with the other half of the sample.

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