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A common practice to sum variables that are on different scales is to standardize them, by subtracting the mean and dividing by the standard deviation. However, this produces negative values, which is sometimes a problem (e.g., here).

I am wondering if subtracting the minimum (instead of the mean) would still make sense, while avoiding negative values, like in the following formula: $$ Index_{i} = \frac{x_{i}-\min_{x}}{SD_{x}} + \frac{y_{i}-\min_{y}}{SD_{y}} + \frac{z_{i}-\min_{z}}{SD_{z}} $$ with variables $x,y,z \geq 0$

See a numeric example here: enter image description here

Would this be a problem in terms of scale or not? I mean, is there a problem if I sum 3 variables normalized in this way? Variance should be taken into account as I divide by SD. But does subtracting the minimum, instead of the mean, produce range problems or other issues?

Would this bias results for the analysis of time series in comparison to classic standardization $ \frac{x_{i}-\mu_{x}}{SD_{x}} $? Take the case in which I measure this indicator ($Index_{i}$), for two persons, every week. However, each week the reference sample changes, with different mean, min and SD.

EDIT

Let me add that summing these variables makes sense in our previous experience, as they are correlated and underlie the same construct (even if they measure different things). Summing them (having subtracted the mean and divided by SD) is something that works well. We already tested it. But negative values are a problem. This is the reason why of my question on the min. Assuming that a sum of standardized values is ok, would it still be ok if I change the mean with the min?

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    $\begingroup$ Make sense for what purpose? Your link notwithstanding, you have not really explained why you regard negative values as a "problem" to begin with. $\endgroup$
    – Ben
    May 31 at 11:11
  • $\begingroup$ I mean if it makes sense as a way to avoid negative values, while having a composite indicator that is not biased. I mean if it is ok to sum three variables that vary on different scales, if normalized in this way. Similarly to regular standardization, which is necessary to sum variables that have different units of measures, different variance and range. So that one does not dominate all the sum. I will add a sentence to the question. $\endgroup$ Jun 1 at 16:46
  • $\begingroup$ Okay, I think the issue here is that it is not usual practice to sum values on different scales, and this causes problems regardless of whether they are standardised. $\endgroup$
    – Ben
    Jun 1 at 23:04
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I think the problem here is that you are proceeding on the basis of a false premise, which is leading you to an XY problem. When dealing with variables on different measurement scales, it is not appropriate (nor common practice) to sum them. Even if variables are standardised to remove their units (rendering them scale-free), it is still unusual to sum them, since this usually does not correspond to any interesting underlying aspect of the distribution of the underlying quantities.

If we want a composite measure that keeps track of multiple different variables, which may be on different scales, we usually form this as a vector composed of one measure for each variable. For example, we might create a vector of standardised values like this:

$$\text{Index}_i = \begin{bmatrix} \frac{x_i - \bar{x}_n}{s_{X,n}^2} \\[6pt] \frac{y_i - \bar{y}_n}{s_{Y,n}^2} \\[6pt] \frac{z_i - \bar{z}_n}{s_{Z,n}^2} \\[6pt] \end{bmatrix}.$$

Vector quantities like this keep track of multiple different variables, which is what you generally want to do. If you want to reduce this to a scalar measure, it is usual to look at things like the norm of the vector, or some other distance measure.

If, contrary to the above, you do indeed want to form a scalar index that sums variables on different measurement scales, you will first need to do some further reflection on what you intend to use this index for. The purpose of the index will determine the appropriate form of the index, but any index that sums variables on different measurement scales is going to involve some kind of amalgamation where there is a loss of information. At the moment it is not clear what your overarching task is, so I see no reason for an index of this kind.

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  • $\begingroup$ Thank you for this comment, which opens for me a new perspective. For the specific case, let me add that summing these variables makes sense in our previous experience, as they are correlated and underlie the same construct (even if they measure different things). Summing them (having subtracted the mean and divided by SD) is something that works well. We already tested it. But negative values are a problem. This is the reason why of my question on the min. Let's say that a sum of standardized values is ok. Would it still be ok if I change the mean with the min? I added this to the question. $\endgroup$ Jun 2 at 8:21
  • $\begingroup$ Also, taking the norm wouldn't fit my case as a negative contribution to the index (if components are standardized as in your fomula) would add a positive value to the final score. $\endgroup$ Jun 2 at 8:34
  • $\begingroup$ Without knowing the full context I cannot offer a view, other than to say it all sounds dubious to me. $\endgroup$
    – Ben
    Jun 2 at 11:13
  • $\begingroup$ Honestly, I don't know what else to add. Would it change things if I say these are intelligence measures (dimensions) that we sum to make the full intelligence construct? $\endgroup$ Jun 4 at 16:56

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