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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?

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    $\begingroup$ 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. $\endgroup$ – Kodiologist Jun 18 '17 at 0:32
  • $\begingroup$ proportional not linearly? please explain a little bit. Your or other definitions $\endgroup$ – Subhash C. Davar Jun 18 '17 at 12:38
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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.

If you have multiple machine settings, you could run them separately.

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.

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  • $\begingroup$ 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 $\rho$. $\endgroup$ – Heteroskedastic Jim Jun 19 '17 at 14:45

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