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?