Confidence intervals for one single-cell bulk sample

The composition and the two technical scales are separated by the Schur complement of the observed Hessian. method chooses the interval for the composition. The default is the inverse ILR chart applied to a Wald interval for $\boldsymbol{\rho}$, using the profile information $\{-\nabla^{2}_{\boldsymbol{\rho}}\ell^{\ast}\}^{-1}$.

Intervals for $(c,\phi)$ in nuisance$interval are Wald intervals on the log scale, mapped back by the exponential, from the Schur complement that removes $\boldsymbol{\rho}$. The rows platform_scale_a ($a=\phi/c$) and cell_count_n ($n=c^{2}/\phi$, a number of cells only when $a$ is known) follow by the delta method on $\log a=\log\phi-\log c$ and $\log n=2\log c-\log\phi$, so all four intervals are positive. nuisance$interval_wald is the symmetric normal interval of $(c,\phi)$ on the original scale. Bootstrap methods replace both with percentile intervals.

ilr: Default. Inverse ILR chart.
wald: Delta method on the composition. Endpoints can leave $[0,1]$.
firth: Wald interval at the penalised profile estimate. interval_ilr maps that estimate through the inverse chart.
profile: Interior likelihood-ratio interval. A face null is lrt_single_cell().
bootstrap: Parametric percentile interval.
reference: Donor bootstrap, or a within-type cell resample that regenerates the bulk. See reference_bootstrap_single_cell().

These intervals treat the reference covariance as known. They do not add a sandwich or higher-order correction for that plug-in.

variance = "reml" corrects for the $\nu=G-J$ residual degrees of freedom of the variance scale: covariances are inflated by $G/\nu$, Wald and ILR quantiles come from $t_{\nu}$ (a multivariate $t$ for the ILR map when $J>2$), $\phi$ gets the chi-square interval $[G\hat\phi/\chi^2_{\nu;1-\alpha/2}, G\hat\phi/\chi^2_{\nu;\alpha/2}]$, and the profile interval uses the cut-off of confint_profile_single_cell(). The maximum-likelihood $\hat\phi$ is biased by about $\nu/G$, so with few genes ($G=10$, $J=2$) the "ml" intervals cover about 0.84 to 0.93 at the 0.95 level; the bootstrap methods are unaffected by variance.

Usage

confint_single_cell(
  y,
  mean_signature_matrix = NULL,
  Sigma = NULL,
  method = c("ilr", "wald", "firth", "profile", "bootstrap", "reference"),
  level = 0.95,
  variance = c("ml", "reml"),
  n_draws = 2000L,
  n_boot = 200L,
  seed = NULL,
  reference_profiles = NULL,
  donor_ids = NULL,
  resample = NULL,
  regenerate_bulk = NULL,
  ...
)

Arguments

y: Bulk vector of length $G$.
mean_signature_matrix: Reference mean, $G\times J$.
Sigma: Cell-level covariances, $G\times G\times J$.
method: Interval for the composition. See Description.
level: Confidence level.
variance: "ml" (default) or "reml"; see Description.
n_draws: Monte Carlo size for the ILR map when $J>2$.
n_boot: Bootstrap size.
seed: Optional seed, restored on exit, for the ILR Monte Carlo map and for the bootstrap methods.
reference_profiles, donor_ids, resample, regenerate_bulk: Passed to reference_bootstrap_single_cell() when method = "reference".
...: epsilon and itmax for the selected method.

Value

A list with method, estimate (the fitted composition), interval and nuisance. Bootstrap results also include the replicate matrices.

Examples

mu <- matrix(c(2, 0.2, 0.2, 2, 0.2, 3, 0.2, 3), nrow = 4)
colnames(mu) <- c("ct1", "ct2")
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)
fit <- confint_single_cell(y, mu, sigma)
fit$interval
#>          2.5%     97.5%
#> ct1 0.7032075 0.7074397
#> ct2 0.2925603 0.2967925
fit$nuisance$interval
#>                          2.5%        97.5%
#> mean_scale       1.998420e+01 2.008519e+01
#> variance_scale   2.652748e-02 4.241083e-01
#> platform_scale_a 1.324079e-03 2.116881e-02
#> cell_count_n     9.464154e+02 1.513110e+04

Usage

Arguments

Value

See also

Examples