Creating an index based on a set of measurements without a target for purpose of rank ordering The problem I'm trying to solve here is very simple but the available data is very limited. That makes it a hard problem to solve.
The available data are as follows:


*

*I have 100 patients and I need to rank order them in terms of how healthy they are.

*I only have 5 measurements for each patient. Each of the five readings is coded as a numeric value, and the rule is that the bigger the reading the healthier is the patient.


Should I have some sort of doctor's "expert judgement based ranking" I could use that as the target variable and fit some sort of an ordinal logistic regression model trying to predict doctor's assessment. However, I don't have that. The only thing I have is (1) and (2).
How would you come up with a simple "scoring" algorithm which would combine those five measurements into a single score which would be good enough (not perfect) in rank ordering patients?
 A: Any function $f: \mathbb{R}^5 \to \mathbb{R}$ that is separately increasing in each of its arguments will work.  For example, you can select positive parameters $\alpha_i$ and any real parameters $\lambda_i$ and rank the data $(x_1, x_2, x_3, x_4, x_5)$ according to the values of
$$\sum_{i=1}^{5} \alpha_i (x_i^{\lambda_i} - 1) / \lambda_i \text{.}$$
Evidently some criterion is needed to select among such a rich set of distinctly different scores.  In particular, the simple obvious solutions (frequently employed, unfortunately) of just summing the scores or first "normalizing" them in some fashion and then summing them will suffer from this lack of grounding in reality.  To put it another way: any answer that does not derive its support from additional information is a pure fabrication.
Because this problem is essentially the same as Creating an index of quality from multiple variables to enable rank ordering, I refer you to the discussion there for more information.
A: I would just simply sum them up, weighting each factor if necessary.
A: How about generating a synthetic binary target variable first and then running a logistic regression model?
The synthetic variable should be something like... "If the observation is in the top decile on all of the input variable distributions flag it as 1 else 0"
Having generated the binary target variable... Run logistic regression to come up with probabilistic metric 0 to 1 assesing how far/close in the tails of multiple distributions observation is?
