predint (version 2.2.1)

neg_bin_pi: Prediction intervals for negative-binomial data

Description

neg_bin_pi() calculates bootstrap calibrated prediction intervals for negative-binomial data.

Usage

neg_bin_pi(
  histdat,
  newdat = NULL,
  newoffset = NULL,
  alternative = "both",
  alpha = 0.05,
  nboot = 10000,
  delta_min = 0.01,
  delta_max = 10,
  tolerance = 0.001,
  traceplot = TRUE,
  n_bisec = 30,
  algorithm = "MS22mod"
)

Value

neg_bin_pi() returns an object of class c("predint", "negativeBinomialPI")

with prediction intervals or limits in the first entry ($prediction).

Arguments

histdat

a data.frame with two columns. The first has to contain the historical observations. The second has to contain the number of experimental units per study (offsets).

newdat

data.frame with two columns. The first has to contain the future observations. The second has to contain the number of experimental units per study (offsets).

newoffset

vector with future number of experimental units per historical study.

alternative

either "both", "upper" or "lower". alternative specifies if a prediction interval or an upper or a lower prediction limit should be computed

alpha

defines the level of confidence (\(1-\alpha\))

nboot

number of bootstraps

delta_min

lower start value for bisection

delta_max

upper start value for bisection

tolerance

tolerance for the coverage probability in the bisection

traceplot

if TRUE: Plot for visualization of the bisection process

n_bisec

maximal number of bisection steps

algorithm

either "MS22" or "MS22mod" (see details)

Details

This function returns bootstrap-calibrated prediction intervals as well as lower or upper prediction limits.

If algorithm is set to "MS22", both limits of the prediction interval are calibrated simultaneously using the algorithm described in Menssen and Schaarschmidt (2022), section 3.2.4. The calibrated prediction interval is given as

$$[l,u]_m = n^*_m \hat{\lambda} \pm q \sqrt{n^*_m \frac{\hat{\lambda} + \hat{\kappa} \bar{n} \hat{\lambda}}{\bar{n} H} + (n^*_m \hat{\lambda} + \hat{\kappa} n^{*2}_m \hat{\lambda}^2) }$$

with \(n^*_m\) as the number of experimental units in the future clusters, \(\hat{\lambda}\) as the estimate for the Poisson mean obtained from the historical data, \(\hat{\kappa}\) as the estimate for the dispersion parameter, \(n_h\) as the number of experimental units per historical cluster and \(\bar{n}=\sum_h^{n_h} n_h / H\).

If algorithm is set to "MS22mod", both limits of the prediction interval are calibrated independently from each other. The resulting prediction interval is given by

$$[l,u] = \Big[n^*_m \hat{\lambda} - q^{calib}_l \sqrt{n^*_m \frac{\hat{\lambda} + \hat{\kappa} \bar{n} \hat{\lambda}}{\bar{n} H} + (n^*_m \hat{\lambda} + \hat{\kappa} n^{*2}_m \hat{\lambda}^2)}, \quad n^*_m \hat{\lambda} + q^{calib}_u \sqrt{n^*_m \frac{\hat{\lambda} + \hat{\kappa} \bar{n} \hat{\lambda}}{\bar{n} H} + (n^*_m \hat{\lambda} + \hat{\kappa} n^{*2}_m \hat{\lambda}^2) } \Big]$$

Please note, that this modification does not affect the calibration procedure, if only prediction limits are of interest.

References

Menssen and Schaarschmidt (2022): Prediction intervals for all of M future observations based on linear random effects models. Statistica Neerlandica, tools:::Rd_expr_doi("10.1111/stan.12260")

Examples

# HCD from the Ames test
ames_HCD

# Prediction interval for one future number of revertant colonies
# obtained in three petridishes
pred_int <- neg_bin_pi(histdat=ames_HCD, newoffset=3, nboot=100)
summary(pred_int)

# Please note that nboot was set to 100 in order to decrease computing time
# of the example. For a valid analysis set nboot=10000.