For the Oats dataset in nlme, page 162 of "Mixed-effects models in S and S-PLUS" provided a way to express multilevel model through single-level expression in nlme.
More specifically, the following is a multilevel model in which Variety is nested in Block specified in the conventional way:
library(nlme)
fm4Oats <- lme( yield ~ nitro, data = Oats, random = ~ 1 | Block/Variety )
Page 162 of "Mixed-effects models in S and S-PLUS" provided the following alternative fitting strategy through single-level representation of the mixed model (R code highlighted in green, math formula also included to facilitate understanding):
I wish ask how to write the code through single-level expression approach (like fm4OatsC) if I want to add a covariate whose random slope effect varies across levels of Variety and additionally assume a compound covariance structure between random intercept and random slope. More specifically, I want the covariance matrix (Psi**_new) to be as follows:
where sigma^2_1 and sigma^2_2 are the same as in the book, and sigma^2_3 is the variance for the random slope of covariate x, and sigma_23 is the covariance between random intercept of Variety and random slope of x.
Note: There is no covariate x in the Oats data. I just wish to see how the formula should have been written in nlme to specify the designated psi**_new.
I like the single-level expression approach since it allows for more flexibility in defining the covariance structure. But I also wish to know if the following code gives the desired covariance matrix:
fm4Oats <- lme( yield ~ nitro, data = Oats, random = ~ x | Block/Variety )
