# TL-DR:

The higher the rate of production on my expendable unit, the less overall product it seems to produce in its lifetime. I want to know how best to model this or 20 pages I could read which would tell me how I should model this; to prove or disprove my hypothesis.

# Full body:

My statistics skills have atrophied in my professional career so feel free to refer me to remedial literature as a solution.
Here is my problem:

I have a unit which refines product and has a limited lifespan.
I suspect that the rate at which the product is produced is inversely proportional to the total quantity of product it produces over it's lifespan(within 2-20%).

The data I have:

• Daily product produced
• Date put online (brand new)
• Date taken offline (due to failure)

My hypothesis is that if the unit produced (for example) 2 units per day it would produce more over all units than if it produced 3 units per day. Since these units run at the whim of production schedules and are subject to other factors such as quality of feed and quality of manufacturing, I am finding it difficult to tease out a solution that seems statistically sound.

My current solution is a simple multi variable regression of with $$Y=\text{life span}$$, $$X_1=\text{average daily usage for those days online}$$, $$X_2=\text{total product produced}$$.
Immediately typing this out I see Y and X2 should be flipped but I feel like I am not properly accounting for the day to day variations of production rates.

You should start with visualization of your data: (and show us the results) scatterplot of $$Y$$ versus x1, then versus x2 maybe a conditioning plot like using R) coplot(Y ~ x1 | x2, data=yourdataframe). Histograms.
In the following just some ideas: For modeling, two options: $$Y$$ is a survival length, so survival analysis for $$Y$$ with x1, x2 as predictors (and possibly others). It could well be the effect is not linear so maybe include the predictors with splines. Or a glm (generalized linear model) for $$Y$$ with a model for positive outcomes, maybe a gamma glm.