PEM vs. PAM split point sensitivity
Last updated: 2019-11-25
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Knit directory: simpamm/
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library(ggplot2)
theme_set(theme_bw())
library(batchtools)Motivation
This is a light-weight simulation study to investigate how sensitive the different approaches (PEM vs. PAM) to the estimation of the baseline-hazard function are to the placement of the interval split points.
Setup
The setup is as follows:
\(n=250\) survival times are simulated from a distribution with log-hazard \(-3.5 + f(8,2)*6\), where \(f(8,2)\) is the density function of the Gamma distribution with respective parameters.
The baseline hazard is estimated by a PEM and PAM respectively
Three different settings are used for the interval split point definition
- “default”: Unique event times from each simulated data set is used
- “fine”: A fine, equidistant grid with interval lengths \(0.2\)
- “rough”: A rough, equidistant grid with interval lengths \(0.5\)
- For each setting, 20 replications are run
Function for data simulation (using pammtools::sim_pexp):
## simulation function
sim_wrapper <- function(data, job, n = 250, 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}
# define function that generates nz exposures z(t_{z,1}), ..., z(t_{z,Q})
sim_pexp(formula = ~ -3.5 + f0(t), data = df, cut = time_grid)
}Function to estimate hazard from simulated data, either by a PEM or PAM
## estimation function
pam_wrapper <- function(data, job, instance,
cut = NA,
bs = "ps",
mod_type = c("pem", "pam") ,
max_time = 10) {
if(is.na(cut)) {
cut <- NULL
} else {
if(cut == "rough") {
cut <- seq(0, max_time, by = 0.5)
} else {
if(cut == "fine") {
cut <- seq(0, max_time, by = 0.2)
}
}
}
ped <- as_ped(data = instance, formula = Surv(time, status) ~ ., cut = cut, id="id")
form <- "ped_status ~ s(tend) + s(x1) + s(x2)"
if(mod_type == "pem") {
form <- ped_status ~ interval
time_var <- "interval"
} else {
form <- ped_status ~ s(tend, bs = bs, k = 10)
time_var <- "tend"
}
mod <- gam(formula = form, data = ped, family = poisson(), offset = offset, method = "REML")
# summary(mod)
make_newdata(ped, tend=unique(tend)) %>%
add_hazard(mod, type="link", se_mult = qnorm(0.975), time_var = time_var) %>%
mutate(truth = -3.5 + dgamma(tend, 8, 2) * 6)
}Setup simulation registry
Setup simulation using batchtools:
if(!checkmate::test_directory_exists("output/sim-pem-vs-pam-registry")) {
reg <- makeExperimentRegistry("output/sim-pem-vs-pam-registry",
packages = c("mgcv", "dplyr", "tidyr", "pammtools"),
seed = 20052018)
reg$cluster.functions = makeClusterFunctionsMulticore(ncpus = 2)
addProblem(name = "pem-vs-pam", fun = sim_wrapper)
addAlgorithm(name = "pem-vs-pam", fun = pam_wrapper)
algo_df <- tidyr::crossing(
cut = c(NA, "fine", "rough"),
mod_type = c("pem", "pam"))
addExperiments(algo.design = list("pem-vs-pam" = algo_df), repls = 20)
submitJobs()
waitForJobs()
}
Warning: The above code chunk cached its results, but it won’t be re-run if previous chunks it depends on are updated. If you need to use caching, it is highly recommended to also set knitr::opts_chunk$set(autodep = TRUE) at the top of the file (in a chunk that is not cached). Alternatively, you can customize the option dependson for each individual chunk that is cached. Using either autodep or dependson will remove this warning. See the knitr cache options for more details.
