Calculate parameter SDs using inverse Hessian
calc_sds_alerts(
pars_opt,
pars_const,
observations,
modelfun,
dose_norm,
log10_trans
)A data.frame with variables `param_name`, `param_sd`, and `sd_alert`, and as many rows as the length of `pars_opt`. `param_name` contains the names of `pars_opt`. `param_sd` contains the parameter standard deviations calculated using the inverse Hessian. `sd_alerts` is a character variable noting any errors encountered while attempting to calculate the parameter SDs.
Named numeric: A vector of parameter values for the parameters that were optimized. For example, you can get this using [coef.pk()] with `include_type = "optim"`.
Named numeric: A vector of parameter values for parameters that were held constant, not optimized (but are necessary to evaluate the model). For example, you can get this using [coef.pk()] with `include_type = "const"`.
The data used to fit the model. For example, you can get this using [get_data.pk()].
The name of the function that evaluates the model (passed to [log_likelihood()]).
Logical: Whether to dose-normalize concentrations before evaluating log-likelihood. Passed to [log_likelihood()].
Logical: Whether to log10-transform concentrations before evaluating log-likelihood. Passed to [log_likelihood()].
Caroline Ring
Calculate parameter SDs using inverse Hessian approach for a single set of parameter values for a single model and a single data set.
This is a workhorse function called by [coef_sd.pk()]. If the length of this vector is \(n\), the Hessian matrix will be \(n \times n\).
The coefficient standard deviations are estimated by computing a numerical approximation to the model Hessian (the matrix of second derivatives of the model objective function with respect to each model parameter) and then attempting to invert it. This procedure yields a variance/covariance matrix for the model parameters. The square root of the diagonal elements of this matrix represent the parameter standard deviations.
A first attempt is made to invert the Hessian using [solve()] (see [hess_sd1()]). If the Hessian is singular, an attempt is made to calculate a pseudovariance matrix, following the procedure outlined in Gill & King (2004) (see [hess_sd2()]). First, the generalized inverse of the Hessian is calculated using [MASS::ginv()]. Then, a generalized Cholesky decomposition (to ensure positive-definiteness) is calculated using [Matrix::Cholesky] with argument `perm = TRUE`. The generalized inverse is reconstructed from the generalized Cholesky factorization. The square root of the diagonal elements of this matrix represent the parameter standard deviations.
If neither of these procedures is successful, then `NA_real_` is returned for all coefficient standard deviations. Record any error messages encountered during the process, and note which method was used to produce the final results. This is a workhorse function called by [coef_sd.pk()].
Gill J, King G. (2004) What to Do When Your Hessian is Not Invertible: Alternatives to Model Respecification in Nonlinear Estimation. Sociological Methods & Research 33(1):54-87. DOI: 10.1177/0049124103262681