What does longevity do?
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 iftime2
is provided) - the right interval
time2
for interval censoring - a vector or scalar
event
indicating whether data are right, left or interval censored. The optioninterval2
, for interval censoring, is useful if bothtime
andtime2
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
andrtrunc
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.
Example
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.
(m1 <- fit_elife(arguments = args,
thresh = 105 * 365.25,
family = "gp",
export = TRUE))
## 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
plot(m1, which.plot = "qq")
References
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.