gibbs_crossgroup: Gibbs sampler for cross-group comparison
In MaStatLab/LTN: Fit LTN model to microbiome count data

A more flexible version of ltnme that allows users to change all hyperparameter specifications in the LTN-based mixed-effects model

gibbs_crossgroup(
  N,
  p,
  g = NULL,
  r,
  YL,
  Y,
  Xtest,
  Xadjust = NULL,
  grouplabel = NULL,
  c0 = 1,
  d0 = 0.001,
  c1 = 1,
  d1 = 0.001,
  nu = 3,
  niter,
  adjust = F,
  reff = F,
  reffcov = 1,
  SEED = 1,
  save_alpha_only = F,
  gprior_m = 1,
  pnull = 0.5,
  a1 = 3,
  a2 = 4,
  a1a2 = "hp",
  lambda_fixed
)

`N`	number of samples
`p`	number of internal nodes
`g`	number of levels of the random effect. e.g., if the random effect is individual, then g is number of individuals
`r`	number of latent factors (unwarranted feature)
`Y`	N*d matrix of y(A)
`Xtest`	N*q1 design matrix of the covariate to test, q1 should be the number of groups-1. For the two-group problem, Xtest is a column vector.
`Xadjust`	N*q2 design matrix of the covariate to adjust, should include a column of 1's
`c0, d0, c1, d1, nu`	hyperparameters
`niter`	number of Gibbs iterations
`SEED`	random seed for initializing parameters
`save_alpha_only`	whether to only return posterior samples of alpha(A)
`gprior_m`	Var(beta2)=mN(X^TX)^-1, gprior_m=1 implies unit information prior
`a1, a2`	hyperparameters in prior on factor loadings (unwarranted feature)
`a1a2`	'hp': use Ga(2,1) hyperprior on a1,a2; 'fix': fix a1, a2
`lambda_fixed`	if >0, use this as the fixed value of lambda; if <=0, adopt a Gamma hyperprior on lambda
`refflabel`	vector of random effect labels with length g, e.g., subject ID.