General Statistics Question on Response Dependence

[#1]

What type of modeling, analysis, and methodology would you use for the following example:

Let's say you have factors A, B, and C in a DOE. You capture responses Y and Z. A combination of B+C increases both Y and Z using a specific regression model, and theoretically (from your discipline and experience), you know that response Y is dependent on Z. How would I best represent the relationship between Y and Z? Only suggestions on what type of analysis would best describe my relationship? Can I use Z as a factor in my model to try to explain Y?

For a more concrete example, let's say I'm a metallurgist exploring the effects of heating media, heat-treatment temperatures, and cooling rates on metals. (Not my exact discipline, but very visual) The responses I'm capturing are the metal's ultimate tensile strength and crystallinity percentage. I know that my maximum tensile strength increases with crystallinity, and that both my responses increase as my cooling rate decreases and my heat-treatment temperatures increase. For this example, let's say the crystallinity rises by 1.5% every 10 degrees, and UTS increases by 8% with each additional 10 degrees of crystallinity, while UTS increases by 3% with each additional 10 degrees of crystallinity. Is this a mathematical algebra problem? Can I drop crystallinity as a factor when it is a captured response? 

Thank you very much for your time and expertise!! 

[#2]

Welcome to the community.  There is no "right" way, but there are some more accepted or more useful ways than others.  Typically you would use multivariate methods, like correlation to see and quantify the relationships between multiple Y's.  Since you can't actually manage crystalinity directly, it wouldn't make sense to have it in your model.  It is a dependent variable, not independent.

[#3]

@statman's reply is spot on. I'd just add the following... 

It depends on what your goal is. If you simply want to know the effect of your A, B, and C factors on outcome Y, then leave Z out (assuming you're fitting a general linear model). But if you're interested in quantifying the "mediated effect" that B and C have through Z, then you'd benefit from fitting a structural equation model (SEM) that captures this exact causal structure:

Each node in the SEM represents a variable and each arrow represents a regression effect. Thus, the SEM decomposes the total causal effects of B and C into direct and indirect effects. If this is of interest, then it's worth looking into it. Here's an example with toy data where you can see all the relevant effects in this type of model:

There are lots of resources for learning how to use the SEM platform for fitting mediation models:

JMP Documentation	Example of Mediation Analysis
JMP Blog	Understanding Simple Mediation Analysis in JMP Pro
JMP Webinar	Path Analysis and Structural Regression
JMP Tutorial	Building SEMs in JMP Pro

HTH,

~Laura