Test to evaluate if two delta values are associated? I have an observational cohort. Subjects have lost weight. I wish to see if the weight they lost is correlated to an increase in break down of amino acids muscle. I have values from 2 time points before and after wight loss for both weight and amino acids break down rate.
Do I calculate the difference in weight and amino acids and perform a normal regression with the delta values?
 A: Your intuition is correct. You are interested in studying the impact of your treatment on the change in weight, not either weight measurement on its own.
Below is a brief simulation in R. The simulation is constructed as simple experiment to clarify the intuition. Even if you do not know R, it shows that by far the best regression model for estimating the true effect (we know the true effect here because it's a simulation) is the regression delta ~ treatment. I compare this regression to a simple regression of the final weight on treatment, and a regression of the final weight on treatment and the baseline weight.
set.seed(961)
require(data.table)

n = 1000

# Control group baseline
C0 <- rnorm(n, mean = 160, sd = 22)

# Treatment group baseline
T0 <- rnorm(n, mean = 160, sd = 22) 

# Add some noise to each observation, as weight shifts over time
# Noise = rnorm(n, mean = 0, sd = 1)
# Treatment effect = rnorm(n, mean = -3, sd = 1)
C1 <- C0 + rnorm(n, mean = 0, sd = 3)
T1 <- T0 + rnorm(n, mean = 0, sd = 3) + rnorm(n, mean = -3, sd = 3)

# Merge all the observations into one data.table
data <- data.table(time0 = c(C0, T0), time1 = c(C1, T1), treatment = c(rep(0, times = n), rep(1, times = n)), delta = c(C1, T1) - c(C0, T0) )

# Model 1: endline weight regressed on treatment
summary(lm(time1 ~ treatment, data = data))

# Model 2: endling weight regressed on treatment and baseline weight
summary(lm(time1 ~ treatment + time0, data = data))

# Model 3: delta regressed on treatment (the best estimate)
summary(lm(delta ~ treatment, data = data))

