Last updated: 2019-11-25
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| Rmd | 4ea7082 | Andreas Bender | 2019-11-25 | Add sim study on confidence intervals |
library(dplyr)
library(ggplot2)
theme_set(theme_bw())
library(patchwork)
library(survival)
library(mgcv)
library(pammtools)
library(batchtools)
knitr::opts_chunk$set(autodep = TRUE)Here we consider three different ways to calculate confidence intervals for hazard rates estimated by PAMMs and derivatives thereof, i.e., the cumulative hazard and survival probability. The three methods compared here are
We simulate data with log-hazard rate \(\log(\lambda(t|x)) = -3.5 + f(8,2) \cdot 6 - 0.5\cdot x_1 + \sqrt{x_2}\), where \(f(8,2)\) is the gamma density function with respective parameters.
The below wrapper creates a data set with \(n = 500\) survival times based on above hazard. The simulation of survival times is performed using function pammtools::sim_pexp.
sim_wrapper <- function(data, job, n = 500, time_grid = seq(0, 10, by = 0.05)) {
# create data set with covariates
df <- tibble::tibble(x1 = runif(n, -3, 3), x2 = runif(n, 0, 6))
# baseline hazard
f0 <- function(t) {dgamma(t, 8, 2) * 6}
ndf <- sim_pexp(
formula = ~ -3.5 + f0(t) -0.5*x1 + sqrt(x2),
data = df, cut = time_grid)
ndf
}The below wrapper - transforms the simulated survival data into the PED format - fits the PAMM - returns a data frame that contains the coverage for each method
Expand to see function source code
ci_wrapper <- function(
data,
job,
instance,
bs = "ps",
k = 10,
ci_type = "default") {
# instance <- sim_wrapper()
ped <- as_ped(
data = instance,
formula = Surv(time, status) ~ x1 + x2,
id = "id")
form <- paste0("ped_status ~ s(tend, bs='", bs, "', k=", k, ") + s(x1) + s(x2)")
mod <- gam(
formula = as.formula(form),
data = ped,
family = poisson(),
offset = offset,
method = "REML")
f0 <- function(t) {dgamma(t, 8, 2) * 6}
# create new data set
nd <- make_newdata(ped, tend = unique(tend), x1 = c(0), x2 = c(3)) %>%
mutate(
true_hazard = exp(-3.5 + f0(tend) -0.5 * x1 + sqrt(x2)),
true_cumu = cumsum(intlen * true_hazard),
true_surv = exp(-true_cumu))
# add hazard, cumulative hazard, survival probability with confidence intervals
# using the 3 different methods
nd_default <- nd %>%
add_hazard(mod, se_mult = qnorm(0.975), ci_type = "default") %>%
add_cumu_hazard(mod, se_mult = qnorm(0.975), ci_type = "default") %>%
add_surv_prob(mod, se_mult = qnorm(0.975), ci_type = "default") %>%
mutate(
hazard = (true_hazard >= ci_lower) & (true_hazard <= ci_upper),
cumu = (true_cumu >= cumu_lower) & (true_cumu <= cumu_upper),
surv = (true_surv >= surv_lower) & (true_surv <= surv_upper)) %>%
select(hazard, cumu, surv) %>%
summarize_all(mean) %>%
mutate(method = "direct")
nd_delta <- nd %>%
add_hazard(mod, se_mult = qnorm(0.975), ci_type = "delta", overwrite = TRUE) %>%
add_cumu_hazard(mod, se_mult = qnorm(0.975), ci_type = "delta", overwrite = TRUE) %>%
add_surv_prob(mod, se_mult = qnorm(0.975), ci_type = "delta", overwrite = TRUE) %>%
mutate(
hazard = (true_hazard >= ci_lower) & (true_hazard <= ci_upper),
cumu = (true_cumu >= cumu_lower) & (true_cumu <= cumu_upper),
surv = (true_surv >= surv_lower) & (true_surv <= surv_upper)) %>%
select(hazard, cumu, surv) %>%
summarize_all(mean) %>%
mutate(method = "delta")
nd_sim <- nd %>%
add_hazard(mod, se_mult = qnorm(0.975), ci_type = "sim", nsim = 500, overwrite = TRUE) %>%
add_cumu_hazard(mod, se_mult = qnorm(0.975), ci_type = "sim", nsim = 500, overwrite = TRUE) %>%
add_surv_prob(mod, se_mult = qnorm(0.975), ci_type = "sim", nsim = 500, overwrite = TRUE) %>%
mutate(
