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I'm constructing an index made by the sum of three variables, which all vary in the range [0,1] (as they have been divided by their theoretical maximum).

Depending on the sample analyzed, it is not always assured that the three variables have the same variances (and st. deviations). In order to attribute them the same importance, one solution, before making the sum, would be to standardize them (subtracting the mean and dividing by the standard deviation). In this case the index would correspond to the sum of standardized scores.

My questions are: - is standardization necessarily required? does it have other drawbacks (apart from changing the scale)? I cannot always assume that the variables follow a specific distribution. - Would you reccommend standardization or is there another more appropriate solution? - Could the sum of unstandardized scores be OK?

(cross posted from here https://www.statalist.org/forums/forum/general-stata-discussion/general/1423416-constructing-an-index-is-it-necessary-to-standardize).

EDIT: I'm adding a data example after Placidia's answer, to provide a better description of my case and make the question more generalizable. Please consider an example for the three variables I want to sum in a final index and their transformations as in the following table. I have ten observations to rank.

enter image description here

The final index can be obtained either by: (1) the sum of raw values divided by the theoretical maximum; (2) or the sum of the rescaled values in [0,1]; (3) or the sum of the standardized values (subtract mean and divide by SD); (4) or the sum of percentile scores. Please see the table and the graph:

enter image description here

Different choices could produce different rankings for the 10 observations. Which solution would be more appropriate? Is there a totally wrong one?

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    $\begingroup$ There are many ways to normalize variables before combining in one index. We don't know your specific data and situation. I would recommend to lie on a sofa and meditate over variants and their implications for you. Two topics I might point to as especially important: (i) do you really nead standardization based on subtraction of the mean? (ii) do you really want to standardize with sample values statistics? $\endgroup$ – ttnphns Dec 22 '17 at 14:27
  • $\begingroup$ Thanks, I would love to receive your help to better understand the implications of these choices. $\endgroup$ – Forinstance Dec 22 '17 at 14:34
  • $\begingroup$ your question is ambiguous. Terms like normalisation, standardisation and standard score (z) or standard normal (0,1) distribution. have different themes and have contextual meanings e.g. measurement theory and mathematical-statistics. You are citing additive model without going into issues of linearity or non-linearity of variables . Rescaling ? for what reasons ? $\endgroup$ – Subhash C. Davar Dec 23 '17 at 9:15
  • $\begingroup$ It is exactly to address these issues that I need your help. Apart from the mathematical solution, what are the things that I need to consider? And what would be appropriate for the data in the example? $\endgroup$ – Forinstance Dec 23 '17 at 12:44
  • $\begingroup$ Rescaling was mostly a test. I think the real choice to make is between dividing by the theoretical maximum (given the sample) or subtracting the mean and dividing by the standard deviation. $\endgroup$ – Forinstance Dec 23 '17 at 12:49
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Since you have divided the variables by the theoretical maximum and the 3 variables now lie in [0,1], you have effectively standardized the varibles. As @ttnphns points out, there are many ways of standardizing, and you have already chosen one. The 3 values are now the same order of magnitude.

Statistical standardization is not always required or recommended. Suppose the index is a consumer price index, and you sample what a group of people spend on rent and what they spend on chocolate. In building the index, you don't want to standardize the variables because in the real world, rent really is more important than chocolate and takes up a larger part of someone's budget. The common element is the time frame of the expenses (1 month, say) and we are interested in the actual dollars spent.

I never like to be in a situation where I have to standardize variables, because it implies that I am grouping together variables that measure different things on different scales. This raises the question of why I am even looking at those quantities together. Sometimes it makes sense to do so, but it often doesn't. One danger of glib standardization is that it may lure people into a false sense of confidence. They believe that a statistical dodge allows them to throw a whole bunch of variables into the hopper without asking themselves whether their project makes sense.

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    $\begingroup$ Thanks a lot @Placidia! I thought that dividing by the maximum was less appropriate then subtracting the average value and dividing by the standard deviation (because if one variable has more variance than the others this will count more in the sum, if I just divide its values by the maximum). Am I wrong? Thanks a lot your answer is very useful. $\endgroup$ – Forinstance Dec 22 '17 at 15:47
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    $\begingroup$ Normalizing (subtracting the mean and dividing by the SD) is a good idea when the data are normalesque (symmetric, unimodal). You said "theoretical maximum", which is why I let it pass. I would need a very good reason before I divided by a sample maximum, since that would make my analysis highly influenced by outliers. It your variables are positive and bounded above, it's almost like you're turning them into proportions of some kind. That could make a lot of sense. $\endgroup$ – Placidia Dec 22 '17 at 15:54
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    $\begingroup$ The percentile method is bad, because you are forcing the variables to be uniformly distributed, thus eliminating all features of interest from the data -- like clusters, modes, outliers, etc. The second and third do not give you a true index. The weights of a true index remain constant for the life of the index and do not change from one sample to the next (which is what will happen here if you add more data). Try a PCA of the correlation matrix to see what weights the data want to have. $\endgroup$ – Placidia Dec 26 '17 at 14:48
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    $\begingroup$ Your results suggest that it makes sense to combine the standardized scores with equal weighting. In a true index, like the consumer price index, the idea is to monitor something over time as new data arrive, so you need to use the same weights. You could do what the QC people do, select an initial sample and use its means and standard deviations as you move forward. Or perhaps this is not a true index and you only want to use these observations for the current study. In that case, you are fine with the z scores. $\endgroup$ – Placidia Dec 28 '17 at 12:55
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    $\begingroup$ If the plan is to collect samples at future time points, you could use the sample z-scores at each location and build the index on that basis. It wouldn't be a true index, but it would be a meaningful measure of something. But if the new data arrive one observation at a time, then you need to find baseline values for the standardization. That's true whether you use min-max or z-scores. Given that the PCA worked, I'm tending towards z-scores. $\endgroup$ – Placidia Dec 28 '17 at 12:59

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