fct_confint: Confidence set for functions of model parameters

Description

Computes confidence sets of functions of model parameters by computing a confidence set of the model parameters and returning the codomain of the provided function given the confidence set of model parameters as domain.

Usage

fct_confint(
  object,
  f,
  which_parm = rep(TRUE, length(coef(object))),
  level = 0.95,
  ...
)
# S3 method for lm
fct_confint(
  object,
  f,
  which_parm = rep(TRUE, length(coef(object))),
  level = 0.95,
  return_beta = FALSE,
  n_grid = NULL,
  k = NULL,
  len = 0.1,
  ...
)
# S3 method for glm
fct_confint(
  object,
  f,
  which_parm = rep(TRUE, length(coef(object))),
  level = 0.95,
  return_beta = FALSE,
  n_grid = NULL,
  k = NULL,
  len = 0.1,
  ...
)
# S3 method for lms
fct_confint(
  object,
  f,
  which_parm = rep(TRUE, length(coef(object))),
  level = 0.95,
  return_beta = FALSE,
  n_grid = NULL,
  k = NULL,
  len = 0.1,
  ...
)
# S3 method for default
fct_confint(
  object,
  f,
  which_parm = rep(TRUE, length(coef(object))),
  level = 0.95,
  ...
)

Value

A tibble with columns estimate, conf.low, and conf.high or if return_beta is TRUE, a list with the tibble and the beta values on the boundary used to calculate the confidence limits.

Arguments

object: A fitted model object.
f: A function taking the parameter vector as its single argument, and returning a numeric vector.
which_parm: Either a logical vector the same length as the coefficient vector, with TRUE indicating a coefficient is used by f, or an integer vector with the indices of the coefficients used by f.
level: The confidence level required.
...: Additional argument(s) passed to methods.
return_beta: Logical, if TRUE returns both the confidence limits and the parameter values used from the boundary of the parameter confidence set.
n_grid: Either NULL or an integer vector of length 1 or the number of TRUE/indices in which_parm. Specifies the number of grid points in each dimension of a grid with endpoints defined by len. If NULL or 0L, will instead sample k points uniformly on a sphere.
k: If n_grid is NULL or 0L, the number of points to sample uniformly from a sphere.
len: numeric, the radius of the sphere or box used to define directions in which to look for boundary points of the parameter confidence set.

Author

KIJA

Details

Assume the response Y and predictors X are given by a generalized linear model, that is, they fulfill the assumptions $$E(Y|X)=\mu(X^T\beta)$$ $$V(Y|X)=\psi \nu(\mu(X^T\beta))$$ $$Y|X\sim\varepsilon(\theta,\nu_{\psi}).$$ Here $\mu$ is the mean value function, $\nu$ is the variance function, and $\psi$ is the dispersion parameter in the exponential dispersion model $\varepsilon(\theta,\nu_{\psi})$, where $\theta$ is the canonical parameter and $\nu_{\psi}$ is the structure measure. Then it follows from the central limit theorem that $$\hat\beta\sim N(\beta, (X^TWX)^{-1})$$ will be a good approximation in large samples, where $X^TWX$ is the Fisher information of the exponential dispersion model.

From this, the combinant $$(\hat\beta-\beta)^TX^TWX(\hat\beta-\beta)$$ is an approximate pivot, with a $\chi_p^2$ distribution. Then $$C_{\beta}=\{\beta|(\hat\beta-\beta)^TX^TWX(\hat\beta-\beta)<\chi_p^2(1-\alpha)\}$$ is an approximate $(1-\alpha)$-confidence set for the parameter vector $\beta$. Similarly, confidence sets for sub-vectors of $\beta$ can be obtained by the fact that marginal distributions of normal distributions are again normally distributed, where the mean vector and covariance matrix are appropriate subvectors and submatrices.

Finally, a confidence set for the transformed parameters $f(\beta)$ is obtained as $$\{f(\beta)|\beta\in C_{\beta}\}$$ Note this is a conservative confidence set, since parameters outside the confidence set of $\beta$ can be mapped to the confidence set of the transformed parameter.

To determine $C_{\beta}$, fct_confint() uses a convex optimization program when f is follows DCP rules. Otherwise, it finds the boundary by taking a number of points around $\hat\beta$ and projecting them onto the boundary. In this case, the confidence set of the transformed parameter will only be valid if the boundary of $C_{\beta}$ is mapped to the boundary of the confidence set for the transformed parameter.

The points projected to the boundary are either laid out in a grid around $\hat\beta$, with the number of points in each direction determined by n_grid, or uniformly at random on a hypersphere, with the number of points determined by k. The radius of the grid/sphere is determined by len.

@section Progress bar:

To print a progress bar with information about the fitting process, wrap the call to fct_confint in with_progress, i.e. progressr::with_progress({result <- fct_confint(object, f)})

@section Specifying a plan to resolve futures:

fct_confint() uses futures to enable parallel processing. Use the future::plan() function to specify how futures are resolved.

Examples

Run this code

data <- 1:5 |>
  purrr::map(
    \(x) {
      name = paste0("cov", x);
      dplyr::tibble("{name}" := rnorm(100, 1))
    }
  ) |>
  purrr::list_cbind() |>
  dplyr::mutate(
  y = rowSums(dplyr::across(dplyr::everything())) + rnorm(100)
  )
lm <- lm(
 as.formula(
  paste0("y ~ 0 + ", paste0(names(data)[names(data) != "y"], collapse = " + "))
 ),
 data
)
fct_confint(lm, sum)
fct_confint(lm, sum, which_parm = 1:3, level = 0.5)

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