Thank you, I hope your defense goes well.

You're welcome!

Thank you!

No problem. The Pearson Correlation equation is like a standardized slope for a linear regression line. It cares about the cov(x,y) and normality of data, so it ends up equaling 0 when the data goes down and up or up and down like that. Spearman Correlation is a more general approach because it ranks the variables and doesn't assume the normality of the data. It cares that the variables are monotonic, that is a one to one type of relationship. The quadratic creates a type of one to two because as the first variable increases the second variable goes up and down or down and up. If you find yourself needing a correlation coefficient for nonlinear models or data, Kendall Correlation would help with that. It is another nonparametric test.

You will not be able to convert a spearman rs to a pearson r. There may be similarities at times such as a 45 degree angle where both are 1 but what they are measuring is different and non-transferable.

Since Y stays at 0 and Spearman looks at the line through a monotonic lens, it would be 1.

Thank you!

If the correlation between the variables go up and down, with your example I would use Pearson's. That will give you more information but it also depends on what type of relationship you are trying to gauge between variables. Both Pearson and Spearman have their uses, one is more sensitive than the other but it also depends on the nature of the data.

There really isn't one as an "ideal" Spearman can take many forms whereas an "ideal" Pearson's correlation is a straight line that is positive or negative 1. Regression lines are still used as they give a comparison within the graph against the Spearman correlation.