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So, I'm stuck: Is there a way to properly parameterize the effects so that the two correlations could be recovered?

So, I'm stuck: Is there a way to recover the two correlations r1 and r2 through variance partitioning or parameterization?

Now Let's switch to a case with 4 variables among which the first and second as well as the third and fourth are correlated. In other words, the four variables are assumed to follow a pentavariate Gaussian $(y_{1}, y_{2}, y_{3}, y_{4})' \sim N((0, 0, 0, 0)', S)$:

r1 <- 0.2; r2 <- 0.8        # correlation value to be recovered
ns <- 2000                  # number of samples
S  <- matrix(c(1,r1,0,0,    # correlation structure of pentavariate data
               r1,1,0,0,    # the first and second variables are correlated
               0,0,1,r2,    # the third and fourth variables are correlated
               0,0,r2,1), nrow=4,ncol=4)

# simulated data
dat <- data.frame(f = c(rep(paste0('P',1:ns), 2), rep(paste0('T',1:ns), 2)),
                  R=c(rep('P',2*ns), rep('T',2*ns)),
                  y = c(mvrnorm(n=ns, mu=rep(0,4), Sigma=S)))

Now Let's switch to a case with 4 variables among which the first and second as well as the third and fourth are correlated. In other words, the four variables are assumed to follow a quadrivariate Gaussian $(y_{1}, y_{2}, y_{3}, y_{4})' \sim N((0, 0, 0, 0)', S)$:

r1 <- 0.2; r2 <- 0.8        # correlation value to be recovered
ns <- 2000                  # number of samples
S  <- matrix(c(1,r1,0,0,    # correlation structure of quadrivariate data
               r1,1,0,0,    # the first and second variables are correlated
               0,0,1,r2,    # the third and fourth variables are correlated
               0,0,r2,1), nrow=4,ncol=4)

# simulated data
dat <- data.frame(f = c(rep(paste0('P',1:ns), 2), rep(paste0('T',1:ns), 2)),
                  R=c(rep('P',2*ns), rep('T',2*ns)),
                  y = c(mvrnorm(n=ns, mu=rep(0,4), Sigma=S)))

One noticeable aspect is that in the model m1 the total variance is conserved in the sense that the sum of the two variances from the modeling result ($0.4916+0.5006\approx 1$) is equal to the the variance (which is 1) of the simulating distribution $N((0, 0, 0)', S)$. In contrast, this is not true for the models m2 and m3: the 3 variances in summary(m2) and summary(m3) do not add up to 1. This may indicate that there is variance leakage happening in the two latter models? Or the models m2 and m3 are not be the right formulation for the retrieval of r1 and r2?

One noticeable aspect is that in the model m1 the total variance is conserved in the sense that the sum of the two variances from the modeling result ($0.4916+0.5006\approx 1$) is equal to the the variance (which is 1) of the simulating distribution $N((0, 0, 0)', S)$. In contrast, this is not true for the models m2 and m3: the 3 variances in summary(m2) and summary(m3) do not add up to 1. This may indicate that there is variance leakage happening in the two latter models? Or the models m2 and m3 are not parameterized properly for retrieving r1 and r2?