The `longevity`

package provides a variety of numerical
routines for parametric and nonparametric models for positive data
subject to non informative censoring and truncation mechanisms. The
package includes functions to estimate various parametric model
parameters via maximum likelihood, produce diagnostic plots accounting
for survival patterns, compare nested models using analysis of deviance,
etc.

The syntax of `longevity`

follows that of the popular
`survival`

package, but forgoes the specification of
`Surv`

type objects: rather, users must specify some of the
following

- the time vector
`time`

(a left interval if`time2`

is provided) - the right interval
`time2`

for interval censoring - a vector or scalar
`event`

indicating whether data are right, left or interval censored. The option`interval2`

, for interval censoring, is useful if both`time`

and`time2`

vectors are provided with (potentially zero or infinite bounds) for censored observations. - the status indicator,
`event`

, with 0 for right censored, 1 for observed event, 2 for left censored and 3 for interval censored. If omitted,`event`

is set to 1 for all subjects. `ltrunc`

and`rtrunc`

for left and right truncation values. If omitted, they are set to 0 and \(\infty\), respectively.

The reason for specifying the `ltrunc`

and
`rtrunc`

vector outside of the usual arguments is to
accomodate instances where there is both interval censoring and interval
truncation; `survival`

supports left-truncation
right-censoring for time-varying covariate models, but this isn’t really
transparent.

We consider Dutch data from CBS; these data were analysed in Einmahl, Einmahl, and Haan (2019). For simplicity, we keep only Dutch people born in the Netherlands, who were at least centenarians when they died and whose death date is known.

```
thresh0 <- 36525
data(dutch, package = "longevity")
dutch1 <- subset(dutch, ndays > thresh0 & !is.na(ndays) & valid == "A")
```

We can fit various parametric models accounting for the fact that
data are interval truncated. First, we create a list to avoid having to
type the name of all arguments repeatedly. These, if not provided
directly to function, are selected from the list through
`arguments`

.

```
args <- with(dutch1, list(
time = ndays, # time vector
ltrunc = ltrunc, # left truncation bound
rtrunc = rtrunc, # right truncation
thresh = thresh0, # threshold (model only exceedances)
family = "gp")) # choice of parametric model
```

The generalized Pareto distribution can be used for extrapolation, provided that the threshold is high enough that shape estimates are more or less stable. To check this, we can produce threshold stability plots, which display point estimates with 95% profile-based pointwise confidence intervals.

```
tstab_c <- tstab(
arguments = args,
family = "gp", # parametric model, here generalized Pareto
thresh = 102:108 * 365.25, # overwrites thresh
method = "wald", # type of interval, Wald or profile-likelihood
plot = FALSE) # by default, calls 'plot' routine
plot(tstab_c,
which.plot = "shape",
xlab = "threshold (age in days)")
```

We can fit various parametric models and compare them using the
`anova`

call, provided they are nested and share the same
data. Diagnostic plots, adapted for survival data, can be used to check
goodness-of-fit. These may be computationally intensive to produce in
large samples, since they require estimation of the nonparametric
maximum likelihood estimator of the distribution function.

```
## Model: generalized Pareto distribution.
## Sampling: interval truncated
## Log-likelihood: -6335.799
##
## Threshold: 38351.25
## Number of exceedances: 886
##
## Estimates
## scale shape
## 572.8722 -0.0748
##
## Standard Errors
## scale shape
## 1.4122 0.0238
##
## Optimization Information
## Convergence: TRUE
```

```
## npar Deviance Df Chisq Pr(>Chisq)
## gp 2 12671.60 NA NA NA
## exp 1 12675.91 1 4.314994 0.03777791
```

Einmahl, Jesson J., John H. J. Einmahl, and Laurens de Haan. 2019.
“Limits to Human Life Span Through Extreme Value Theory.”
*Journal of the American Statistical Association* 114 (527):
1075–80. https://doi.org/10.1080/01621459.2018.1537912.