block diagonal covariance matrix by group of variable - r

Question

If this is my dataset, 3 subjects, measured at two time points t1,t2 , at each time point each subject is measured twice
ID Time Observation Y
1 t1 1 3.1
1 t1 2 4.5
1 t2 1 4.2
1 t2 2 1.7
2 t1 1 2.3
2 t1 2 3.1
2 t2 1 1.5
2 t2 2 2.2
3 t1 1 4.1
3 t1 2 4.9
3 t2 1 3.5
3 t2 2 3.2
What is the block diagonal covariance matrix for this dataset ? I am assuming the matrix would have three blocks, each block representing 2x2 variance-covariance matrix of subject i with t1 and t2. Is this correct ?

Answer 1

The code below computes the covariance matrices for each ID, then creates a block matrix out of them.
make_block_matrix <- function(x, y, fill = NA) {
u <- matrix(fill, nrow = dim(x)[1], ncol = dim(y)[2])
v <- matrix(fill, nrow = dim(y)[1], ncol = dim(x)[2])
u <- cbind(x, u)
v <- cbind(v, y)
rbind(u, v)
}
# compute covariance matrices for each ID
sp <- split(df1, df1$ID)
cov_list <- lapply(sp, \(x) {
y <- reshape(x[-1], direction = "wide", idvar = "Time", timevar = "Observation")
cov(y[-1])
})
# no longer needed, tidy up
rm(sp)
# fill with NA's, simple call
# res <- Reduce(make_block_matrix, cov_list)
# fill with zeros
res <- Reduce(\(x, y) make_block_matrix(x, y, fill = 0), cov_list)
# finally, the results' row and column names
nms <- paste(sapply(cov_list, rownames), names(cov_list), sep = ".")
dimnames(res) <- list(nms, nms)
res
#> Y.1.1 Y.2.2 Y.1.3 Y.2.1 Y.1.2 Y.2.3
#> Y.1.1 0.605 -1.54 0.00 0.000 0.00 0.000
#> Y.2.2 -1.540 3.92 0.00 0.000 0.00 0.000
#> Y.1.3 0.000 0.00 0.32 0.360 0.00 0.000
#> Y.2.1 0.000 0.00 0.36 0.405 0.00 0.000
#> Y.1.2 0.000 0.00 0.00 0.000 0.18 0.510
#> Y.2.3 0.000 0.00 0.00 0.000 0.51 1.445
Created on 2023-02-20 with reprex v2.0.2
Data
df1 <- "ID Time Observation Y
1 t1 1 3.1
1 t1 2 4.5
1 t2 1 4.2
1 t2 2 1.7
2 t1 1 2.3
2 t1 2 3.1
2 t2 1 1.5
2 t2 2 2.2
3 t1 1 4.1
3 t1 2 4.9
3 t2 1 3.5
3 t2 2 3.2"
df1 <- read.table(text = df1, header = TRUE)
Created on 2023-02-20 with reprex v2.0.2