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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.

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- #2

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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##.

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