In many pharmaceutical and biomedical applications such as assay validation, assessment of historical control data or the detection of anti-drug antibodies, prediction intervals are of use. The package predint provides functions to calculate bootstrap calibrated prediction intervals for one or more future observations based on overdispersed binomial data, overdispersed poisson data, as well as data that is modeled by linear random effects models fitted with lme4::lmer(). The main functions are:

`beta_bin_pi()`

for beta-binomial data (overdispersion differs between clusters)

`quasi_bin_pi()`

for quasi-binomial data (constant overdispersion)

`quasi_pois_pi()`

for quasi-poisson data (constant overdispersion)`lmer_pi()`

for data that is modeled by a linear random effects model

For all of these functions, it is assumed that the historical, as well as the actual (or future) data descend from the same data generating process.

You can install the released version of predint from CRAN with:

And the development version from GitHub with:

The following examples are based on the scenario described in Menssen and Schaarschmidt 2019: Based on historical control data for the mortality of male B6C3F1-mice obtained in long term studies at the National Toxicology Program (NTP 2017), prediction intervals (PI) can be computed in order to validate the observed mortality of an actual (or future) trial. In this scenario prediction intervals can be computed for two different purposes.

On the one hand, a PI for one future observation can be computed in order to validate the outcome of one actual (or future) untreated control group that is compared with several groups treated with the compound of interest.

On the other hand, in some cases it might be useful to validate the outcome of the complete actual (or future) study including the treatment groups, based on the knowledge gained from historical control data.

Similarly to Menssen and Schaarschmidt 2019, it is assumed that the data is overdispersed binomial. Hence, we will use the `quasi_bin_pi()`

function in the following two examples.

In this scenario, it is of interest to validate the control group of an actual (or future) study that is comprised of 30 mice instead of 50 mice as in the historical data. For this purpose a single prediction interval for one future observation is computed.

```
# load predint
library(predint)
# data set (Table 1 of the supplementary material of Menssen and Schaarschmidt 2019)
dat_real <- data.frame("dead"=c(15, 10, 12, 12, 13, 11, 19, 11, 14, 21),
"alive"=c(35, 40, 38, 38, 37, 39, 31, 39, 36, 29))
# PI for one future control group comprised of 50 mice
pi_m1 <- quasi_bin_pi(histdat=dat_real,
newsize=30,
traceplot = FALSE,
alpha=0.05)
pi_m1
#> total hist_prob quant_calib pred_se lower upper
#> 1 30 0.276 1.014854 5.6 2.59682 13.96318
```

The historical binomial probability of success (historical mortality rate) is 0.276, the bootstrap calibrated coefficient is 1.01485 and the standard error of the prediction is 5.6. The lower limit of the bootstrap calibrated asymptotic prediction interval is 2.59682 and its upper limit is given by 13.96318.

If the mortality is lower than 2.59682 (practically spoken lower than 3) it can be treated as unusual low. Consequently, mean comparisons between the control group might result in too many differences that are considered as significant and the compound of interest might be treated as more hazardous than it actually is.

On the other hand, the compound of interest might be treated as less hazardous if the mortality in the untreated control group is unusual high. This might be the case, if its mortality exceeds 13.96318 (practically spoken higher than 13).

Let us assume, there is a study in which one untreated control group comprised of 50 male mice is compared to three treatment groups comprised of 30 mice each. If the whole study should be compared with the historical knowledge, four prediction intervals have to be computed. Hence `newsize`

is set to `c(50, 30, 30, 30)`

.

```
pi_m4 <- quasi_bin_pi(histdat=dat_real,
newsize=c(50, 30, 30, 30),
traceplot = FALSE,
alpha=0.05)
pi_m4
#> total hist_prob quant_calib pred_se lower upper
#> 1 50 0.276 1.288018 8.854377 2.395406 25.20459
#> 2 30 0.276 1.288018 5.600000 1.067102 15.49290
#> 3 30 0.276 1.288018 5.600000 1.067102 15.49290
#> 4 30 0.276 1.288018 5.600000 1.067102 15.49290
```

In this case, the untreated control group is in line with the historical control data if its mortality falls between 2.39541 and 25.20459. Similarly, the groups treated with the compound of interest are in line with the historical knowledge regarding untreated control groups if their mortality ranges between 1.0671 and 15.4929. This means that the compound of interest might not have an effect on mortality, if the observed moralities of some (or all) of the treatment groups fall into their corresponding prediction interval.

Menssen M, Schaarschmidt F.: Prediction intervals for overdispersed binomial data with application to historical controls. Statistics in Medicine. 2019;38:2652-2663. https://doi.org/10.1002/sim.8124

NTP 2017: Tables of historical controls: pathology tables by route/vehicle. https://ntp.niehs.nih.gov/results/dbsearch/historical/index.html. Accessed May 17, 2017.