# Which test to use: hypothesis that a variable is proportional to another variable (not necessarily linear)

Question 1: Assume that in a manufacturing process, I change some parameter (X) and observe some output (Y). For example, I change temperature (X) and observe strength of part (Y).

How do I test a hypothesis that (increasing X causes an increase in Y)?

I thought of testing for (correlation - Rho), but isn't that just limited to assuming an underlying linear relationship between the variables? What is they are proportional but not linearly?

What is a good statistical test for this? Pearson? Spearman? Kendall?

Question 2: For the same case above, If I have m multiple data sets, what do I do? In other words, suppose that I have datasets 1,...,m. Each data set represents some specific and distinct machine setting for example, and each data set has values for X and Y. Do I just run the test for each data set, or is there a way to combine data sets together?

• A proportional relationship (i.e., $Y$ varies with $aX$ for some constant $a$) is necessarily also a linear relationship (i.e., $Y$ varies with $aX + b$ for some constants $a$ and $b$). Perhaps you're confusing proportionality with monotonicity. – Kodiologist Jun 18 '17 at 0:32
• proportional not linearly? please explain a little bit. Your or other definitions – Subhash C. Davar Jun 18 '17 at 12:38

You can use Spearman's $\rho$. It can capture monotonicity properly i.e. as one increases, the other increases; however, the increments need not be linear, only that both variables change in the same direction.
However, if your primary question is the relation between X and Y, and you do not care for the machine settings, then you can calculate $\rho$ in one analysis, but you would also need to account for the fact that X and Y measurements are clustered within machine settings. I answered a similar question here.
• If you check the answer I linked to -alongside the comments, you can see how. The Fisher transformation can also be applied to Spearman's $\rho$. – Heteroskedastic Jim Jun 19 '17 at 14:45