Confidence intervals — conf_intervals • chandwich

Calculates confidence intervals for individual parameters.

conf_intervals(
  object,
  which_pars = NULL,
  init = NULL,
  conf = 95,
  mult = 1.5,
  num = 10,
  type = c("vertical", "cholesky", "spectral", "none"),
  profile = TRUE,
  ...
)

Arguments

object

An object of class "chandwich" returned by adjust_loglik.

which_pars

A vector specifying the (unfixed) parameters for which confidence intervals are required. Can be either a numeric vector, specifying indices of the components of the full parameter vector, or a character vector of parameter names, which must be a subset of those supplied in par_names in the call to adjust_loglik that produced object.

which_pars must not have any parameters in common with attr(object, "fixed_pars"). which_pars must not contain all of the unfixed parameters, i.e. there is no point in profiling over all the unfixed parameters.

If missing, all parameters are included.

init

A numeric vector of initial estimates of the values of the parameters that are not fixed and are not in which_pars. Should have length attr(object, "p_current") - length(which_pars). If init is NULL or is of the wrong length then the relevant components from the MLE stored in object are used.

conf

A numeric scalar in (0, 100). Confidence level for the intervals.

mult

A numeric vector of length 1 or the same length as which_pars. The search for the profile loglikelihood-based confidence limits is conducted over the corresponding symmetric confidence intervals (based on approximate normal theory), extended by a factor of the corresponding component of mult.

num

A numeric scalar. The number of values at which to evaluate the profile loglikelihood either side of the MLE. Increasing num increases the accuracy of the confidence limits, but the code will take longer to run.

type

A character scalar. The argument type to the function returned by adjust_loglik, that is, the type of adjustment made to the independence loglikelihood function.

profile

A logical scalar. If FALSE then only intervals based on approximate large sample normal theory, which are symmetric about the MLE, are returned (in sym_CI) and prof_CI in the returned object will contain NAs.

...

Further arguments to be passed to optim. These may include gr, method, lower, upper or control.

Value

An object of class "confint", a list with components

conf: The argument conf.
cutoff: A numeric scalar. For values inside the confidence interval the profile loglikelihood lies above cutoff.
parameter_vals, prof_loglik_vals: 2 * num + 1 by length(which_pars) numeric matrices. Column i of parameter_vals contains the profiled values of parameter which_par[i]. Column i of prof_loglik_vals contains the corresponding values of the profile loglikelihood.
sym_CI, prof_CI: length(which_pars) by 2 numeric matrices. Row i of sym_CI (prof_CI) contains the symmetric (profile loglikelihood-based) confidence intervals for parameter which_pars[i].

If a value in

prof_CI is NA then this means that the search for the confidence limit did no extend far enough. A remedy is to increase the value of mult, or the relevant component of mult, and perhaps also increase num.

max_loglik: The value of the adjusted loglikelihood at its maximum, stored in object$max_loglik.
type: The argument type supplied in the call to conf_intervals, i.e. the type of loglikelihood adjustment.
which_pars: The argument which_pars.
name: A character scalar. The name of the model, stored in attr(object, "name").
p_current: The number of free parameters in the current model.
fixed_pars, fixed_at: attr(object, "fixed_pars") and attr(object, "fixed_at"), the arguments fixed_pars and fixed_at to adjust_loglik, if these were supplied.

Details

Calculates (profile, if necessary) likelihood-based confidence intervals for individual parameters, and also provides symmetric intervals based on a normal approximation to the sampling distribution of the estimator. See also the S3 confint method confint.chandwich.

Examples

# ------------------------- Binomial model, rats data ----------------------

# Contributions to the independence loglikelihood
binom_loglik <- function(prob, data) {
  if (prob < 0 || prob > 1) {
    return(-Inf)
  }
  return(dbinom(data[, "y"], data[, "n"], prob, log = TRUE))
}
rat_res <- adjust_loglik(loglik = binom_loglik, data = rats, par_names = "p")

# 95% likelihood-based confidence intervals, vertically adjusted
ci <- conf_intervals(rat_res)
#> Waiting for profiling to be done...
plot(ci)

# Unadjusted
conf_intervals(rat_res, type = "none")
#> Waiting for profiling to be done...
#> Model: binom_loglik 
#> 
#> 95% confidence interval, independence loglikelihood
#> 
#> Symmetric:
#>    lower   upper 
#> p  0.1366  0.1705
#> 
#> Likelihood-based:
#>    lower   upper 
#> p  0.1372  0.1710

