rotate_model: QR rotation to independent models.
In dcgerard/vicar: Various Ideas for Confounder Adjustment in Regression

Description Usage Arguments Value Author(s)

QR rotation to independent models.

rotate_model(
  Y,
  X,
  k,
  cov_of_interest = ncol(X),
  include_intercept = TRUE,
  limmashrink = FALSE,
  fa_func = pca_naive,
  fa_args = list(),
  do_factor = TRUE
)

`Y`	A matrix of numerics. These are the response variables where each column has its own variance. In a gene expression study, the rows are the individuals and the columns are the genes.
`X`	A matrix of numerics. The covariates of interest.
`k`	A non-negative integer.The number of unobserved confounders. If not specified and the R package sva is installed, then this function will estimate the number of hidden confounders using the methods of Buja and Eyuboglu (1992).
`cov_of_interest`	A vector of positive integers. The column numbers of the covariates in X whose coefficients you are interested in. The rest are considered nuisance parameters and are regressed out by OLS.
`include_intercept`	A logical. If `TRUE`, then it will check `X` to see if it has an intercept term. If not, then it will add an intercept term. If `FALSE`, then `X` will be unchanged.
`limmashrink`	A logical. Should we apply hierarchical shrinkage to the variances (`TRUE`) or not (`FALSE`)? If `degrees_freedom = NULL` and `limmashrink = TRUE` and `likelihood = "t"`, then we'll also use the limma returned degrees of freedom.
`fa_func`	A factor analysis function. The function must have as inputs a numeric matrix `Y` and a rank (numeric scalar) `r`. It must output numeric matrices `alpha` and `Z` and a numeric vector `sig_diag`. `alpha` is the estimate of the coefficients of the unobserved confounders, so it must be an `r` by `ncol(Y)` matrix. `Z` must be an `r` by `nrow(Y)` matrix. `sig_diag` is the estimate of the column-wise variances so it must be of length `ncol(Y)`. The default is the function `pca_naive` that just uses the first `r` singular vectors as the estimate of `alpha`. The estimated variances are just the column-wise mean square.
`fa_args`	A list. Additional arguments you want to pass to fa_func.
`do_factor`	A logical. Should we do the factor analysis or just rotation?

A list that contains some of the following elements.

betahat_ols: The OLS estimates of the coefficients of interest.
Q: The Q from the QR decomposition of X
R11: The top left element of the R matrix from the QR decomposition of X
R12: The top right element of the R matrix from the QR decomposition of X
R22: The bottom right element of the R matrix from the QR decomposition of X
Y1: The rows of the rotated Y matrix that correspond to the uninteresting covariates in X
Y2: The rows of the rotated Y matrix that correspond to the interesting covariates in X
Y3: The rows of the rotated Y matrix that correspond to the nullspace of X
k: The number of latent factors. Estimated with num.sv if not provided in the function arguments.
X: The provided X matrix. This might differ from the original X matrix up to a permutation of the columns. The interesting covariates are placed at the end of X and the uninteresting covariates are placed at the start of X.
Y_tilde: The rotated Y matrix.
alpha: The estimates of the coefficients of the unobserved confounders. Only returned if do_factor = TRUE
Z3: The estimates of the unobserved confounders (third part). Only returned if do_factor = TRUE
sig_diag: The estimates of the column-specific variances (NOT standard deviations). A vector. Only returned if do_factor = TRUE
prior_df: The estimated prior degrees of freedom. Returns NULL if limmashrink = FALSE. Only returned if do_factor = TRUE