hazard = (true_hazard >= ci_lower) & (true_hazard <= ci_upper),
cumu = (true_cumu >= cumu_lower) & (true_cumu <= cumu_upper),
surv = (true_surv >= surv_lower) & (true_surv <= surv_upper)) %>%
select(hazard, cumu, surv) %>%
summarize_all(mean) %>%
mutate(method = "simulation")
rbind(nd_default, nd_delta, nd_sim)
}We simulate 100 data sets and obtain respective estimates using package batchtools:
Expand to see
batchtools code
if (!checkmate::test_directory_exists("output/sim-conf-int-registry")) {
reg <- makeExperimentRegistry(
"output/sim-conf-int-registry",
packages = c("mgcv", "dplyr", "tidyr", "pammtools"),
seed = 20052018)
reg <- loadRegistry("output/sim-conf-int-registry", writeable = TRUE)
# reg$cluster.functions = makeClusterFunctionsInteractive()
addProblem(name = "ci", fun = sim_wrapper)
addAlgorithm(name = "ci", fun = ci_wrapper)
algo_df <- data.frame(bs = c("tp", "ps"), stringsAsFactors = FALSE)
addExperiments(algo.design = list(ci = algo_df), repls = 300)
submitJobs(findNotDone())
# waitForJobs()
}Below the RMSE and coverage are calculated for the different methods to estimate the confidence intervals
reg <- loadRegistry("output/sim-conf-int-registry", writeable = TRUE)
ids_res <- findExperiments(prob.name = "ci", algo.name = "ci")
pars <- unwrap(getJobPars()) %>% as_tibble()
res <- reduceResultsDataTable(ids=findDone(ids_res)) %>%
as_tibble() %>%
tidyr::unnest(cols = c(result)) %>%
left_join(pars)# RMSE and coverage hazard
res %>%
group_by(bs, method) %>%
summarize(
"coverage hazard" = mean(hazard),
"coverage cumulative hazard" = mean(cumu),
"coverage survival probability" = mean(surv)) %>%
ungroup() %>%
mutate_if(is.numeric, ~round(., 3)) %>%
rename("basis" = "bs") %>%
knitr::kable()| basis | method | coverage hazard | coverage cumulative hazard | coverage survival probability |
|---|---|---|---|---|
| ps | delta | 0.911 | 0.904 | 0.907 |
| ps | direct | 0.914 | 0.965 | 0.965 |
| ps | simulation | 0.891 | 0.904 | 0.907 |
| tp | delta | 0.936 | 0.925 | 0.933 |
| tp | direct | 0.938 | 0.978 | 0.978 |
| tp | simulation | 0.918 | 0.933 | 0.933 |
sessionInfo()R version 3.6.1 (2019-07-05)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 18.04.3 LTS
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.7.1
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.7.1
locale:
[1] LC_CTYPE=en_US.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_US.UTF-8 LC_COLLATE=en_US.UTF-8
[5] LC_MONETARY=en_US.UTF-8 LC_MESSAGES=en_US.UTF-8
[7] LC_PAPER=en_US.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] tidyr_1.0.0 batchtools_0.9.11 data.table_1.12.6
[4] pammtools_0.1.14 mgcv_1.8-31 nlme_3.1-142
[7] survival_3.1-7 patchwork_0.0.1.9000 ggplot2_3.2.1
[10] dplyr_0.8.3
loaded via a namespace (and not attached):
[1] progress_1.2.2 tidyselect_0.2.5 xfun_0.11 purrr_0.3.3
[5] splines_3.6.1 lattice_0.20-38 colorspace_1.4-1 vctrs_0.2.0
[9] expm_0.999-4 htmltools_0.4.0 yaml_2.2.0 rlang_0.4.2
[13] later_1.0.0 pillar_1.4.2 glue_1.3.1 withr_2.1.2
[17] rappdirs_0.3.1 lifecycle_0.1.0 stringr_1.4.0 munsell_0.5.0
[21] gtable_0.3.0 workflowr_1.5.0 mvtnorm_1.0-11 evaluate_0.14
[25] knitr_1.26 httpuv_1.5.2 highr_0.8 Rcpp_1.0.3
[29] promises_1.1.0 scales_1.1.0 backports_1.1.5 checkmate_1.9.4
[33] fs_1.3.1 brew_1.0-6 hms_0.5.2 digest_0.6.23
[37] stringi_1.4.3 msm_1.6.7 grid_3.6.1 rprojroot_1.3-2
[41] tools_3.6.1 magrittr_1.5 base64url_1.4 lazyeval_0.2.2
[45] tibble_2.1.3 Formula_1.2-3 crayon_1.3.4 whisker_0.4
[49] pkgconfig_2.0.3 zeallot_0.1.0 Matrix_1.2-17 prettyunits_1.0.2
[53] assertthat_0.2.1 rmarkdown_1.17 R6_2.4.1 git2r_0.26.1
[57] compiler_3.6.1