Computes the local-shift sensitivity \(\partial \hat{\beta}/\partial y\)
(and the analogous quantity for the spherical random effects \(\hat{u}\))
by implicit differentiation of the joint \((\beta, u)\)-scoring
equations at the fit. The returned list is a building block for
caseweightIF and vcov_sandwich; users who
only want robust standard errors should call vcov(object, type = "sandwich").
implicitIF(fit, use.expected = FALSE)A list with components IF_beta, IF_u, IF_b,
the Jacobian blocks Jbb, Jbu, Jub, S,
and intermediate rotated-design matrices used by the cluster
sandwich. IF_beta is the local-shift sensitivity matrix
\(p \times n\).
An rlmerMod object.
If TRUE, use \(E[\psi']\) for the diagonal
blocks of the Jacobian; the default (FALSE) uses the actual
\(\psi'\) values at the fitted residuals (the empirical version).
Prior weights and offsets are supported: the score is evaluated in the same prior-weight-whitened coordinates the estimator iterates in (\(U_e = \mathrm{diag}(\sqrt{w})\), residuals offset-corrected).
This is the partial (\(\sigma, \theta\) held fixed) version. It is
not the Hampel influence function; the Hampel object is
caseweightIF, which reuses the same Jacobian but with the
score value \(\psi\) on the right-hand side rather than
\(\partial_y \psi\). See Koller (2026, Paper 1) for a discussion.