# -------------------------- GEV model, owtemps data -----------------------
# ------------ following Section 5.2 of Chandler and Bate (2007) -----------

gev_loglik <- function(pars, data) {
  o_pars <- pars[c(1, 3, 5)] + pars[c(2, 4, 6)]
  w_pars <- pars[c(1, 3, 5)] - pars[c(2, 4, 6)]
  if (isTRUE(o_pars[2] <= 0 | w_pars[2] <= 0)) return(-Inf)
  o_data <- data[, "Oxford"]
  w_data <- data[, "Worthing"]
  check <- 1 + o_pars[3] * (o_data - o_pars[1]) / o_pars[2]
  if (isTRUE(any(check <= 0))) return(-Inf)
  check <- 1 + w_pars[3] * (w_data - w_pars[1]) / w_pars[2]
  if (isTRUE(any(check <= 0))) return(-Inf)
  o_loglik <- log_gev(o_data, o_pars[1], o_pars[2], o_pars[3])
  w_loglik <- log_gev(w_data, w_pars[1], w_pars[2], w_pars[3])
  return(o_loglik + w_loglik)
}

# Initial estimates (method of moments for the Gumbel case)
sigma <- as.numeric(sqrt(6 * diag(var(owtemps))) / pi)
mu <- as.numeric(colMeans(owtemps) - 0.57722 * sigma)
init <- c(mean(mu), -diff(mu) / 2, mean(sigma), -diff(sigma) / 2, 0, 0)

# Log-likelihood adjustment of the full model
par_names <- c("mu[0]", "mu[1]", "sigma[0]", "sigma[1]", "xi[0]", "xi[1]")
large <- adjust_loglik(gev_loglik, data = owtemps, init = init,
                       par_names = par_names)

# 95% likelihood-based confidence intervals, vertically adjusted
large_v <- conf_intervals(large, which_pars = c("xi[0]", "xi[1]"))
#> Waiting for profiling to be done...
large_v
#> Model: gev_loglik 
#> 
#> 95% confidence intervals, adjusted loglikelihod with type = ''vertical'' 
#> 
#> Symmetric:
#>        lower     upper   
#> xi[0]  -0.27623  -0.12166
#> xi[1]  -0.15939  -0.01731
#> 
#> Profile likelihood-based:
#>        lower     upper   
#> xi[0]  -0.27410  -0.11572
#> xi[1]  -0.16519  -0.02002
plot(large_v)

plot(large_v, which_par = "xi[1]")

# \donttest{
# Unadjusted
large_none <- conf_intervals(large, which_pars = c("xi[0]", "xi[1]"),
                             type = "none")
#> Waiting for profiling to be done...
large_none
#> Model: gev_loglik 
#> 
#> 95% confidence intervals, independence loglikelihood
#> 
#> Symmetric:
#>        lower      upper    
#> xi[0]  -0.295708  -0.102188
#> xi[1]  -0.185107   0.008412
#> 
#> Profile likelihood-based:
#>        lower      upper    
#> xi[0]  -0.289574  -0.094328
#> xi[1]  -0.188502   0.008628
plot(large_v, large_none)

plot(large_v, large_none, which_par = "xi[1]")

# }

# --------- Misspecified Poisson model for negative binomial data ----------

# ... following Section 5.1 of the "Object-Oriented Computation of Sandwich
# Estimators" vignette of the sandwich package
# https://cran.r-project.org/web/packages/sandwich/vignettes/sandwich-OOP.pdf

# Simulate data
set.seed(123)
x <- rnorm(250)
y <- rnbinom(250, mu = exp(1 + x), size = 1)
# Fit misspecified Poisson model
fm_pois <- glm(y ~ x + I(x^2), family = poisson)
summary(fm_pois)$coefficients
#>                Estimate Std. Error   z value      Pr(>|z|)
#> (Intercept)  1.06326821 0.04135723 25.709367 9.184267e-146
#> x            0.99607219 0.05353446 18.606186  2.862861e-77
#> I(x^2)      -0.04912373 0.02314608 -2.122335  3.380961e-02

# Contributions to the independence loglikelihood
pois_glm_loglik <- function(pars, y, x) {
  log_mu <- pars[1] + pars[2] * x + pars[3] * x ^ 2
  return(dpois(y, lambda = exp(log_mu), log = TRUE))
}
pars <- c("alpha", "beta", "gamma")
pois_quad <- adjust_loglik(pois_glm_loglik, y = y, x = x, par_names = pars)
conf_intervals(pois_quad)
#> Waiting for profiling to be done...
#> Model: pois_glm_loglik 
#> 
#> 95% confidence intervals, adjusted loglikelihod with type = ''vertical'' 
#> 
#> Symmetric:
#>        lower     upper   
#> alpha   0.89907   1.22747
#> beta    0.78986   1.20231
#> gamma  -0.12024   0.02199
#> 
#> Profile likelihood-based:
#>        lower    upper  
#> alpha   0.8954   1.2232
#> beta    0.7877   1.1991
#> gamma  -0.1198   0.0222