Observed information of one single-cell fit

At a fitted composition, the covariance of the ILR coordinate \(\hat{\boldsymbol{\rho}}\) is the inverse of the negative profile Hessian hessian_profile_loglik_single_cell_ilr(). The delta method carries it to \(\hat{\boldsymbol{p}}\) with \(\mathbf{J}_{\psi}\). The two technical scales use the other Schur complement of the joint observed Hessian in \((\boldsymbol{\rho},c,\phi)\): estimating the composition widens their variances, and \(I_{c\phi}\) is not assumed to make that complement diagonal.

Both matrices are observed information, not expected Fisher information. They treat the reference moments as known.

Usage

vcov_single_cell(
  p,
  y,
  mean_signature_matrix,
  Sigma,
  mean_scale = NULL,
  variance_scale = NULL,
  scales_known = FALSE,
  warn = TRUE
)

Arguments

p

Fitted composition. A boundary coordinate makes the ILR chart undefined.

y

Bulk vector of length \(G\).

mean_signature_matrix

Reference mean, \(G\times J\).

Sigma

Cell-level covariances, \(G\times G\times J\).

mean_scale, variance_scale

Scales at p. NULL recomputes the closed forms.

scales_known

If TRUE, the scales are treated as known: the composition covariance inverts the fixed-scale observed Hessian hessian_loglik_single_cell_ilr() at mean_scale and variance_scale, and vcov_nuisance is NA. This is the covariance of a "fixed_scales" fit. For a "two_stage" fit it ignores the uncertainty of the start scales, so its intervals are too narrow.

warn

If TRUE, warn when either information matrix is not positive definite.

Value

A list with rho, vcov_rho, vcov (delta method for \(\boldsymbol{p}\)), mean_scale, variance_scale and vcov_nuisance (order \(c\), \(\phi\)). Singular blocks are filled with NA.

See also

deconvolute_single_cell(), confint_single_cell()

Examples

mu <- matrix(c(2, 0.2, 0.2, 2, 0.2, 3, 0.2, 3), nrow = 4)
sigma <- array(rep(diag(0.05, 4), 2), c(4, 4, 2))
y <- 20 * drop(mu %*% c(0.7, 0.3)) + c(0.3, -0.2, 0.1, -0.1)
vcov_single_cell(c(0.7, 0.3), y, mu, sigma)
#> Warning: The profile information is not positive definite.
#> Warning: The nuisance information is not positive definite.
#> $rho
#> [1] 0.5991301
#> 
#> $vcov_rho
#>      [,1]
#> [1,]   NA
#> 
#> $vcov
#>      [,1] [,2]
#> [1,]   NA   NA
#> [2,]   NA   NA
#> 
#> $mean_scale
#> [1] 20.00234
#> 
#> $variance_scale
#> [1] 0.7497659
#> 
#> $vcov_nuisance
#>                mean_scale variance_scale
#> mean_scale             NA             NA
#> variance_scale         NA             NA
#>