# Determining the weights of known parameters in a formula

I have a formula of the following form:

$a_1*w + a_2*x + a_3*y + a_4*z$

In the above formula, the $a_i$s can be thought of as weights to the corresponding parameters. The values of the parameters $w, x, y,$ and $z$ are known whereas the $a_i$s are unknown.

The above formula aims to assign 'values' to certain objects after the $a_i$s become known. After making an initial guess about the $a_i$s, I am able to assign the values(There are about 50 such objects whose values have to be assigned with the above formula) and rank the objects in a decreasing order based on the values obtained and compare the obtained ranking of the objects with the ranking they should have obtained (which is determined by other means) through various statistical parameters such as covariance, corelation etc.

PROBLEM:

Now, I want to determine which $a_i$ (or for that matter, which term) in the above equation contributed the most to the ranking order obtained from the formula (i.e., which $a_i$ was the most significant in getting the obtained ranking).
Is there any statistical way to determine this?

• if any mathematical method(s) exist(s) for the above problem instead of statistical, you are also welcome to point that(those) method(s) too. Jan 30, 2015 at 10:12
• It will depend on what you mean by "most significant". You also have other issues: e.g. multiplying all the $a_i$ by a positive constant will give the same ranking; if the $w,x,y,z$ are not comparable in some sense (similar units or similar dispersion) then comparisons of the $a_i$ may not be meaningful. Jan 30, 2015 at 11:06
• @Henry Hello, by "most significant" I mean that the term that accounts "the most" for the ranking. (The weights are the same for all objects but the values of $x, y, z, w$ are different for different objects). Assuming that $x, y, z, w$ are comparable, what can be the possible approach to this? Jan 30, 2015 at 14:34