Evaluate Simulation
reg <- loadRegistry("output/sim-pem-vs-pam-registry", writeable = TRUE)
ids_pam <- findExperiments(prob.name="pem-vs-pam", algo.name="pem-vs-pam")
pars <- unwrap(getJobPars()) %>% as_tibble()
res <- reduceResultsDataTable(ids=findDone(ids_pam)) %>%
as_tibble() %>%
tidyr::unnest() %>%
left_join(pars) %>%
mutate(cut = case_when(is.na(cut) ~ "default", TRUE ~ cut))
res %>%
mutate(
sq_error = (truth - hazard)^2,
covered = (truth >= ci_lower) & (truth <= ci_upper)) %>%
group_by(job.id, mod_type, cut) %>%
summarize(
RMSE = sqrt(mean(sq_error)),
coverage = mean(covered)) %>%
group_by(mod_type, cut) %>%
summarize(
RMSE = mean(RMSE),
coverage = mean(coverage))# A tibble: 6 x 4
# Groups: mod_type [2]
mod_type cut RMSE coverage
<chr> <chr> <dbl> <dbl>
1 pam default 0.223 0.881
2 pam fine 0.268 0.916
3 pam rough 0.281 0.92
4 pem default 1.38 0.870
5 pem fine 6.84 0.966
6 pem rough 3.02 0.92 Visualize Estimations
ggplot(res, aes(x=tend, y = hazard)) +
geom_step(aes(group = job.id), alpha = 0.3) +
geom_line(aes(y = truth, col = "truth"), lwd = 2) +
facet_grid(cut ~ mod_type) +
coord_cartesian(ylim=c(-5, 0)) +
geom_smooth(aes(col="average estimate"), method="gam", formula = y ~ s(x),
se=FALSE) +
scale_color_brewer("", palette = "Dark2") +
xlab("time")
Conclusion
For the PAM, the RMSE has about the same magnitude for all three split point settings
For the PEM, the RMSE is highly dependent on the RMSE, partly because even for the “rough” split point setting, in some simulations some intervals have no events and the hazard is estimated close to zero (very small log-hazard values) and for the “default” setting, where each interval contains at least one event, appears to overestimate the hazard on average
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] pammtools_0.1.14 tidyr_1.0.0 dplyr_0.8.3 mgcv_1.8-31
[5] nlme_3.1-142 batchtools_0.9.11 data.table_1.12.6 ggplot2_3.2.1
loaded via a namespace (and not attached):
[1] progress_1.2.2 tidyselect_0.2.5 xfun_0.11 reshape2_1.4.3
[5] purrr_0.3.3 splines_3.6.1 lattice_0.20-38 expm_0.999-4
[9] colorspace_1.4-1 vctrs_0.2.0 htmltools_0.4.0 yaml_2.2.0
[13] utf8_1.1.4 survival_3.1-7 rlang_0.4.2 later_1.0.0
[17] pillar_1.4.2 glue_1.3.1 withr_2.1.2 rappdirs_0.3.1
[21] RColorBrewer_1.1-2 plyr_1.8.4 lifecycle_0.1.0 stringr_1.4.0
[25] munsell_0.5.0 gtable_0.3.0 workflowr_1.5.0 mvtnorm_1.0-11
[29] evaluate_0.14 labeling_0.3 knitr_1.26 httpuv_1.5.2
[33] fansi_0.4.0 Rcpp_1.0.3 promises_1.1.0 scales_1.1.0
[37] backports_1.1.5 checkmate_1.9.4 farver_2.0.1 fs_1.3.1
[41] brew_1.0-6 hms_0.5.2 digest_0.6.23 stringi_1.4.3
[45] msm_1.6.7 grid_3.6.1 rprojroot_1.3-2 cli_1.1.0
[49] tools_3.6.1 magrittr_1.5 base64url_1.4 lazyeval_0.2.2
[53] tibble_2.1.3 Formula_1.2-3 crayon_1.3.4 whisker_0.4
[57] pkgconfig_2.0.3 zeallot_0.1.0 Matrix_1.2-17 prettyunits_1.0.2
[61] assertthat_0.2.1 rmarkdown_1.17 R6_2.4.1 git2r_0.26.1
[65] compiler_3.6.1