First-order correction of sampling covariance for 2S-PA estimates

Usage

vcov_corrected(
  tspa_fit,
  vfsLT,
  which_free = NULL,
  engine = c("analytic", "fd"),
  ...
)

Arguments

tspa_fit: A fit from tspa() with fsT and fsL supplied (multi-factor measurement model, single- or multi-group), so that it carries the fsT, fsL, and tspa_args attributes. A fit corrected in place via tspa(corrected_se = TRUE) (attribute tspa_corrected = TRUE) is rejected.
vfsLT: The sampling covariance matrix of the free fsL/fsT elements, taken (or sub-matrixed) from the vfsLT attribute of a get_fs() result fitted with vfsLT = TRUE. Its row and column order is the same fsL-then-fsT order described by which_free. When which_free = NULL (the default) this is the full square matrix returned by get_fs(); when which_free is a subset of length k, this must be the matching k x k principal submatrix, i.e. vfsLT_full[which_free, which_free].
which_free: An optional numeric vector of positions selecting which fsL/fsT elements to treat as free (and therefore to propagate through the Jacobian). Positions run over the fsL matrix (column-major order) followed by the lower-triangular part of fsT including the diagonal (column-major order). For a two-factor model fsL and fsT are both 2 x 2: fsL occupies positions 1:4 and fsT positions 5:7, so the two error variances are 5 and 7 and the error covariance between the two factor scores (the [2, 1] element of fsT) is position 6. For a multigroup fit the positions run per group, in group order: group 1's full fsL (column-major), then group 2's full fsL, ..., then group 1's lower-triangular fsT (column-major), then group 2's, i.e. all loadings across groups first, then all error-variance elements (matching the order of the vfsLT attribute from get_fs(vfsLT = TRUE)). NULL (the default) treats every per-group fsL/fsT element as free. A non-NULL which_free of length k requires vfsLT to be the matching k x k principal submatrix (see vfsLT).
engine: The engine used to evaluate the Jacobian J = d(thetahat)/d(eta). "analytic" (the default) uses a refit-free, deterministic influence-function closed form (PLAN 16, sections 2.4 and 4.3): J = -H^{-1} C, with H (the log-likelihood Hessian over the free params) and C (the cross-derivative w.r.t. the fixed fsL/fsT entries) obtained by central-differencing the analytic log-likelihood score. It covers single- and multi-group models, saturated and restricted (df > 0) structural models, and mean-structure models, and is a pure function of the base fit + vfsLT (bit-reproducible, no refits). "fd" uses central finite differences (one stage-2 refit on each side of each free element) and is retained as the A/B reference; the analytic path transparently falls back to it only for a shape it cannot handle (an unrecognised free parameter or unequal per-group free-param counts). The two agree to the finite-difference noise floor whenever "analytic" applies.
...: Currently not used.

Value

A corrected covariance matrix in the same dimension as vcov(tspa_fit) (symmetric).

Details

vcov_corrected() applies the first-order (delta-method) two-stage approximation: the stage-2 covariance vcov(tspa_fit) is augmented by J %*% vfsLT %*% t(J), where J is the Jacobian of the stage-2 estimates with respect to the selected fsL/fsT free elements. With the default engine = "analytic", J is evaluated refit-free via a closed form (see the engine argument below); with engine = "fd" it is instead estimated by central differences, one full stage-2 refit on each side of each free element while reusing the base fit's coefficients, so the cost is 2 x (number of free elements) stage-2 refits.

The correction is partial by design. It propagates only the sampling uncertainty of the stage-1 estimates of fsL (loadings/cross-loadings) and fsT (score error variance-covariance, including its off-diagonal). It does not account for the sampling uncertainty of the factor-score values (the fs_<name> columns fitted in stage 2), their standard errors (se_fs), or the score intercepts (fsb); these are held fixed. The two-stage approximation also treats the stages as independent, so the cross-covariance between the stage-1 estimates and the stage-2 data is not modelled.

The same correction can also be requested in place via tspa(..., corrected_se = TRUE, vfsLT = <matrix>); tspa() then sets the tspa_corrected = TRUE attribute on the returned fit, and standardizedSolution() on that fit reports corrected standard SEs. Such an already-corrected fit is rejected by vcov_corrected() (the correction is never applied twice).

The engine argument selects how the Jacobian is evaluated. The default "analytic" is refit-free and deterministic: an influence-function closed form (PLAN 16, sections 2.4 and 4.3), J = -H^{-1} C, with H (the log-likelihood Hessian over the free params) and C (the cross-derivative w.r.t. the fixed fsL/fsT entries) obtained by central-differencing the analytic log-likelihood score. It covers single- and multi-group models, saturated and restricted (df > 0) structural models, and mean-structure models, and is a pure function of the base fit + vfsLT (bit-reproducible, no refits). "fd" (central finite differences, one stage-2 refit per side of each free element) is retained as the A/B reference; the analytic path falls back to it only for a shape it cannot handle. The two agree to the finite-difference noise floor whenever "analytic" applies.

Examples