How to check if a function doesn't depend on a variable?

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• kelly0303
In summary, the conversation discusses experimental data points and how a variable can be written in terms of two other variables. The question arises of whether the addition of a nonzero coefficient is beneficial, and if there is a way to test this experimentally. It is also mentioned that the value of the coefficient may depend on the relationship between the two variables. Regression can be used to determine if the model has a statistically significant non-zero coefficient, but without being able to measure one of the variables, it may be difficult to determine. The use of the Inverse/Implicit function theorems in Calculus may be helpful in this situation.

kelly0303

Hello! I have some experimental data points ##(z_i,dz_i)## and I know that in the most general case this variable can be written in terms of 2 other variables as ##z_i = ay_i+bx_i##. Beside ##z_i## I can also measure, for each point, ##x_i## (we can assume that the uncertainty in ##x_i## is negligible), but not ##y_i##. I suspect, based on some calculations, that (at least at the level of the experimental uncertainties, ##dz_i##) the ##bx_i## term will be negligible i.e. ##b\sim 0## given my uncertainties. Is there a way to test this experimentally, given my current data and the expected functional form? Thank you!

An important question is whether the ##x_i##s and ##y_i##s are independent or are correlated. If they are independent, then you can consider whether ##b \approx 0## without regard to the value of ##a##. But if they are related, you must consider the value of ##a## to understand whether the addition of a nonzero ##b## is beneficial.
Since you can not measure the ##y_i##s, I am afraid that the best you can do is to use regression to determine if the model ##z_i = a x_i## has a statistically significant non-zero ##a##.

In Calculus, the Inverse/Implicit function theorems are usually used with this purpose, when given an expression in terms of ##x,y ##.