lmer_pi_futvec {predint}

Prediction intervals for future observations based on linear random effects models

Description

lmer_pi_futvec() calculates a bootstrap calibrated prediction interval for one or more future observation(s) based on linear random effects models. With this approach, the experimental design of the future data is taken into account (see below).

Usage

lmer_pi_futvec(
  model,
  futvec,
  newdat = NULL,
  alternative = "both",
  alpha = 0.05,
  nboot = 10000,
  delta_min = 0.01,
  delta_max = 10,
  tolerance = 0.001,
  traceplot = TRUE,
  n_bisec = 30,
  algorithm = "MS22"
)

Arguments

model

a random effects model of class lmerMod

futvec

an integer vector that defines the structure of the future data based on the row numbers of the historical data. If length(futvec) is one, a PI for one future observation is computed

newdat

a data.frame with the same column names as the historical data on which model depends

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] = \hat{\mu} \pm q^{calib} \sqrt{\widehat{var}(\hat{\mu}) + \sum_{c=1}^{C+1} \hat{\sigma}^2_c}

with \hat{\mu} as the expected future observation (historical mean) and \hat{\sigma}^2_c as the c=1, 2, ..., C variance components and \hat{\sigma}^2_{C+1} as the residual variance obtained from the random effects model fitted with lme4::lmer() and q^{calib} as the as the bootstrap-calibrated coefficient used for interval calculation.

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[\hat{\mu} - q^{calib}_l \sqrt{\widehat{var}(\hat{\mu}) + \sum_{c=1}^{C+1} \hat{\sigma}^2_c}, \quad \hat{\mu} + q^{calib}_u \sqrt{\widehat{var}(\hat{\mu}) + \sum_{c=1}^{C+1} \hat{\sigma}^2_c} \Big].

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

Be aware that the sampling structure of the historical data must contain the structure of the future data. This means that the observations per random factor must be less or equal in the future data compared to the historical data.

This function is an implementation of the PI given in Menssen and Schaarschmidt 2022 section 3.2.4 except that the bootstrap calibration values are drawn from bootstrap samples that mimic the future data.

Value

lmer_pi_futvec() returns an object of class c("predint", "normalPI") with prediction intervals or limits in the first entry ($prediction).

References

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

Examples

# loading lme4
library(lme4)

# Fitting a random effects model based on c2_dat1
fit <- lmer(y_ijk~(1|a)+(1|b)+(1|a:b), c2_dat1)
summary(fit)

#----------------------------------------------------------------------------

### Prediction interval using c2_dat3 as future data
# without printing c2_dat3 in the output

# Row numbers of the historical data c2_dat1 that define the structure of
# the future data c2_dat3
futvec <- c(1, 2, 4, 5, 10, 11, 13, 14)

# Calculating the PI
pred_int <- lmer_pi_futvec(model=fit, futvec=futvec, nboot=100)
summary(pred_int)

#----------------------------------------------------------------------------

### Calculating the PI with c2_dat3 printed in the output
pred_int_new <- lmer_pi_futvec(model=fit, futvec=futvec, newdat=c2_dat3, nboot=100)
summary(pred_int_new)

#----------------------------------------------------------------------------

### Upper prediction limit for m=1 future observation
pred_u <- lmer_pi_futvec(model=fit, futvec=1, alternative="upper", nboot=100)
summary(pred_u)

#----------------------------------------------------------------------------

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